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--- Day 20: A Regular Map ---
While you were learning about instruction pointers, the Elves made considerable progress. When you look up, you discover that the North Pole base construction project has completely surrounded you.
The area you are in is made up entirely of rooms and doors. The rooms are arranged in a grid, and rooms only connect to adjacent rooms when a door is present between them.
For example, drawing rooms as ., walls as #, doors as | or -, your current position as X, and where north is up, the area you're in might look like this:
#####
#.|.#
#-###
#.|X#
#####
You get the attention of a passing construction Elf and ask for a map. "I don't have time to draw out a map of this place - it's huge. Instead, I can give you directions to every room in the facility!" He writes down some directions on a piece of parchment and runs off. In the example above, the instructions might have been ^WNE$, a regular expression or "regex" (your puzzle input).
The regex matches routes (like WNE for "west, north, east") that will take you from your current room through various doors in the facility. In aggregate, the routes will take you through every door in the facility at least once; mapping out all of these routes will let you build a proper map and find your way around.
^ and $ are at the beginning and end of your regex; these just mean that the regex doesn't match anything outside the routes it describes. (Specifically, ^ matches the start of the route, and $ matches the end of it.) These characters will not appear elsewhere in the regex.
The rest of the regex matches various sequences of the characters N (north), S (south), E (east), and W (west). In the example above, ^WNE$ matches only one route, WNE, which means you can move west, then north, then east from your current position. Sequences of letters like this always match that exact route in the same order.
Sometimes, the route can branch. A branch is given by a list of options separated by pipes (|) and wrapped in parentheses. So, ^N(E|W)N$ contains a branch: after going north, you must choose to go either east or west before finishing your route by going north again. By tracing out the possible routes after branching, you can determine where the doors are and, therefore, where the rooms are in the facility.
For example, consider this regex: ^ENWWW(NEEE|SSE(EE|N))$
This regex begins with ENWWW, which means that from your current position, all routes must begin by moving east, north, and then west three times, in that order. After this, there is a branch. Before you consider the branch, this is what you know about the map so far, with doors you aren't sure about marked with a ?:
#?#?#?#?#
?.|.|.|.?
#?#?#?#-#
?X|.?
#?#?#
After this point, there is (NEEE|SSE(EE|N)). This gives you exactly two options: NEEE and SSE(EE|N). By following NEEE, the map now looks like this:
#?#?#?#?#
?.|.|.|.?
#-#?#?#?#
?.|.|.|.?
#?#?#?#-#
?X|.?
#?#?#
Now, only SSE(EE|N) remains. Because it is in the same parenthesized group as NEEE, it starts from the same room NEEE started in. It states that starting from that point, there exist doors which will allow you to move south twice, then east; this ends up at another branch. After that, you can either move east twice or north once. This information fills in the rest of the doors:
#?#?#?#?#
?.|.|.|.?
#-#?#?#?#
?.|.|.|.?
#-#?#?#-#
?.?.?X|.?
#-#-#?#?#
?.|.|.|.?
#?#?#?#?#
Once you've followed all possible routes, you know the remaining unknown parts are all walls, producing a finished map of the facility:
#########
#.|.|.|.#
#-#######
#.|.|.|.#
#-#####-#
#.#.#X|.#
#-#-#####
#.|.|.|.#
#########
Sometimes, a list of options can have an empty option, like (NEWS|WNSE|). This means that routes at this point could effectively skip the options in parentheses and move on immediately. For example, consider this regex and the corresponding map:
^ENNWSWW(NEWS|)SSSEEN(WNSE|)EE(SWEN|)NNN$
###########
#.|.#.|.#.#
#-###-#-#-#
#.|.|.#.#.#
#-#####-#-#
#.#.#X|.#.#
#-#-#####-#
#.#.|.|.|.#
#-###-###-#
#.|.|.#.|.#
###########
This regex has one main route which, at three locations, can optionally include additional detours and be valid: (NEWS|), (WNSE|), and (SWEN|). Regardless of which option is taken, the route continues from the position it is left at after taking those steps. So, for example, this regex matches all of the following routes (and more that aren't listed here):
ENNWSWWSSSEENEENNN
ENNWSWWNEWSSSSEENEENNN
ENNWSWWNEWSSSSEENEESWENNNN
ENNWSWWSSSEENWNSEEENNN
By following the various routes the regex matches, a full map of all of the doors and rooms in the facility can be assembled.
To get a sense for the size of this facility, you'd like to determine which room is furthest from you: specifically, you would like to find the room for which the shortest path to that room would require passing through the most doors.
In the first example (^WNE$), this would be the north-east corner 3 doors away.
In the second example (^ENWWW(NEEE|SSE(EE|N))$), this would be the south-east corner 10 doors away.
In the third example (^ENNWSWW(NEWS|)SSSEEN(WNSE|)EE(SWEN|)NNN$), this would be the north-east corner 18 doors away.
Here are a few more examples:
Regex: ^ESSWWN(E|NNENN(EESS(WNSE|)SSS|WWWSSSSE(SW|NNNE)))$
Furthest room requires passing 23 doors
#############
#.|.|.|.|.|.#
#-#####-###-#
#.#.|.#.#.#.#
#-#-###-#-#-#
#.#.#.|.#.|.#
#-#-#-#####-#
#.#.#.#X|.#.#
#-#-#-###-#-#
#.|.#.|.#.#.#
###-#-###-#-#
#.|.#.|.|.#.#
#############
Regex: ^WSSEESWWWNW(S|NENNEEEENN(ESSSSW(NWSW|SSEN)|WSWWN(E|WWS(E|SS))))$
Furthest room requires passing 31 doors
###############
#.|.|.|.#.|.|.#
#-###-###-#-#-#
#.|.#.|.|.#.#.#
#-#########-#-#
#.#.|.|.|.|.#.#
#-#-#########-#
#.#.#.|X#.|.#.#
###-#-###-#-#-#
#.|.#.#.|.#.|.#
#-###-#####-###
#.|.#.|.|.#.#.#
#-#-#####-#-#-#
#.#.|.|.|.#.|.#
###############
What is the largest number of doors you would be required to pass through to reach a room? That is, find the room for which the shortest path from your starting location to that room would require passing through the most doors; what is the fewest doors you can pass through to reach it?
Your puzzle answer was 3721.
--- Part Two ---
Okay, so the facility is big.
How many rooms have a shortest path from your current location that pass through at least 1000 doors?
| 400
|
--- Day 15: Beacon Exclusion Zone ---
You feel the ground rumble again as the distress signal leads you to a large network of subterranean tunnels. You don't have time to search them all, but you don't need to: your pack contains a set of deployable sensors that you imagine were originally built to locate lost Elves.
The sensors aren't very powerful, but that's okay; your handheld device indicates that you're close enough to the source of the distress signal to use them. You pull the emergency sensor system out of your pack, hit the big button on top, and the sensors zoom off down the tunnels.
Once a sensor finds a spot it thinks will give it a good reading, it attaches itself to a hard surface and begins monitoring for the nearest signal source beacon. Sensors and beacons always exist at integer coordinates. Each sensor knows its own position and can determine the position of a beacon precisely; however, sensors can only lock on to the one beacon closest to the sensor as measured by the Manhattan distance. (There is never a tie where two beacons are the same distance to a sensor.)
It doesn't take long for the sensors to report back their positions and closest beacons (your puzzle input). For example:
Sensor at x=2, y=18: closest beacon is at x=-2, y=15
Sensor at x=9, y=16: closest beacon is at x=10, y=16
Sensor at x=13, y=2: closest beacon is at x=15, y=3
Sensor at x=12, y=14: closest beacon is at x=10, y=16
Sensor at x=10, y=20: closest beacon is at x=10, y=16
Sensor at x=14, y=17: closest beacon is at x=10, y=16
Sensor at x=8, y=7: closest beacon is at x=2, y=10
Sensor at x=2, y=0: closest beacon is at x=2, y=10
Sensor at x=0, y=11: closest beacon is at x=2, y=10
Sensor at x=20, y=14: closest beacon is at x=25, y=17
Sensor at x=17, y=20: closest beacon is at x=21, y=22
Sensor at x=16, y=7: closest beacon is at x=15, y=3
Sensor at x=14, y=3: closest beacon is at x=15, y=3
Sensor at x=20, y=1: closest beacon is at x=15, y=3
So, consider the sensor at 2,18; the closest beacon to it is at -2,15. For the sensor at 9,16, the closest beacon to it is at 10,16.
Drawing sensors as S and beacons as B, the above arrangement of sensors and beacons looks like this:
1 1 2 2
0 5 0 5 0 5
0 ....S.......................
1 ......................S.....
2 ...............S............
3 ................SB..........
4 ............................
5 ............................
6 ............................
7 ..........S.......S.........
8 ............................
9 ............................
10 ....B.......................
11 ..S.........................
12 ............................
13 ............................
14 ..............S.......S.....
15 B...........................
16 ...........SB...............
17 ................S..........B
18 ....S.......................
19 ............................
20 ............S......S........
21 ............................
22 .......................B....
This isn't necessarily a comprehensive map of all beacons in the area, though. Because each sensor only identifies its closest beacon, if a sensor detects a beacon, you know there are no other beacons that close or closer to that sensor. There could still be beacons that just happen to not be the closest beacon to any sensor. Consider the sensor at 8,7:
1 1 2 2
0 5 0 5 0 5
-2 ..........#.................
-1 .........###................
0 ....S...#####...............
1 .......#######........S.....
2 ......#########S............
3 .....###########SB..........
4 ....#############...........
5 ...###############..........
6 ..#################.........
7 .#########S#######S#........
8 ..#################.........
9 ...###############..........
10 ....B############...........
11 ..S..###########............
12 ......#########.............
13 .......#######..............
14 ........#####.S.......S.....
15 B........###................
16 ..........#SB...............
17 ................S..........B
18 ....S.......................
19 ............................
20 ............S......S........
21 ............................
22 .......................B....
This sensor's closest beacon is at 2,10, and so you know there are no beacons that close or closer (in any positions marked #).
None of the detected beacons seem to be producing the distress signal, so you'll need to work out where the distress beacon is by working out where it isn't. For now, keep things simple by counting the positions where a beacon cannot possibly be along just a single row.
So, suppose you have an arrangement of beacons and sensors like in the example above and, just in the row where y=10, you'd like to count the number of positions a beacon cannot possibly exist. The coverage from all sensors near that row looks like this:
1 1 2 2
0 5 0 5 0 5
9 ...#########################...
10 ..####B######################..
11 .###S#############.###########.
In this example, in the row where y=10, there are 26 positions where a beacon cannot be present.
Consult the report from the sensors you just deployed. In the row where y=2000000, how many positions cannot contain a beacon?
Your puzzle answer was 4861076.
--- Part Two ---
Your handheld device indicates that the distress signal is coming from a beacon nearby. The distress beacon is not detected by any sensor, but the distress beacon must have x and y coordinates each no lower than 0 and no larger than 4000000.
To isolate the distress beacon's signal, you need to determine its tuning frequency, which can be found by multiplying its x coordinate by 4000000 and then adding its y coordinate.
In the example above, the search space is smaller: instead, the x and y coordinates can each be at most 20. With this reduced search area, there is only a single position that could have a beacon: x=14, y=11. The tuning frequency for this distress beacon is 56000011.
Find the only possible position for the distress beacon. What is its tuning frequency?
| 401
|
--- Day 15: Timing is Everything ---
The halls open into an interior plaza containing a large kinetic sculpture. The sculpture is in a sealed enclosure and seems to involve a set of identical spherical capsules that are carried to the top and allowed to bounce through the maze of spinning pieces.
Part of the sculpture is even interactive! When a button is pressed, a capsule is dropped and tries to fall through slots in a set of rotating discs to finally go through a little hole at the bottom and come out of the sculpture. If any of the slots aren't aligned with the capsule as it passes, the capsule bounces off the disc and soars away. You feel compelled to get one of those capsules.
The discs pause their motion each second and come in different sizes; they seem to each have a fixed number of positions at which they stop. You decide to call the position with the slot 0, and count up for each position it reaches next.
Furthermore, the discs are spaced out so that after you push the button, one second elapses before the first disc is reached, and one second elapses as the capsule passes from one disc to the one below it. So, if you push the button at time=100, then the capsule reaches the top disc at time=101, the second disc at time=102, the third disc at time=103, and so on.
The button will only drop a capsule at an integer time - no fractional seconds allowed.
For example, at time=0, suppose you see the following arrangement:
Disc #1 has 5 positions; at time=0, it is at position 4.
Disc #2 has 2 positions; at time=0, it is at position 1.
If you press the button exactly at time=0, the capsule would start to fall; it would reach the first disc at time=1. Since the first disc was at position 4 at time=0, by time=1 it has ticked one position forward. As a five-position disc, the next position is 0, and the capsule falls through the slot.
Then, at time=2, the capsule reaches the second disc. The second disc has ticked forward two positions at this point: it started at position 1, then continued to position 0, and finally ended up at position 1 again. Because there's only a slot at position 0, the capsule bounces away.
If, however, you wait until time=5 to push the button, then when the capsule reaches each disc, the first disc will have ticked forward 5+1 = 6 times (to position 0), and the second disc will have ticked forward 5+2 = 7 times (also to position 0). In this case, the capsule would fall through the discs and come out of the machine.
However, your situation has more than two discs; you've noted their positions in your puzzle input. What is the first time you can press the button to get a capsule?
Your puzzle answer was 121834.
--- Part Two ---
After getting the first capsule (it contained a star! what great fortune!), the machine detects your success and begins to rearrange itself.
When it's done, the discs are back in their original configuration as if it were time=0 again, but a new disc with 11 positions and starting at position 0 has appeared exactly one second below the previously-bottom disc.
With this new disc, and counting again starting from time=0 with the configuration in your puzzle input, what is the first time you can press the button to get another capsule?
| 402
|
--- Day 12: Digital Plumber ---
Walking along the memory banks of the stream, you find a small village that is experiencing a little confusion: some programs can't communicate with each other.
Programs in this village communicate using a fixed system of pipes. Messages are passed between programs using these pipes, but most programs aren't connected to each other directly. Instead, programs pass messages between each other until the message reaches the intended recipient.
For some reason, though, some of these messages aren't ever reaching their intended recipient, and the programs suspect that some pipes are missing. They would like you to investigate.
You walk through the village and record the ID of each program and the IDs with which it can communicate directly (your puzzle input). Each program has one or more programs with which it can communicate, and these pipes are bidirectional; if 8 says it can communicate with 11, then 11 will say it can communicate with 8.
You need to figure out how many programs are in the group that contains program ID 0.
For example, suppose you go door-to-door like a travelling salesman and record the following list:
0 <-> 2
1 <-> 1
2 <-> 0, 3, 4
3 <-> 2, 4
4 <-> 2, 3, 6
5 <-> 6
6 <-> 4, 5
In this example, the following programs are in the group that contains program ID 0:
Program 0 by definition.
Program 2, directly connected to program 0.
Program 3 via program 2.
Program 4 via program 2.
Program 5 via programs 6, then 4, then 2.
Program 6 via programs 4, then 2.
Therefore, a total of 6 programs are in this group; all but program 1, which has a pipe that connects it to itself.
How many programs are in the group that contains program ID 0?
Your puzzle answer was 128.
--- Part Two ---
There are more programs than just the ones in the group containing program ID 0. The rest of them have no way of reaching that group, and still might have no way of reaching each other.
A group is a collection of programs that can all communicate via pipes either directly or indirectly. The programs you identified just a moment ago are all part of the same group. Now, they would like you to determine the total number of groups.
In the example above, there were 2 groups: one consisting of programs 0,2,3,4,5,6, and the other consisting solely of program 1.
How many groups are there in total?
| 403
|
--- Day 1: Historian Hysteria ---
The Chief Historian is always present for the big Christmas sleigh launch, but nobody has seen him in months! Last anyone heard, he was visiting locations that are historically significant to the North Pole; a group of Senior Historians has asked you to accompany them as they check the places they think he was most likely to visit.
As each location is checked, they will mark it on their list with a star. They figure the Chief Historian must be in one of the first fifty places they'll look, so in order to save Christmas, you need to help them get fifty stars on their list before Santa takes off on December 25th.
Collect stars by solving puzzles. Two puzzles will be made available on each day in the Advent calendar; the second puzzle is unlocked when you complete the first. Each puzzle grants one star. Good luck!
You haven't even left yet and the group of Elvish Senior Historians has already hit a problem: their list of locations to check is currently empty. Eventually, someone decides that the best place to check first would be the Chief Historian's office.
Upon pouring into the office, everyone confirms that the Chief Historian is indeed nowhere to be found. Instead, the Elves discover an assortment of notes and lists of historically significant locations! This seems to be the planning the Chief Historian was doing before he left. Perhaps these notes can be used to determine which locations to search?
Throughout the Chief's office, the historically significant locations are listed not by name but by a unique number called the location ID. To make sure they don't miss anything, The Historians split into two groups, each searching the office and trying to create their own complete list of location IDs.
There's just one problem: by holding the two lists up side by side (your puzzle input), it quickly becomes clear that the lists aren't very similar. Maybe you can help The Historians reconcile their lists?
For example:
3 4
4 3
2 5
1 3
3 9
3 3
Maybe the lists are only off by a small amount! To find out, pair up the numbers and measure how far apart they are. Pair up the smallest number in the left list with the smallest number in the right list, then the second-smallest left number with the second-smallest right number, and so on.
Within each pair, figure out how far apart the two numbers are; you'll need to add up all of those distances. For example, if you pair up a 3 from the left list with a 7 from the right list, the distance apart is 4; if you pair up a 9 with a 3, the distance apart is 6.
In the example list above, the pairs and distances would be as follows:
The smallest number in the left list is 1, and the smallest number in the right list is 3. The distance between them is 2.
The second-smallest number in the left list is 2, and the second-smallest number in the right list is another 3. The distance between them is 1.
The third-smallest number in both lists is 3, so the distance between them is 0.
The next numbers to pair up are 3 and 4, a distance of 1.
The fifth-smallest numbers in each list are 3 and 5, a distance of 2.
Finally, the largest number in the left list is 4, while the largest number in the right list is 9; these are a distance 5 apart.
To find the total distance between the left list and the right list, add up the distances between all of the pairs you found. In the example above, this is 2 + 1 + 0 + 1 + 2 + 5, a total distance of 11!
Your actual left and right lists contain many location IDs. What is the total distance between your lists?
Your puzzle answer was 2430334.
--- Part Two ---
Your analysis only confirmed what everyone feared: the two lists of location IDs are indeed very different.
Or are they?
The Historians can't agree on which group made the mistakes or how to read most of the Chief's handwriting, but in the commotion you notice an interesting detail: a lot of location IDs appear in both lists! Maybe the other numbers aren't location IDs at all but rather misinterpreted handwriting.
This time, you'll need to figure out exactly how often each number from the left list appears in the right list. Calculate a total similarity score by adding up each number in the left list after multiplying it by the number of times that number appears in the right list.
Here are the same example lists again:
3 4
4 3
2 5
1 3
3 9
3 3
For these example lists, here is the process of finding the similarity score:
The first number in the left list is 3. It appears in the right list three times, so the similarity score increases by 3 * 3 = 9.
The second number in the left list is 4. It appears in the right list once, so the similarity score increases by 4 * 1 = 4.
The third number in the left list is 2. It does not appear in the right list, so the similarity score does not increase (2 * 0 = 0).
The fourth number, 1, also does not appear in the right list.
The fifth number, 3, appears in the right list three times; the similarity score increases by 9.
The last number, 3, appears in the right list three times; the similarity score again increases by 9.
So, for these example lists, the similarity score at the end of this process is 31 (9 + 4 + 0 + 0 + 9 + 9).
Once again consider your left and right lists. What is their similarity score?
| 404
|
--- Day 16: Aunt Sue ---
Your Aunt Sue has given you a wonderful gift, and you'd like to send her a thank you card. However, there's a small problem: she signed it "From, Aunt Sue".
You have 500 Aunts named "Sue".
So, to avoid sending the card to the wrong person, you need to figure out which Aunt Sue (which you conveniently number 1 to 500, for sanity) gave you the gift. You open the present and, as luck would have it, good ol' Aunt Sue got you a My First Crime Scene Analysis Machine! Just what you wanted. Or needed, as the case may be.
The My First Crime Scene Analysis Machine (MFCSAM for short) can detect a few specific compounds in a given sample, as well as how many distinct kinds of those compounds there are. According to the instructions, these are what the MFCSAM can detect:
children, by human DNA age analysis.
cats. It doesn't differentiate individual breeds.
Several seemingly random breeds of dog: samoyeds, pomeranians, akitas, and vizslas.
goldfish. No other kinds of fish.
trees, all in one group.
cars, presumably by exhaust or gasoline or something.
perfumes, which is handy, since many of your Aunts Sue wear a few kinds.
In fact, many of your Aunts Sue have many of these. You put the wrapping from the gift into the MFCSAM. It beeps inquisitively at you a few times and then prints out a message on ticker tape:
children: 3
cats: 7
samoyeds: 2
pomeranians: 3
akitas: 0
vizslas: 0
goldfish: 5
trees: 3
cars: 2
perfumes: 1
You make a list of the things you can remember about each Aunt Sue. Things missing from your list aren't zero - you simply don't remember the value.
What is the number of the Sue that got you the gift?
| 405
|
--- Day 1: The Tyranny of the Rocket Equation ---
Santa has become stranded at the edge of the Solar System while delivering presents to other planets! To accurately calculate his position in space, safely align his warp drive, and return to Earth in time to save Christmas, he needs you to bring him measurements from fifty stars.
Collect stars by solving puzzles. Two puzzles will be made available on each day in the Advent calendar; the second puzzle is unlocked when you complete the first. Each puzzle grants one star. Good luck!
The Elves quickly load you into a spacecraft and prepare to launch.
At the first Go / No Go poll, every Elf is Go until the Fuel Counter-Upper. They haven't determined the amount of fuel required yet.
Fuel required to launch a given module is based on its mass. Specifically, to find the fuel required for a module, take its mass, divide by three, round down, and subtract 2.
For example:
For a mass of 12, divide by 3 and round down to get 4, then subtract 2 to get 2.
For a mass of 14, dividing by 3 and rounding down still yields 4, so the fuel required is also 2.
For a mass of 1969, the fuel required is 654.
For a mass of 100756, the fuel required is 33583.
The Fuel Counter-Upper needs to know the total fuel requirement. To find it, individually calculate the fuel needed for the mass of each module (your puzzle input), then add together all the fuel values.
What is the sum of the fuel requirements for all of the modules on your spacecraft?
| 406
|
--- Day 7: Amplification Circuit ---
Based on the navigational maps, you're going to need to send more power to your ship's thrusters to reach Santa in time. To do this, you'll need to configure a series of amplifiers already installed on the ship.
There are five amplifiers connected in series; each one receives an input signal and produces an output signal. They are connected such that the first amplifier's output leads to the second amplifier's input, the second amplifier's output leads to the third amplifier's input, and so on. The first amplifier's input value is 0, and the last amplifier's output leads to your ship's thrusters.
O-------O O-------O O-------O O-------O O-------O
0 ->| Amp A |->| Amp B |->| Amp C |->| Amp D |->| Amp E |-> (to thrusters)
O-------O O-------O O-------O O-------O O-------O
The Elves have sent you some Amplifier Controller Software (your puzzle input), a program that should run on your existing Intcode computer. Each amplifier will need to run a copy of the program.
When a copy of the program starts running on an amplifier, it will first use an input instruction to ask the amplifier for its current phase setting (an integer from 0 to 4). Each phase setting is used exactly once, but the Elves can't remember which amplifier needs which phase setting.
The program will then call another input instruction to get the amplifier's input signal, compute the correct output signal, and supply it back to the amplifier with an output instruction. (If the amplifier has not yet received an input signal, it waits until one arrives.)
Your job is to find the largest output signal that can be sent to the thrusters by trying every possible combination of phase settings on the amplifiers. Make sure that memory is not shared or reused between copies of the program.
For example, suppose you want to try the phase setting sequence 3,1,2,4,0, which would mean setting amplifier A to phase setting 3, amplifier B to setting 1, C to 2, D to 4, and E to 0. Then, you could determine the output signal that gets sent from amplifier E to the thrusters with the following steps:
Start the copy of the amplifier controller software that will run on amplifier A. At its first input instruction, provide it the amplifier's phase setting, 3. At its second input instruction, provide it the input signal, 0. After some calculations, it will use an output instruction to indicate the amplifier's output signal.
Start the software for amplifier B. Provide it the phase setting (1) and then whatever output signal was produced from amplifier A. It will then produce a new output signal destined for amplifier C.
Start the software for amplifier C, provide the phase setting (2) and the value from amplifier B, then collect its output signal.
Run amplifier D's software, provide the phase setting (4) and input value, and collect its output signal.
Run amplifier E's software, provide the phase setting (0) and input value, and collect its output signal.
The final output signal from amplifier E would be sent to the thrusters. However, this phase setting sequence may not have been the best one; another sequence might have sent a higher signal to the thrusters.
Here are some example programs:
Max thruster signal 43210 (from phase setting sequence 4,3,2,1,0):
3,15,3,16,1002,16,10,16,1,16,15,15,4,15,99,0,0
Max thruster signal 54321 (from phase setting sequence 0,1,2,3,4):
3,23,3,24,1002,24,10,24,1002,23,-1,23,
101,5,23,23,1,24,23,23,4,23,99,0,0
Max thruster signal 65210 (from phase setting sequence 1,0,4,3,2):
3,31,3,32,1002,32,10,32,1001,31,-2,31,1007,31,0,33,
1002,33,7,33,1,33,31,31,1,32,31,31,4,31,99,0,0,0
Try every combination of phase settings on the amplifiers. What is the highest signal that can be sent to the thrusters?
| 407
|
--- Day 17: Conway Cubes ---
As your flight slowly drifts through the sky, the Elves at the Mythical Information Bureau at the North Pole contact you. They'd like some help debugging a malfunctioning experimental energy source aboard one of their super-secret imaging satellites.
The experimental energy source is based on cutting-edge technology: a set of Conway Cubes contained in a pocket dimension! When you hear it's having problems, you can't help but agree to take a look.
The pocket dimension contains an infinite 3-dimensional grid. At every integer 3-dimensional coordinate (x,y,z), there exists a single cube which is either active or inactive.
In the initial state of the pocket dimension, almost all cubes start inactive. The only exception to this is a small flat region of cubes (your puzzle input); the cubes in this region start in the specified active (#) or inactive (.) state.
The energy source then proceeds to boot up by executing six cycles.
Each cube only ever considers its neighbors: any of the 26 other cubes where any of their coordinates differ by at most 1. For example, given the cube at x=1,y=2,z=3, its neighbors include the cube at x=2,y=2,z=2, the cube at x=0,y=2,z=3, and so on.
During a cycle, all cubes simultaneously change their state according to the following rules:
If a cube is active and exactly 2 or 3 of its neighbors are also active, the cube remains active. Otherwise, the cube becomes inactive.
If a cube is inactive but exactly 3 of its neighbors are active, the cube becomes active. Otherwise, the cube remains inactive.
The engineers responsible for this experimental energy source would like you to simulate the pocket dimension and determine what the configuration of cubes should be at the end of the six-cycle boot process.
For example, consider the following initial state:
.#.
..#
###
Even though the pocket dimension is 3-dimensional, this initial state represents a small 2-dimensional slice of it. (In particular, this initial state defines a 3x3x1 region of the 3-dimensional space.)
Simulating a few cycles from this initial state produces the following configurations, where the result of each cycle is shown layer-by-layer at each given z coordinate (and the frame of view follows the active cells in each cycle):
Before any cycles:
z=0
.#.
..#
###
After 1 cycle:
z=-1
#..
..#
.#.
z=0
#.#
.##
.#.
z=1
#..
..#
.#.
After 2 cycles:
z=-2
.....
.....
..#..
.....
.....
z=-1
..#..
.#..#
....#
.#...
.....
z=0
##...
##...
#....
....#
.###.
z=1
..#..
.#..#
....#
.#...
.....
z=2
.....
.....
..#..
.....
.....
After 3 cycles:
z=-2
.......
.......
..##...
..###..
.......
.......
.......
z=-1
..#....
...#...
#......
.....##
.#...#.
..#.#..
...#...
z=0
...#...
.......
#......
.......
.....##
.##.#..
...#...
z=1
..#....
...#...
#......
.....##
.#...#.
..#.#..
...#...
z=2
.......
.......
..##...
..###..
.......
.......
.......
After the full six-cycle boot process completes, 112 cubes are left in the active state.
Starting with your given initial configuration, simulate six cycles. How many cubes are left in the active state after the sixth cycle?
Your puzzle answer was 271.
--- Part Two ---
For some reason, your simulated results don't match what the experimental energy source engineers expected. Apparently, the pocket dimension actually has four spatial dimensions, not three.
The pocket dimension contains an infinite 4-dimensional grid. At every integer 4-dimensional coordinate (x,y,z,w), there exists a single cube (really, a hypercube) which is still either active or inactive.
Each cube only ever considers its neighbors: any of the 80 other cubes where any of their coordinates differ by at most 1. For example, given the cube at x=1,y=2,z=3,w=4, its neighbors include the cube at x=2,y=2,z=3,w=3, the cube at x=0,y=2,z=3,w=4, and so on.
The initial state of the pocket dimension still consists of a small flat region of cubes. Furthermore, the same rules for cycle updating still apply: during each cycle, consider the number of active neighbors of each cube.
For example, consider the same initial state as in the example above. Even though the pocket dimension is 4-dimensional, this initial state represents a small 2-dimensional slice of it. (In particular, this initial state defines a 3x3x1x1 region of the 4-dimensional space.)
Simulating a few cycles from this initial state produces the following configurations, where the result of each cycle is shown layer-by-layer at each given z and w coordinate:
Before any cycles:
z=0, w=0
.#.
..#
###
After 1 cycle:
z=-1, w=-1
#..
..#
.#.
z=0, w=-1
#..
..#
.#.
z=1, w=-1
#..
..#
.#.
z=-1, w=0
#..
..#
.#.
z=0, w=0
#.#
.##
.#.
z=1, w=0
#..
..#
.#.
z=-1, w=1
#..
..#
.#.
z=0, w=1
#..
..#
.#.
z=1, w=1
#..
..#
.#.
After 2 cycles:
z=-2, w=-2
.....
.....
..#..
.....
.....
z=-1, w=-2
.....
.....
.....
.....
.....
z=0, w=-2
###..
##.##
#...#
.#..#
.###.
z=1, w=-2
.....
.....
.....
.....
.....
z=2, w=-2
.....
.....
..#..
.....
.....
z=-2, w=-1
.....
.....
.....
.....
.....
z=-1, w=-1
.....
.....
.....
.....
.....
z=0, w=-1
.....
.....
.....
.....
.....
z=1, w=-1
.....
.....
.....
.....
.....
z=2, w=-1
.....
.....
.....
.....
.....
z=-2, w=0
###..
##.##
#...#
.#..#
.###.
z=-1, w=0
.....
.....
.....
.....
.....
z=0, w=0
.....
.....
.....
.....
.....
z=1, w=0
.....
.....
.....
.....
.....
z=2, w=0
###..
##.##
#...#
.#..#
.###.
z=-2, w=1
.....
.....
.....
.....
.....
z=-1, w=1
.....
.....
.....
.....
.....
z=0, w=1
.....
.....
.....
.....
.....
z=1, w=1
.....
.....
.....
.....
.....
z=2, w=1
.....
.....
.....
.....
.....
z=-2, w=2
.....
.....
..#..
.....
.....
z=-1, w=2
.....
.....
.....
.....
.....
z=0, w=2
###..
##.##
#...#
.#..#
.###.
z=1, w=2
.....
.....
.....
.....
.....
z=2, w=2
.....
.....
..#..
.....
.....
After the full six-cycle boot process completes, 848 cubes are left in the active state.
Starting with your given initial configuration, simulate six cycles in a 4-dimensional space. How many cubes are left in the active state after the sixth cycle?
| 408
|
--- Day 24: Lobby Layout ---
Your raft makes it to the tropical island; it turns out that the small crab was an excellent navigator. You make your way to the resort.
As you enter the lobby, you discover a small problem: the floor is being renovated. You can't even reach the check-in desk until they've finished installing the new tile floor.
The tiles are all hexagonal; they need to be arranged in a hex grid with a very specific color pattern. Not in the mood to wait, you offer to help figure out the pattern.
The tiles are all white on one side and black on the other. They start with the white side facing up. The lobby is large enough to fit whatever pattern might need to appear there.
A member of the renovation crew gives you a list of the tiles that need to be flipped over (your puzzle input). Each line in the list identifies a single tile that needs to be flipped by giving a series of steps starting from a reference tile in the very center of the room. (Every line starts from the same reference tile.)
Because the tiles are hexagonal, every tile has six neighbors: east, southeast, southwest, west, northwest, and northeast. These directions are given in your list, respectively, as e, se, sw, w, nw, and ne. A tile is identified by a series of these directions with no delimiters; for example, esenee identifies the tile you land on if you start at the reference tile and then move one tile east, one tile southeast, one tile northeast, and one tile east.
Each time a tile is identified, it flips from white to black or from black to white. Tiles might be flipped more than once. For example, a line like esew flips a tile immediately adjacent to the reference tile, and a line like nwwswee flips the reference tile itself.
Here is a larger example:
sesenwnenenewseeswwswswwnenewsewsw
neeenesenwnwwswnenewnwwsewnenwseswesw
seswneswswsenwwnwse
nwnwneseeswswnenewneswwnewseswneseene
swweswneswnenwsewnwneneseenw
eesenwseswswnenwswnwnwsewwnwsene
sewnenenenesenwsewnenwwwse
wenwwweseeeweswwwnwwe
wsweesenenewnwwnwsenewsenwwsesesenwne
neeswseenwwswnwswswnw
nenwswwsewswnenenewsenwsenwnesesenew
enewnwewneswsewnwswenweswnenwsenwsw
sweneswneswneneenwnewenewwneswswnese
swwesenesewenwneswnwwneseswwne
enesenwswwswneneswsenwnewswseenwsese
wnwnesenesenenwwnenwsewesewsesesew
nenewswnwewswnenesenwnesewesw
eneswnwswnwsenenwnwnwwseeswneewsenese
neswnwewnwnwseenwseesewsenwsweewe
wseweeenwnesenwwwswnew
In the above example, 10 tiles are flipped once (to black), and 5 more are flipped twice (to black, then back to white). After all of these instructions have been followed, a total of 10 tiles are black.
Go through the renovation crew's list and determine which tiles they need to flip. After all of the instructions have been followed, how many tiles are left with the black side up?
Your puzzle answer was 411.
--- Part Two ---
The tile floor in the lobby is meant to be a living art exhibit. Every day, the tiles are all flipped according to the following rules:
Any black tile with zero or more than 2 black tiles immediately adjacent to it is flipped to white.
Any white tile with exactly 2 black tiles immediately adjacent to it is flipped to black.
Here, tiles immediately adjacent means the six tiles directly touching the tile in question.
The rules are applied simultaneously to every tile; put another way, it is first determined which tiles need to be flipped, then they are all flipped at the same time.
In the above example, the number of black tiles that are facing up after the given number of days has passed is as follows:
Day 1: 15
Day 2: 12
Day 3: 25
Day 4: 14
Day 5: 23
Day 6: 28
Day 7: 41
Day 8: 37
Day 9: 49
Day 10: 37
Day 20: 132
Day 30: 259
Day 40: 406
Day 50: 566
Day 60: 788
Day 70: 1106
Day 80: 1373
Day 90: 1844
Day 100: 2208
After executing this process a total of 100 times, there would be 2208 black tiles facing up.
How many tiles will be black after 100 days?
| 409
|
--- Day 8: I Heard You Like Registers ---
You receive a signal directly from the CPU. Because of your recent assistance with jump instructions, it would like you to compute the result of a series of unusual register instructions.
Each instruction consists of several parts: the register to modify, whether to increase or decrease that register's value, the amount by which to increase or decrease it, and a condition. If the condition fails, skip the instruction without modifying the register. The registers all start at 0. The instructions look like this:
b inc 5 if a > 1
a inc 1 if b < 5
c dec -10 if a >= 1
c inc -20 if c == 10
These instructions would be processed as follows:
Because a starts at 0, it is not greater than 1, and so b is not modified.
a is increased by 1 (to 1) because b is less than 5 (it is 0).
c is decreased by -10 (to 10) because a is now greater than or equal to 1 (it is 1).
c is increased by -20 (to -10) because c is equal to 10.
After this process, the largest value in any register is 1.
You might also encounter <= (less than or equal to) or != (not equal to). However, the CPU doesn't have the bandwidth to tell you what all the registers are named, and leaves that to you to determine.
What is the largest value in any register after completing the instructions in your puzzle input?
Your puzzle answer was 6611.
--- Part Two ---
To be safe, the CPU also needs to know the highest value held in any register during this process so that it can decide how much memory to allocate to these operations. For example, in the above instructions, the highest value ever held was 10 (in register c after the third instruction was evaluated).
| 410
|
--- Day 12: Leonardo's Monorail ---
You finally reach the top floor of this building: a garden with a slanted glass ceiling. Looks like there are no more stars to be had.
While sitting on a nearby bench amidst some tiger lilies, you manage to decrypt some of the files you extracted from the servers downstairs.
According to these documents, Easter Bunny HQ isn't just this building - it's a collection of buildings in the nearby area. They're all connected by a local monorail, and there's another building not far from here! Unfortunately, being night, the monorail is currently not operating.
You remotely connect to the monorail control systems and discover that the boot sequence expects a password. The password-checking logic (your puzzle input) is easy to extract, but the code it uses is strange: it's assembunny code designed for the new computer you just assembled. You'll have to execute the code and get the password.
The assembunny code you've extracted operates on four registers (a, b, c, and d) that start at 0 and can hold any integer. However, it seems to make use of only a few instructions:
cpy x y copies x (either an integer or the value of a register) into register y.
inc x increases the value of register x by one.
dec x decreases the value of register x by one.
jnz x y jumps to an instruction y away (positive means forward; negative means backward), but only if x is not zero.
The jnz instruction moves relative to itself: an offset of -1 would continue at the previous instruction, while an offset of 2 would skip over the next instruction.
For example:
cpy 41 a
inc a
inc a
dec a
jnz a 2
dec a
The above code would set register a to 41, increase its value by 2, decrease its value by 1, and then skip the last dec a (because a is not zero, so the jnz a 2 skips it), leaving register a at 42. When you move past the last instruction, the program halts.
After executing the assembunny code in your puzzle input, what value is left in register a?
Your puzzle answer was 318083.
--- Part Two ---
As you head down the fire escape to the monorail, you notice it didn't start; register c needs to be initialized to the position of the ignition key.
If you instead initialize register c to be 1, what value is now left in register a?
| 411
|
--- Day 17: Trick Shot ---
You finally decode the Elves' message. HI, the message says. You continue searching for the sleigh keys.
Ahead of you is what appears to be a large ocean trench. Could the keys have fallen into it? You'd better send a probe to investigate.
The probe launcher on your submarine can fire the probe with any integer velocity in the x (forward) and y (upward, or downward if negative) directions. For example, an initial x,y velocity like 0,10 would fire the probe straight up, while an initial velocity like 10,-1 would fire the probe forward at a slight downward angle.
The probe's x,y position starts at 0,0. Then, it will follow some trajectory by moving in steps. On each step, these changes occur in the following order:
The probe's x position increases by its x velocity.
The probe's y position increases by its y velocity.
Due to drag, the probe's x velocity changes by 1 toward the value 0; that is, it decreases by 1 if it is greater than 0, increases by 1 if it is less than 0, or does not change if it is already 0.
Due to gravity, the probe's y velocity decreases by 1.
For the probe to successfully make it into the trench, the probe must be on some trajectory that causes it to be within a target area after any step. The submarine computer has already calculated this target area (your puzzle input). For example:
target area: x=20..30, y=-10..-5
This target area means that you need to find initial x,y velocity values such that after any step, the probe's x position is at least 20 and at most 30, and the probe's y position is at least -10 and at most -5.
Given this target area, one initial velocity that causes the probe to be within the target area after any step is 7,2:
.............#....#............
.......#..............#........
...............................
S........................#.....
...............................
...............................
...........................#...
...............................
....................TTTTTTTTTTT
....................TTTTTTTTTTT
....................TTTTTTTT#TT
....................TTTTTTTTTTT
....................TTTTTTTTTTT
....................TTTTTTTTTTT
In this diagram, S is the probe's initial position, 0,0. The x coordinate increases to the right, and the y coordinate increases upward. In the bottom right, positions that are within the target area are shown as T. After each step (until the target area is reached), the position of the probe is marked with #. (The bottom-right # is both a position the probe reaches and a position in the target area.)
Another initial velocity that causes the probe to be within the target area after any step is 6,3:
...............#..#............
...........#........#..........
...............................
......#..............#.........
...............................
...............................
S....................#.........
...............................
...............................
...............................
.....................#.........
....................TTTTTTTTTTT
....................TTTTTTTTTTT
....................TTTTTTTTTTT
....................TTTTTTTTTTT
....................T#TTTTTTTTT
....................TTTTTTTTTTT
Another one is 9,0:
S........#.....................
.................#.............
...............................
........................#......
...............................
....................TTTTTTTTTTT
....................TTTTTTTTTT#
....................TTTTTTTTTTT
....................TTTTTTTTTTT
....................TTTTTTTTTTT
....................TTTTTTTTTTT
One initial velocity that doesn't cause the probe to be within the target area after any step is 17,-4:
S..............................................................
...............................................................
...............................................................
...............................................................
.................#.............................................
....................TTTTTTTTTTT................................
....................TTTTTTTTTTT................................
....................TTTTTTTTTTT................................
....................TTTTTTTTTTT................................
....................TTTTTTTTTTT..#.............................
....................TTTTTTTTTTT................................
...............................................................
...............................................................
...............................................................
...............................................................
................................................#..............
...............................................................
...............................................................
...............................................................
...............................................................
...............................................................
...............................................................
..............................................................#
The probe appears to pass through the target area, but is never within it after any step. Instead, it continues down and to the right - only the first few steps are shown.
If you're going to fire a highly scientific probe out of a super cool probe launcher, you might as well do it with style. How high can you make the probe go while still reaching the target area?
In the above example, using an initial velocity of 6,9 is the best you can do, causing the probe to reach a maximum y position of 45. (Any higher initial y velocity causes the probe to overshoot the target area entirely.)
Find the initial velocity that causes the probe to reach the highest y position and still eventually be within the target area after any step. What is the highest y position it reaches on this trajectory?
Your puzzle answer was 8646.
--- Part Two ---
Maybe a fancy trick shot isn't the best idea; after all, you only have one probe, so you had better not miss.
To get the best idea of what your options are for launching the probe, you need to find every initial velocity that causes the probe to eventually be within the target area after any step.
In the above example, there are 112 different initial velocity values that meet these criteria:
23,-10 25,-9 27,-5 29,-6 22,-6 21,-7 9,0 27,-7 24,-5
25,-7 26,-6 25,-5 6,8 11,-2 20,-5 29,-10 6,3 28,-7
8,0 30,-6 29,-8 20,-10 6,7 6,4 6,1 14,-4 21,-6
26,-10 7,-1 7,7 8,-1 21,-9 6,2 20,-7 30,-10 14,-3
20,-8 13,-2 7,3 28,-8 29,-9 15,-3 22,-5 26,-8 25,-8
25,-6 15,-4 9,-2 15,-2 12,-2 28,-9 12,-3 24,-6 23,-7
25,-10 7,8 11,-3 26,-7 7,1 23,-9 6,0 22,-10 27,-6
8,1 22,-8 13,-4 7,6 28,-6 11,-4 12,-4 26,-9 7,4
24,-10 23,-8 30,-8 7,0 9,-1 10,-1 26,-5 22,-9 6,5
7,5 23,-6 28,-10 10,-2 11,-1 20,-9 14,-2 29,-7 13,-3
23,-5 24,-8 27,-9 30,-7 28,-5 21,-10 7,9 6,6 21,-5
27,-10 7,2 30,-9 21,-8 22,-7 24,-9 20,-6 6,9 29,-5
8,-2 27,-8 30,-5 24,-7
How many distinct initial velocity values cause the probe to be within the target area after any step?
| 412
|
--- Day 15: Dueling Generators ---
Here, you encounter a pair of dueling generators. The generators, called generator A and generator B, are trying to agree on a sequence of numbers. However, one of them is malfunctioning, and so the sequences don't always match.
As they do this, a judge waits for each of them to generate its next value, compares the lowest 16 bits of both values, and keeps track of the number of times those parts of the values match.
The generators both work on the same principle. To create its next value, a generator will take the previous value it produced, multiply it by a factor (generator A uses 16807; generator B uses 48271), and then keep the remainder of dividing that resulting product by 2147483647. That final remainder is the value it produces next.
To calculate each generator's first value, it instead uses a specific starting value as its "previous value" (as listed in your puzzle input).
For example, suppose that for starting values, generator A uses 65, while generator B uses 8921. Then, the first five pairs of generated values are:
--Gen. A-- --Gen. B--
1092455 430625591
1181022009 1233683848
245556042 1431495498
1744312007 137874439
1352636452 285222916
In binary, these pairs are (with generator A's value first in each pair):
00000000000100001010101101100111
00011001101010101101001100110111
01000110011001001111011100111001
01001001100010001000010110001000
00001110101000101110001101001010
01010101010100101110001101001010
01100111111110000001011011000111
00001000001101111100110000000111
01010000100111111001100000100100
00010001000000000010100000000100
Here, you can see that the lowest (here, rightmost) 16 bits of the third value match: 1110001101001010. Because of this one match, after processing these five pairs, the judge would have added only 1 to its total.
To get a significant sample, the judge would like to consider 40 million pairs. (In the example above, the judge would eventually find a total of 588 pairs that match in their lowest 16 bits.)
After 40 million pairs, what is the judge's final count?
Your puzzle answer was 594.
--- Part Two ---
In the interest of trying to align a little better, the generators get more picky about the numbers they actually give to the judge.
They still generate values in the same way, but now they only hand a value to the judge when it meets their criteria:
Generator A looks for values that are multiples of 4.
Generator B looks for values that are multiples of 8.
Each generator functions completely independently: they both go through values entirely on their own, only occasionally handing an acceptable value to the judge, and otherwise working through the same sequence of values as before until they find one.
The judge still waits for each generator to provide it with a value before comparing them (using the same comparison method as before). It keeps track of the order it receives values; the first values from each generator are compared, then the second values from each generator, then the third values, and so on.
Using the example starting values given above, the generators now produce the following first five values each:
--Gen. A-- --Gen. B--
1352636452 1233683848
1992081072 862516352
530830436 1159784568
1980017072 1616057672
740335192 412269392
These values have the following corresponding binary values:
01010000100111111001100000100100
01001001100010001000010110001000
01110110101111001011111010110000
00110011011010001111010010000000
00011111101000111101010001100100
01000101001000001110100001111000
01110110000001001010100110110000
01100000010100110001010101001000
00101100001000001001111001011000
00011000100100101011101101010000
Unfortunately, even though this change makes more bits similar on average, none of these values' lowest 16 bits match. Now, it's not until the 1056th pair that the judge finds the first match:
--Gen. A-- --Gen. B--
1023762912 896885216
00111101000001010110000111100000
00110101011101010110000111100000
This change makes the generators much slower, and the judge is getting impatient; it is now only willing to consider 5 million pairs. (Using the values from the example above, after five million pairs, the judge would eventually find a total of 309 pairs that match in their lowest 16 bits.)
After 5 million pairs, but using this new generator logic, what is the judge's final count?
| 413
|
--- Day 8: Treetop Tree House ---
The expedition comes across a peculiar patch of tall trees all planted carefully in a grid. The Elves explain that a previous expedition planted these trees as a reforestation effort. Now, they're curious if this would be a good location for a tree house.
First, determine whether there is enough tree cover here to keep a tree house hidden. To do this, you need to count the number of trees that are visible from outside the grid when looking directly along a row or column.
The Elves have already launched a quadcopter to generate a map with the height of each tree (your puzzle input). For example:
30373
25512
65332
33549
35390
Each tree is represented as a single digit whose value is its height, where 0 is the shortest and 9 is the tallest.
A tree is visible if all of the other trees between it and an edge of the grid are shorter than it. Only consider trees in the same row or column; that is, only look up, down, left, or right from any given tree.
All of the trees around the edge of the grid are visible - since they are already on the edge, there are no trees to block the view. In this example, that only leaves the interior nine trees to consider:
The top-left 5 is visible from the left and top. (It isn't visible from the right or bottom since other trees of height 5 are in the way.)
The top-middle 5 is visible from the top and right.
The top-right 1 is not visible from any direction; for it to be visible, there would need to only be trees of height 0 between it and an edge.
The left-middle 5 is visible, but only from the right.
The center 3 is not visible from any direction; for it to be visible, there would need to be only trees of at most height 2 between it and an edge.
The right-middle 3 is visible from the right.
In the bottom row, the middle 5 is visible, but the 3 and 4 are not.
With 16 trees visible on the edge and another 5 visible in the interior, a total of 21 trees are visible in this arrangement.
Consider your map; how many trees are visible from outside the grid?
| 414
|
--- Day 2: Cube Conundrum ---
You're launched high into the atmosphere! The apex of your trajectory just barely reaches the surface of a large island floating in the sky. You gently land in a fluffy pile of leaves. It's quite cold, but you don't see much snow. An Elf runs over to greet you.
The Elf explains that you've arrived at Snow Island and apologizes for the lack of snow. He'll be happy to explain the situation, but it's a bit of a walk, so you have some time. They don't get many visitors up here; would you like to play a game in the meantime?
As you walk, the Elf shows you a small bag and some cubes which are either red, green, or blue. Each time you play this game, he will hide a secret number of cubes of each color in the bag, and your goal is to figure out information about the number of cubes.
To get information, once a bag has been loaded with cubes, the Elf will reach into the bag, grab a handful of random cubes, show them to you, and then put them back in the bag. He'll do this a few times per game.
You play several games and record the information from each game (your puzzle input). Each game is listed with its ID number (like the 11 in Game 11: ...) followed by a semicolon-separated list of subsets of cubes that were revealed from the bag (like 3 red, 5 green, 4 blue).
For example, the record of a few games might look like this:
Game 1: 3 blue, 4 red; 1 red, 2 green, 6 blue; 2 green
Game 2: 1 blue, 2 green; 3 green, 4 blue, 1 red; 1 green, 1 blue
Game 3: 8 green, 6 blue, 20 red; 5 blue, 4 red, 13 green; 5 green, 1 red
Game 4: 1 green, 3 red, 6 blue; 3 green, 6 red; 3 green, 15 blue, 14 red
Game 5: 6 red, 1 blue, 3 green; 2 blue, 1 red, 2 green
In game 1, three sets of cubes are revealed from the bag (and then put back again). The first set is 3 blue cubes and 4 red cubes; the second set is 1 red cube, 2 green cubes, and 6 blue cubes; the third set is only 2 green cubes.
The Elf would first like to know which games would have been possible if the bag contained only 12 red cubes, 13 green cubes, and 14 blue cubes?
In the example above, games 1, 2, and 5 would have been possible if the bag had been loaded with that configuration. However, game 3 would have been impossible because at one point the Elf showed you 20 red cubes at once; similarly, game 4 would also have been impossible because the Elf showed you 15 blue cubes at once. If you add up the IDs of the games that would have been possible, you get 8.
Determine which games would have been possible if the bag had been loaded with only 12 red cubes, 13 green cubes, and 14 blue cubes. What is the sum of the IDs of those games?
Your puzzle answer was 2207.
--- Part Two ---
The Elf says they've stopped producing snow because they aren't getting any water! He isn't sure why the water stopped; however, he can show you how to get to the water source to check it out for yourself. It's just up ahead!
As you continue your walk, the Elf poses a second question: in each game you played, what is the fewest number of cubes of each color that could have been in the bag to make the game possible?
Again consider the example games from earlier:
Game 1: 3 blue, 4 red; 1 red, 2 green, 6 blue; 2 green
Game 2: 1 blue, 2 green; 3 green, 4 blue, 1 red; 1 green, 1 blue
Game 3: 8 green, 6 blue, 20 red; 5 blue, 4 red, 13 green; 5 green, 1 red
Game 4: 1 green, 3 red, 6 blue; 3 green, 6 red; 3 green, 15 blue, 14 red
Game 5: 6 red, 1 blue, 3 green; 2 blue, 1 red, 2 green
In game 1, the game could have been played with as few as 4 red, 2 green, and 6 blue cubes. If any color had even one fewer cube, the game would have been impossible.
Game 2 could have been played with a minimum of 1 red, 3 green, and 4 blue cubes.
Game 3 must have been played with at least 20 red, 13 green, and 6 blue cubes.
Game 4 required at least 14 red, 3 green, and 15 blue cubes.
Game 5 needed no fewer than 6 red, 3 green, and 2 blue cubes in the bag.
The power of a set of cubes is equal to the numbers of red, green, and blue cubes multiplied together. The power of the minimum set of cubes in game 1 is 48. In games 2-5 it was 12, 1560, 630, and 36, respectively. Adding up these five powers produces the sum 2286.
For each game, find the minimum set of cubes that must have been present. What is the sum of the power of these sets?
| 415
|
--- Day 9: Sensor Boost ---
You've just said goodbye to the rebooted rover and left Mars when you receive a faint distress signal coming from the asteroid belt. It must be the Ceres monitoring station!
In order to lock on to the signal, you'll need to boost your sensors. The Elves send up the latest BOOST program - Basic Operation Of System Test.
While BOOST (your puzzle input) is capable of boosting your sensors, for tenuous safety reasons, it refuses to do so until the computer it runs on passes some checks to demonstrate it is a complete Intcode computer.
Your existing Intcode computer is missing one key feature: it needs support for parameters in relative mode.
Parameters in mode 2, relative mode, behave very similarly to parameters in position mode: the parameter is interpreted as a position. Like position mode, parameters in relative mode can be read from or written to.
The important difference is that relative mode parameters don't count from address 0. Instead, they count from a value called the relative base. The relative base starts at 0.
The address a relative mode parameter refers to is itself plus the current relative base. When the relative base is 0, relative mode parameters and position mode parameters with the same value refer to the same address.
For example, given a relative base of 50, a relative mode parameter of -7 refers to memory address 50 + -7 = 43.
The relative base is modified with the relative base offset instruction:
Opcode 9 adjusts the relative base by the value of its only parameter. The relative base increases (or decreases, if the value is negative) by the value of the parameter.
For example, if the relative base is 2000, then after the instruction 109,19, the relative base would be 2019. If the next instruction were 204,-34, then the value at address 1985 would be output.
Your Intcode computer will also need a few other capabilities:
The computer's available memory should be much larger than the initial program. Memory beyond the initial program starts with the value 0 and can be read or written like any other memory. (It is invalid to try to access memory at a negative address, though.)
The computer should have support for large numbers. Some instructions near the beginning of the BOOST program will verify this capability.
Here are some example programs that use these features:
109,1,204,-1,1001,100,1,100,1008,100,16,101,1006,101,0,99 takes no input and produces a copy of itself as output.
1102,34915192,34915192,7,4,7,99,0 should output a 16-digit number.
104,1125899906842624,99 should output the large number in the middle.
The BOOST program will ask for a single input; run it in test mode by providing it the value 1. It will perform a series of checks on each opcode, output any opcodes (and the associated parameter modes) that seem to be functioning incorrectly, and finally output a BOOST keycode.
Once your Intcode computer is fully functional, the BOOST program should report no malfunctioning opcodes when run in test mode; it should only output a single value, the BOOST keycode. What BOOST keycode does it produce?
Your puzzle answer was 2204990589.
--- Part Two ---
You now have a complete Intcode computer.
Finally, you can lock on to the Ceres distress signal! You just need to boost your sensors using the BOOST program.
The program runs in sensor boost mode by providing the input instruction the value 2. Once run, it will boost the sensors automatically, but it might take a few seconds to complete the operation on slower hardware. In sensor boost mode, the program will output a single value: the coordinates of the distress signal.
Run the BOOST program in sensor boost mode. What are the coordinates of the distress signal?
| 416
|
--- Day 17: Reservoir Research ---
You arrive in the year 18. If it weren't for the coat you got in 1018, you would be very cold: the North Pole base hasn't even been constructed.
Rather, it hasn't been constructed yet. The Elves are making a little progress, but there's not a lot of liquid water in this climate, so they're getting very dehydrated. Maybe there's more underground?
You scan a two-dimensional vertical slice of the ground nearby and discover that it is mostly sand with veins of clay. The scan only provides data with a granularity of square meters, but it should be good enough to determine how much water is trapped there. In the scan, x represents the distance to the right, and y represents the distance down. There is also a spring of water near the surface at x=500, y=0. The scan identifies which square meters are clay (your puzzle input).
For example, suppose your scan shows the following veins of clay:
x=495, y=2..7
y=7, x=495..501
x=501, y=3..7
x=498, y=2..4
x=506, y=1..2
x=498, y=10..13
x=504, y=10..13
y=13, x=498..504
Rendering clay as #, sand as ., and the water spring as +, and with x increasing to the right and y increasing downward, this becomes:
44444455555555
99999900000000
45678901234567
0 ......+.......
1 ............#.
2 .#..#.......#.
3 .#..#..#......
4 .#..#..#......
5 .#.....#......
6 .#.....#......
7 .#######......
8 ..............
9 ..............
10 ....#.....#...
11 ....#.....#...
12 ....#.....#...
13 ....#######...
The spring of water will produce water forever. Water can move through sand, but is blocked by clay. Water always moves down when possible, and spreads to the left and right otherwise, filling space that has clay on both sides and falling out otherwise.
For example, if five squares of water are created, they will flow downward until they reach the clay and settle there. Water that has come to rest is shown here as ~, while sand through which water has passed (but which is now dry again) is shown as |:
......+.......
......|.....#.
.#..#.|.....#.
.#..#.|#......
.#..#.|#......
.#....|#......
.#~~~~~#......
.#######......
..............
..............
....#.....#...
....#.....#...
....#.....#...
....#######...
Two squares of water can't occupy the same location. If another five squares of water are created, they will settle on the first five, filling the clay reservoir a little more:
......+.......
......|.....#.
.#..#.|.....#.
.#..#.|#......
.#..#.|#......
.#~~~~~#......
.#~~~~~#......
.#######......
..............
..............
....#.....#...
....#.....#...
....#.....#...
....#######...
Water pressure does not apply in this scenario. If another four squares of water are created, they will stay on the right side of the barrier, and no water will reach the left side:
......+.......
......|.....#.
.#..#.|.....#.
.#..#~~#......
.#..#~~#......
.#~~~~~#......
.#~~~~~#......
.#######......
..............
..............
....#.....#...
....#.....#...
....#.....#...
....#######...
At this point, the top reservoir overflows. While water can reach the tiles above the surface of the water, it cannot settle there, and so the next five squares of water settle like this:
......+.......
......|.....#.
.#..#||||...#.
.#..#~~#|.....
.#..#~~#|.....
.#~~~~~#|.....
.#~~~~~#|.....
.#######|.....
........|.....
........|.....
....#...|.#...
....#...|.#...
....#~~~~~#...
....#######...
Note especially the leftmost |: the new squares of water can reach this tile, but cannot stop there. Instead, eventually, they all fall to the right and settle in the reservoir below.
After 10 more squares of water, the bottom reservoir is also full:
......+.......
......|.....#.
.#..#||||...#.
.#..#~~#|.....
.#..#~~#|.....
.#~~~~~#|.....
.#~~~~~#|.....
.#######|.....
........|.....
........|.....
....#~~~~~#...
....#~~~~~#...
....#~~~~~#...
....#######...
Finally, while there is nowhere left for the water to settle, it can reach a few more tiles before overflowing beyond the bottom of the scanned data:
......+....... (line not counted: above minimum y value)
......|.....#.
.#..#||||...#.
.#..#~~#|.....
.#..#~~#|.....
.#~~~~~#|.....
.#~~~~~#|.....
.#######|.....
........|.....
...|||||||||..
...|#~~~~~#|..
...|#~~~~~#|..
...|#~~~~~#|..
...|#######|..
...|.......|.. (line not counted: below maximum y value)
...|.......|.. (line not counted: below maximum y value)
...|.......|.. (line not counted: below maximum y value)
How many tiles can be reached by the water? To prevent counting forever, ignore tiles with a y coordinate smaller than the smallest y coordinate in your scan data or larger than the largest one. Any x coordinate is valid. In this example, the lowest y coordinate given is 1, and the highest is 13, causing the water spring (in row 0) and the water falling off the bottom of the render (in rows 14 through infinity) to be ignored.
So, in the example above, counting both water at rest (~) and other sand tiles the water can hypothetically reach (|), the total number of tiles the water can reach is 57.
How many tiles can the water reach within the range of y values in your scan?
Your puzzle answer was 31861.
--- Part Two ---
After a very long time, the water spring will run dry. How much water will be retained?
In the example above, water that won't eventually drain out is shown as ~, a total of 29 tiles.
How many water tiles are left after the water spring stops producing water and all remaining water not at rest has drained?
| 417
|
--- Day 3: Mull It Over ---
"Our computers are having issues, so I have no idea if we have any Chief Historians in stock! You're welcome to check the warehouse, though," says the mildly flustered shopkeeper at the North Pole Toboggan Rental Shop. The Historians head out to take a look.
The shopkeeper turns to you. "Any chance you can see why our computers are having issues again?"
The computer appears to be trying to run a program, but its memory (your puzzle input) is corrupted. All of the instructions have been jumbled up!
It seems like the goal of the program is just to multiply some numbers. It does that with instructions like mul(X,Y), where X and Y are each 1-3 digit numbers. For instance, mul(44,46) multiplies 44 by 46 to get a result of 2024. Similarly, mul(123,4) would multiply 123 by 4.
However, because the program's memory has been corrupted, there are also many invalid characters that should be ignored, even if they look like part of a mul instruction. Sequences like mul(4*, mul(6,9!, ?(12,34), or mul ( 2 , 4 ) do nothing.
For example, consider the following section of corrupted memory:
xmul(2,4)%&mul[3,7]!@^do_not_mul(5,5)+mul(32,64]then(mul(11,8)mul(8,5))
Only the four highlighted sections are real mul instructions. Adding up the result of each instruction produces 161 (2*4 + 5*5 + 11*8 + 8*5).
Scan the corrupted memory for uncorrupted mul instructions. What do you get if you add up all of the results of the multiplications?
| 418
|
--- Day 18: Lavaduct Lagoon ---
Thanks to your efforts, the machine parts factory is one of the first factories up and running since the lavafall came back. However, to catch up with the large backlog of parts requests, the factory will also need a large supply of lava for a while; the Elves have already started creating a large lagoon nearby for this purpose.
However, they aren't sure the lagoon will be big enough; they've asked you to take a look at the dig plan (your puzzle input). For example:
R 6 (#70c710)
D 5 (#0dc571)
L 2 (#5713f0)
D 2 (#d2c081)
R 2 (#59c680)
D 2 (#411b91)
L 5 (#8ceee2)
U 2 (#caa173)
L 1 (#1b58a2)
U 2 (#caa171)
R 2 (#7807d2)
U 3 (#a77fa3)
L 2 (#015232)
U 2 (#7a21e3)
The digger starts in a 1 meter cube hole in the ground. They then dig the specified number of meters up (U), down (D), left (L), or right (R), clearing full 1 meter cubes as they go. The directions are given as seen from above, so if "up" were north, then "right" would be east, and so on. Each trench is also listed with the color that the edge of the trench should be painted as an RGB hexadecimal color code.
When viewed from above, the above example dig plan would result in the following loop of trench (#) having been dug out from otherwise ground-level terrain (.):
#######
#.....#
###...#
..#...#
..#...#
###.###
#...#..
##..###
.#....#
.######
At this point, the trench could contain 38 cubic meters of lava. However, this is just the edge of the lagoon; the next step is to dig out the interior so that it is one meter deep as well:
#######
#######
#######
..#####
..#####
#######
#####..
#######
.######
.######
Now, the lagoon can contain a much more respectable 62 cubic meters of lava. While the interior is dug out, the edges are also painted according to the color codes in the dig plan.
The Elves are concerned the lagoon won't be large enough; if they follow their dig plan, how many cubic meters of lava could it hold?
| 419
|
--- Day 17: No Such Thing as Too Much ---
The elves bought too much eggnog again - 150 liters this time. To fit it all into your refrigerator, you'll need to move it into smaller containers. You take an inventory of the capacities of the available containers.
For example, suppose you have containers of size 20, 15, 10, 5, and 5 liters. If you need to store 25 liters, there are four ways to do it:
15 and 10
20 and 5 (the first 5)
20 and 5 (the second 5)
15, 5, and 5
Filling all containers entirely, how many different combinations of containers can exactly fit all 150 liters of eggnog?
| 420
|
--- Day 22: Mode Maze ---
This is it, your final stop: the year -483. It's snowing and dark outside; the only light you can see is coming from a small cottage in the distance. You make your way there and knock on the door.
A portly man with a large, white beard answers the door and invites you inside. For someone living near the North Pole in -483, he must not get many visitors, but he doesn't act surprised to see you. Instead, he offers you some milk and cookies.
After talking for a while, he asks a favor of you. His friend hasn't come back in a few hours, and he's not sure where he is. Scanning the region briefly, you discover one life signal in a cave system nearby; his friend must have taken shelter there. The man asks if you can go there to retrieve his friend.
The cave is divided into square regions which are either dominantly rocky, narrow, or wet (called its type). Each region occupies exactly one coordinate in X,Y format where X and Y are integers and zero or greater. (Adjacent regions can be the same type.)
The scan (your puzzle input) is not very detailed: it only reveals the depth of the cave system and the coordinates of the target. However, it does not reveal the type of each region. The mouth of the cave is at 0,0.
The man explains that due to the unusual geology in the area, there is a method to determine any region's type based on its erosion level. The erosion level of a region can be determined from its geologic index. The geologic index can be determined using the first rule that applies from the list below:
The region at 0,0 (the mouth of the cave) has a geologic index of 0.
The region at the coordinates of the target has a geologic index of 0.
If the region's Y coordinate is 0, the geologic index is its X coordinate times 16807.
If the region's X coordinate is 0, the geologic index is its Y coordinate times 48271.
Otherwise, the region's geologic index is the result of multiplying the erosion levels of the regions at X-1,Y and X,Y-1.
A region's erosion level is its geologic index plus the cave system's depth, all modulo 20183. Then:
If the erosion level modulo 3 is 0, the region's type is rocky.
If the erosion level modulo 3 is 1, the region's type is wet.
If the erosion level modulo 3 is 2, the region's type is narrow.
For example, suppose the cave system's depth is 510 and the target's coordinates are 10,10. Using % to represent the modulo operator, the cavern would look as follows:
At 0,0, the geologic index is 0. The erosion level is (0 + 510) % 20183 = 510. The type is 510 % 3 = 0, rocky.
At 1,0, because the Y coordinate is 0, the geologic index is 1 * 16807 = 16807. The erosion level is (16807 + 510) % 20183 = 17317. The type is 17317 % 3 = 1, wet.
At 0,1, because the X coordinate is 0, the geologic index is 1 * 48271 = 48271. The erosion level is (48271 + 510) % 20183 = 8415. The type is 8415 % 3 = 0, rocky.
At 1,1, neither coordinate is 0 and it is not the coordinate of the target, so the geologic index is the erosion level of 0,1 (8415) times the erosion level of 1,0 (17317), 8415 * 17317 = 145722555. The erosion level is (145722555 + 510) % 20183 = 1805. The type is 1805 % 3 = 2, narrow.
At 10,10, because they are the target's coordinates, the geologic index is 0. The erosion level is (0 + 510) % 20183 = 510. The type is 510 % 3 = 0, rocky.
Drawing this same cave system with rocky as ., wet as =, narrow as |, the mouth as M, the target as T, with 0,0 in the top-left corner, X increasing to the right, and Y increasing downward, the top-left corner of the map looks like this:
M=.|=.|.|=.|=|=.
.|=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||...|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Before you go in, you should determine the risk level of the area. For the rectangle that has a top-left corner of region 0,0 and a bottom-right corner of the region containing the target, add up the risk level of each individual region: 0 for rocky regions, 1 for wet regions, and 2 for narrow regions.
In the cave system above, because the mouth is at 0,0 and the target is at 10,10, adding up the risk level of all regions with an X coordinate from 0 to 10 and a Y coordinate from 0 to 10, this total is 114.
What is the total risk level for the smallest rectangle that includes 0,0 and the target's coordinates?
Your puzzle answer was 9899.
--- Part Two ---
Okay, it's time to go rescue the man's friend.
As you leave, he hands you some tools: a torch and some climbing gear. You can't equip both tools at once, but you can choose to use neither.
Tools can only be used in certain regions:
In rocky regions, you can use the climbing gear or the torch. You cannot use neither (you'll likely slip and fall).
In wet regions, you can use the climbing gear or neither tool. You cannot use the torch (if it gets wet, you won't have a light source).
In narrow regions, you can use the torch or neither tool. You cannot use the climbing gear (it's too bulky to fit).
You start at 0,0 (the mouth of the cave) with the torch equipped and must reach the target coordinates as quickly as possible. The regions with negative X or Y are solid rock and cannot be traversed. The fastest route might involve entering regions beyond the X or Y coordinate of the target.
You can move to an adjacent region (up, down, left, or right; never diagonally) if your currently equipped tool allows you to enter that region. Moving to an adjacent region takes one minute. (For example, if you have the torch equipped, you can move between rocky and narrow regions, but cannot enter wet regions.)
You can change your currently equipped tool or put both away if your new equipment would be valid for your current region. Switching to using the climbing gear, torch, or neither always takes seven minutes, regardless of which tools you start with. (For example, if you are in a rocky region, you can switch from the torch to the climbing gear, but you cannot switch to neither.)
Finally, once you reach the target, you need the torch equipped before you can find him in the dark. The target is always in a rocky region, so if you arrive there with climbing gear equipped, you will need to spend seven minutes switching to your torch.
For example, using the same cave system as above, starting in the top left corner (0,0) and moving to the bottom right corner (the target, 10,10) as quickly as possible, one possible route is as follows, with your current position marked X:
Initially:
X=.|=.|.|=.|=|=.
.|=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||...|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Down:
M=.|=.|.|=.|=|=.
X|=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||...|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Right:
M=.|=.|.|=.|=|=.
.X=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||...|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Switch from using the torch to neither tool:
M=.|=.|.|=.|=|=.
.X=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||...|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Right 3:
M=.|=.|.|=.|=|=.
.|=|X|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||...|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Switch from using neither tool to the climbing gear:
M=.|=.|.|=.|=|=.
.|=|X|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||...|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Down 7:
M=.|=.|.|=.|=|=.
.|=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..X==..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||...|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Right:
M=.|=.|.|=.|=|=.
.|=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..=X=..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||...|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Down 3:
M=.|=.|.|=.|=|=.
.|=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||.X.|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Right:
M=.|=.|.|=.|=|=.
.|=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||..X|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Down:
M=.|=.|.|=.|=|=.
.|=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||...|==..|=.|
=.=|=.X..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Right 4:
M=.|=.|.|=.|=|=.
.|=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===T===||
=|||...|==..|=.|
=.=|=.=..=X||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Up 2:
M=.|=.|.|=.|=|=.
.|=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===X===||
=|||...|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
Switch from using the climbing gear to the torch:
M=.|=.|.|=.|=|=.
.|=|=|||..|.=...
.==|....||=..|==
=.|....|.==.|==.
=|..==...=.|==..
=||.=.=||=|=..|=
|.=.===|||..=..|
|..==||=.|==|===
.=..===..=|.|||.
.======|||=|=.|=
.===|=|===X===||
=|||...|==..|=.|
=.=|=.=..=.||==|
||=|=...|==.=|==
|=.=||===.|||===
||.|==.|.|.||=||
This is tied with other routes as the fastest way to reach the target: 45 minutes. In it, 21 minutes are spent switching tools (three times, seven minutes each) and the remaining 24 minutes are spent moving.
What is the fewest number of minutes you can take to reach the target?
| 421
|
--- Day 12: JSAbacusFramework.io ---
Santa's Accounting-Elves need help balancing the books after a recent order. Unfortunately, their accounting software uses a peculiar storage format. That's where you come in.
They have a JSON document which contains a variety of things: arrays ([1,2,3]), objects ({"a":1, "b":2}), numbers, and strings. Your first job is to simply find all of the numbers throughout the document and add them together.
For example:
[1,2,3] and {"a":2,"b":4} both have a sum of 6.
[[[3]]] and {"a":{"b":4},"c":-1} both have a sum of 3.
{"a":[-1,1]} and [-1,{"a":1}] both have a sum of 0.
[] and {} both have a sum of 0.
You will not encounter any strings containing numbers.
What is the sum of all numbers in the document?
Your puzzle answer was 111754.
--- Part Two ---
Uh oh - the Accounting-Elves have realized that they double-counted everything red.
Ignore any object (and all of its children) which has any property with the value "red". Do this only for objects ({...}), not arrays ([...]).
[1,2,3] still has a sum of 6.
[1,{"c":"red","b":2},3] now has a sum of 4, because the middle object is ignored.
{"d":"red","e":[1,2,3,4],"f":5} now has a sum of 0, because the entire structure is ignored.
[1,"red",5] has a sum of 6, because "red" in an array has no effect.
| 422
|
--- Day 10: Balance Bots ---
You come upon a factory in which many robots are zooming around handing small microchips to each other.
Upon closer examination, you notice that each bot only proceeds when it has two microchips, and once it does, it gives each one to a different bot or puts it in a marked "output" bin. Sometimes, bots take microchips from "input" bins, too.
Inspecting one of the microchips, it seems like they each contain a single number; the bots must use some logic to decide what to do with each chip. You access the local control computer and download the bots' instructions (your puzzle input).
Some of the instructions specify that a specific-valued microchip should be given to a specific bot; the rest of the instructions indicate what a given bot should do with its lower-value or higher-value chip.
For example, consider the following instructions:
value 5 goes to bot 2
bot 2 gives low to bot 1 and high to bot 0
value 3 goes to bot 1
bot 1 gives low to output 1 and high to bot 0
bot 0 gives low to output 2 and high to output 0
value 2 goes to bot 2
Initially, bot 1 starts with a value-3 chip, and bot 2 starts with a value-2 chip and a value-5 chip.
Because bot 2 has two microchips, it gives its lower one (2) to bot 1 and its higher one (5) to bot 0.
Then, bot 1 has two microchips; it puts the value-2 chip in output 1 and gives the value-3 chip to bot 0.
Finally, bot 0 has two microchips; it puts the 3 in output 2 and the 5 in output 0.
In the end, output bin 0 contains a value-5 microchip, output bin 1 contains a value-2 microchip, and output bin 2 contains a value-3 microchip. In this configuration, bot number 2 is responsible for comparing value-5 microchips with value-2 microchips.
Based on your instructions, what is the number of the bot that is responsible for comparing value-61 microchips with value-17 microchips?
| 423
|
--- Day 22: Slam Shuffle ---
There isn't much to do while you wait for the droids to repair your ship. At least you're drifting in the right direction. You decide to practice a new card shuffle you've been working on.
Digging through the ship's storage, you find a deck of space cards! Just like any deck of space cards, there are 10007 cards in the deck numbered 0 through 10006. The deck must be new - they're still in factory order, with 0 on the top, then 1, then 2, and so on, all the way through to 10006 on the bottom.
You've been practicing three different techniques that you use while shuffling. Suppose you have a deck of only 10 cards (numbered 0 through 9):
To deal into new stack, create a new stack of cards by dealing the top card of the deck onto the top of the new stack repeatedly until you run out of cards:
Top Bottom
0 1 2 3 4 5 6 7 8 9 Your deck
New stack
1 2 3 4 5 6 7 8 9 Your deck
0 New stack
2 3 4 5 6 7 8 9 Your deck
1 0 New stack
3 4 5 6 7 8 9 Your deck
2 1 0 New stack
Several steps later...
9 Your deck
8 7 6 5 4 3 2 1 0 New stack
Your deck
9 8 7 6 5 4 3 2 1 0 New stack
Finally, pick up the new stack you've just created and use it as the deck for the next technique.
To cut N cards, take the top N cards off the top of the deck and move them as a single unit to the bottom of the deck, retaining their order. For example, to cut 3:
Top Bottom
0 1 2 3 4 5 6 7 8 9 Your deck
3 4 5 6 7 8 9 Your deck
0 1 2 Cut cards
3 4 5 6 7 8 9 Your deck
0 1 2 Cut cards
3 4 5 6 7 8 9 0 1 2 Your deck
You've also been getting pretty good at a version of this technique where N is negative! In that case, cut (the absolute value of) N cards from the bottom of the deck onto the top. For example, to cut -4:
Top Bottom
0 1 2 3 4 5 6 7 8 9 Your deck
0 1 2 3 4 5 Your deck
6 7 8 9 Cut cards
0 1 2 3 4 5 Your deck
6 7 8 9 Cut cards
6 7 8 9 0 1 2 3 4 5 Your deck
To deal with increment N, start by clearing enough space on your table to lay out all of the cards individually in a long line. Deal the top card into the leftmost position. Then, move N positions to the right and deal the next card there. If you would move into a position past the end of the space on your table, wrap around and keep counting from the leftmost card again. Continue this process until you run out of cards.
For example, to deal with increment 3:
0 1 2 3 4 5 6 7 8 9 Your deck
. . . . . . . . . . Space on table
^ Current position
Deal the top card to the current position:
1 2 3 4 5 6 7 8 9 Your deck
0 . . . . . . . . . Space on table
^ Current position
Move the current position right 3:
1 2 3 4 5 6 7 8 9 Your deck
0 . . . . . . . . . Space on table
^ Current position
Deal the top card:
2 3 4 5 6 7 8 9 Your deck
0 . . 1 . . . . . . Space on table
^ Current position
Move right 3 and deal:
3 4 5 6 7 8 9 Your deck
0 . . 1 . . 2 . . . Space on table
^ Current position
Move right 3 and deal:
4 5 6 7 8 9 Your deck
0 . . 1 . . 2 . . 3 Space on table
^ Current position
Move right 3, wrapping around, and deal:
5 6 7 8 9 Your deck
0 . 4 1 . . 2 . . 3 Space on table
^ Current position
And so on:
0 7 4 1 8 5 2 9 6 3 Space on table
Positions on the table which already contain cards are still counted; they're not skipped. Of course, this technique is carefully designed so it will never put two cards in the same position or leave a position empty.
Finally, collect the cards on the table so that the leftmost card ends up at the top of your deck, the card to its right ends up just below the top card, and so on, until the rightmost card ends up at the bottom of the deck.
The complete shuffle process (your puzzle input) consists of applying many of these techniques. Here are some examples that combine techniques; they all start with a factory order deck of 10 cards:
deal with increment 7
deal into new stack
deal into new stack
Result: 0 3 6 9 2 5 8 1 4 7
cut 6
deal with increment 7
deal into new stack
Result: 3 0 7 4 1 8 5 2 9 6
deal with increment 7
deal with increment 9
cut -2
Result: 6 3 0 7 4 1 8 5 2 9
deal into new stack
cut -2
deal with increment 7
cut 8
cut -4
deal with increment 7
cut 3
deal with increment 9
deal with increment 3
cut -1
Result: 9 2 5 8 1 4 7 0 3 6
Positions within the deck count from 0 at the top, then 1 for the card immediately below the top card, and so on to the bottom. (That is, cards start in the position matching their number.)
After shuffling your factory order deck of 10007 cards, what is the position of card 2019?
Your puzzle answer was 3589.
--- Part Two ---
After a while, you realize your shuffling skill won't improve much more with merely a single deck of cards. You ask every 3D printer on the ship to make you some more cards while you check on the ship repairs. While reviewing the work the droids have finished so far, you think you see Halley's Comet fly past!
When you get back, you discover that the 3D printers have combined their power to create for you a single, giant, brand new, factory order deck of 119315717514047 space cards.
Finally, a deck of cards worthy of shuffling!
You decide to apply your complete shuffle process (your puzzle input) to the deck 101741582076661 times in a row.
You'll need to be careful, though - one wrong move with this many cards and you might overflow your entire ship!
After shuffling your new, giant, factory order deck that many times, what number is on the card that ends up in position 2020?
| 424
|
--- Day 3: Rucksack Reorganization ---
One Elf has the important job of loading all of the rucksacks with supplies for the jungle journey. Unfortunately, that Elf didn't quite follow the packing instructions, and so a few items now need to be rearranged.
Each rucksack has two large compartments. All items of a given type are meant to go into exactly one of the two compartments. The Elf that did the packing failed to follow this rule for exactly one item type per rucksack.
The Elves have made a list of all of the items currently in each rucksack (your puzzle input), but they need your help finding the errors. Every item type is identified by a single lowercase or uppercase letter (that is, a and A refer to different types of items).
The list of items for each rucksack is given as characters all on a single line. A given rucksack always has the same number of items in each of its two compartments, so the first half of the characters represent items in the first compartment, while the second half of the characters represent items in the second compartment.
For example, suppose you have the following list of contents from six rucksacks:
vJrwpWtwJgWrhcsFMMfFFhFp
jqHRNqRjqzjGDLGLrsFMfFZSrLrFZsSL
PmmdzqPrVvPwwTWBwg
wMqvLMZHhHMvwLHjbvcjnnSBnvTQFn
ttgJtRGJQctTZtZT
CrZsJsPPZsGzwwsLwLmpwMDw
The first rucksack contains the items vJrwpWtwJgWrhcsFMMfFFhFp, which means its first compartment contains the items vJrwpWtwJgWr, while the second compartment contains the items hcsFMMfFFhFp. The only item type that appears in both compartments is lowercase p.
The second rucksack's compartments contain jqHRNqRjqzjGDLGL and rsFMfFZSrLrFZsSL. The only item type that appears in both compartments is uppercase L.
The third rucksack's compartments contain PmmdzqPrV and vPwwTWBwg; the only common item type is uppercase P.
The fourth rucksack's compartments only share item type v.
The fifth rucksack's compartments only share item type t.
The sixth rucksack's compartments only share item type s.
To help prioritize item rearrangement, every item type can be converted to a priority:
Lowercase item types a through z have priorities 1 through 26.
Uppercase item types A through Z have priorities 27 through 52.
In the above example, the priority of the item type that appears in both compartments of each rucksack is 16 (p), 38 (L), 42 (P), 22 (v), 20 (t), and 19 (s); the sum of these is 157.
Find the item type that appears in both compartments of each rucksack. What is the sum of the priorities of those item types?
| 425
|
--- Day 10: Knot Hash ---
You come across some programs that are trying to implement a software emulation of a hash based on knot-tying. The hash these programs are implementing isn't very strong, but you decide to help them anyway. You make a mental note to remind the Elves later not to invent their own cryptographic functions.
This hash function simulates tying a knot in a circle of string with 256 marks on it. Based on the input to be hashed, the function repeatedly selects a span of string, brings the ends together, and gives the span a half-twist to reverse the order of the marks within it. After doing this many times, the order of the marks is used to build the resulting hash.
4--5 pinch 4 5 4 1
/ 5,0,1 / / twist / / 3 0 --> 3 0 --> 3 X 0
/ / / / /
2--1 2 1 2 5
To achieve this, begin with a list of numbers from 0 to 255, a current position which begins at 0 (the first element in the list), a skip size (which starts at 0), and a sequence of lengths (your puzzle input). Then, for each length:
Reverse the order of that length of elements in the list, starting with the element at the current position.
Move the current position forward by that length plus the skip size.
Increase the skip size by one.
The list is circular; if the current position and the length try to reverse elements beyond the end of the list, the operation reverses using as many extra elements as it needs from the front of the list. If the current position moves past the end of the list, it wraps around to the front. Lengths larger than the size of the list are invalid.
Here's an example using a smaller list:
Suppose we instead only had a circular list containing five elements, 0, 1, 2, 3, 4, and were given input lengths of 3, 4, 1, 5.
The list begins as [0] 1 2 3 4 (where square brackets indicate the current position).
The first length, 3, selects ([0] 1 2) 3 4 (where parentheses indicate the sublist to be reversed).
After reversing that section (0 1 2 into 2 1 0), we get ([2] 1 0) 3 4.
Then, the current position moves forward by the length, 3, plus the skip size, 0: 2 1 0 [3] 4. Finally, the skip size increases to 1.
The second length, 4, selects a section which wraps: 2 1) 0 ([3] 4.
The sublist 3 4 2 1 is reversed to form 1 2 4 3: 4 3) 0 ([1] 2.
The current position moves forward by the length plus the skip size, a total of 5, causing it not to move because it wraps around: 4 3 0 [1] 2. The skip size increases to 2.
The third length, 1, selects a sublist of a single element, and so reversing it has no effect.
The current position moves forward by the length (1) plus the skip size (2): 4 [3] 0 1 2. The skip size increases to 3.
The fourth length, 5, selects every element starting with the second: 4) ([3] 0 1 2. Reversing this sublist (3 0 1 2 4 into 4 2 1 0 3) produces: 3) ([4] 2 1 0.
Finally, the current position moves forward by 8: 3 4 2 1 [0]. The skip size increases to 4.
In this example, the first two numbers in the list end up being 3 and 4; to check the process, you can multiply them together to produce 12.
However, you should instead use the standard list size of 256 (with values 0 to 255) and the sequence of lengths in your puzzle input. Once this process is complete, what is the result of multiplying the first two numbers in the list?
| 426
|
--- Day 20: Firewall Rules ---
You'd like to set up a small hidden computer here so you can use it to get back into the network later. However, the corporate firewall only allows communication with certain external IP addresses.
You've retrieved the list of blocked IPs from the firewall, but the list seems to be messy and poorly maintained, and it's not clear which IPs are allowed. Also, rather than being written in dot-decimal notation, they are written as plain 32-bit integers, which can have any value from 0 through 4294967295, inclusive.
For example, suppose only the values 0 through 9 were valid, and that you retrieved the following blacklist:
5-8
0-2
4-7
The blacklist specifies ranges of IPs (inclusive of both the start and end value) that are not allowed. Then, the only IPs that this firewall allows are 3 and 9, since those are the only numbers not in any range.
Given the list of blocked IPs you retrieved from the firewall (your puzzle input), what is the lowest-valued IP that is not blocked?
Your puzzle answer was 17348574.
--- Part Two ---
How many IPs are allowed by the blacklist?
| 427
|
--- Day 10: Knot Hash ---
You come across some programs that are trying to implement a software emulation of a hash based on knot-tying. The hash these programs are implementing isn't very strong, but you decide to help them anyway. You make a mental note to remind the Elves later not to invent their own cryptographic functions.
This hash function simulates tying a knot in a circle of string with 256 marks on it. Based on the input to be hashed, the function repeatedly selects a span of string, brings the ends together, and gives the span a half-twist to reverse the order of the marks within it. After doing this many times, the order of the marks is used to build the resulting hash.
4--5 pinch 4 5 4 1
/ 5,0,1 / / twist / / 3 0 --> 3 0 --> 3 X 0
/ / / / /
2--1 2 1 2 5
To achieve this, begin with a list of numbers from 0 to 255, a current position which begins at 0 (the first element in the list), a skip size (which starts at 0), and a sequence of lengths (your puzzle input). Then, for each length:
Reverse the order of that length of elements in the list, starting with the element at the current position.
Move the current position forward by that length plus the skip size.
Increase the skip size by one.
The list is circular; if the current position and the length try to reverse elements beyond the end of the list, the operation reverses using as many extra elements as it needs from the front of the list. If the current position moves past the end of the list, it wraps around to the front. Lengths larger than the size of the list are invalid.
Here's an example using a smaller list:
Suppose we instead only had a circular list containing five elements, 0, 1, 2, 3, 4, and were given input lengths of 3, 4, 1, 5.
The list begins as [0] 1 2 3 4 (where square brackets indicate the current position).
The first length, 3, selects ([0] 1 2) 3 4 (where parentheses indicate the sublist to be reversed).
After reversing that section (0 1 2 into 2 1 0), we get ([2] 1 0) 3 4.
Then, the current position moves forward by the length, 3, plus the skip size, 0: 2 1 0 [3] 4. Finally, the skip size increases to 1.
The second length, 4, selects a section which wraps: 2 1) 0 ([3] 4.
The sublist 3 4 2 1 is reversed to form 1 2 4 3: 4 3) 0 ([1] 2.
The current position moves forward by the length plus the skip size, a total of 5, causing it not to move because it wraps around: 4 3 0 [1] 2. The skip size increases to 2.
The third length, 1, selects a sublist of a single element, and so reversing it has no effect.
The current position moves forward by the length (1) plus the skip size (2): 4 [3] 0 1 2. The skip size increases to 3.
The fourth length, 5, selects every element starting with the second: 4) ([3] 0 1 2. Reversing this sublist (3 0 1 2 4 into 4 2 1 0 3) produces: 3) ([4] 2 1 0.
Finally, the current position moves forward by 8: 3 4 2 1 [0]. The skip size increases to 4.
In this example, the first two numbers in the list end up being 3 and 4; to check the process, you can multiply them together to produce 12.
However, you should instead use the standard list size of 256 (with values 0 to 255) and the sequence of lengths in your puzzle input. Once this process is complete, what is the result of multiplying the first two numbers in the list?
Your puzzle answer was 40132.
--- Part Two ---
The logic you've constructed forms a single round of the Knot Hash algorithm; running the full thing requires many of these rounds. Some input and output processing is also required.
First, from now on, your input should be taken not as a list of numbers, but as a string of bytes instead. Unless otherwise specified, convert characters to bytes using their ASCII codes. This will allow you to handle arbitrary ASCII strings, and it also ensures that your input lengths are never larger than 255. For example, if you are given 1,2,3, you should convert it to the ASCII codes for each character: 49,44,50,44,51.
Once you have determined the sequence of lengths to use, add the following lengths to the end of the sequence: 17, 31, 73, 47, 23. For example, if you are given 1,2,3, your final sequence of lengths should be 49,44,50,44,51,17,31,73,47,23 (the ASCII codes from the input string combined with the standard length suffix values).
Second, instead of merely running one round like you did above, run a total of 64 rounds, using the same length sequence in each round. The current position and skip size should be preserved between rounds. For example, if the previous example was your first round, you would start your second round with the same length sequence (3, 4, 1, 5, 17, 31, 73, 47, 23, now assuming they came from ASCII codes and include the suffix), but start with the previous round's current position (4) and skip size (4).
Once the rounds are complete, you will be left with the numbers from 0 to 255 in some order, called the sparse hash. Your next task is to reduce these to a list of only 16 numbers called the dense hash. To do this, use numeric bitwise XOR to combine each consecutive block of 16 numbers in the sparse hash (there are 16 such blocks in a list of 256 numbers). So, the first element in the dense hash is the first sixteen elements of the sparse hash XOR'd together, the second element in the dense hash is the second sixteen elements of the sparse hash XOR'd together, etc.
For example, if the first sixteen elements of your sparse hash are as shown below, and the XOR operator is ^, you would calculate the first output number like this:
65 ^ 27 ^ 9 ^ 1 ^ 4 ^ 3 ^ 40 ^ 50 ^ 91 ^ 7 ^ 6 ^ 0 ^ 2 ^ 5 ^ 68 ^ 22 = 64
Perform this operation on each of the sixteen blocks of sixteen numbers in your sparse hash to determine the sixteen numbers in your dense hash.
Finally, the standard way to represent a Knot Hash is as a single hexadecimal string; the final output is the dense hash in hexadecimal notation. Because each number in your dense hash will be between 0 and 255 (inclusive), always represent each number as two hexadecimal digits (including a leading zero as necessary). So, if your first three numbers are 64, 7, 255, they correspond to the hexadecimal numbers 40, 07, ff, and so the first six characters of the hash would be 4007ff. Because every Knot Hash is sixteen such numbers, the hexadecimal representation is always 32 hexadecimal digits (0-f) long.
Here are some example hashes:
The empty string becomes a2582a3a0e66e6e86e3812dcb672a272.
AoC 2017 becomes 33efeb34ea91902bb2f59c9920caa6cd.
1,2,3 becomes 3efbe78a8d82f29979031a4aa0b16a9d.
1,2,4 becomes 63960835bcdc130f0b66d7ff4f6a5a8e.
Treating your puzzle input as a string of ASCII characters, what is the Knot Hash of your puzzle input? Ignore any leading or trailing whitespace you might encounter.
| 428
|
--- Day 7: Bridge Repair ---
The Historians take you to a familiar rope bridge over a river in the middle of a jungle. The Chief isn't on this side of the bridge, though; maybe he's on the other side?
When you go to cross the bridge, you notice a group of engineers trying to repair it. (Apparently, it breaks pretty frequently.) You won't be able to cross until it's fixed.
You ask how long it'll take; the engineers tell you that it only needs final calibrations, but some young elephants were playing nearby and stole all the operators from their calibration equations! They could finish the calibrations if only someone could determine which test values could possibly be produced by placing any combination of operators into their calibration equations (your puzzle input).
For example:
190: 10 19
3267: 81 40 27
83: 17 5
156: 15 6
7290: 6 8 6 15
161011: 16 10 13
192: 17 8 14
21037: 9 7 18 13
292: 11 6 16 20
Each line represents a single equation. The test value appears before the colon on each line; it is your job to determine whether the remaining numbers can be combined with operators to produce the test value.
Operators are always evaluated left-to-right, not according to precedence rules. Furthermore, numbers in the equations cannot be rearranged. Glancing into the jungle, you can see elephants holding two different types of operators: add (+) and multiply (*).
Only three of the above equations can be made true by inserting operators:
190: 10 19 has only one position that accepts an operator: between 10 and 19. Choosing + would give 29, but choosing * would give the test value (10 * 19 = 190).
3267: 81 40 27 has two positions for operators. Of the four possible configurations of the operators, two cause the right side to match the test value: 81 + 40 * 27 and 81 * 40 + 27 both equal 3267 (when evaluated left-to-right)!
292: 11 6 16 20 can be solved in exactly one way: 11 + 6 * 16 + 20.
The engineers just need the total calibration result, which is the sum of the test values from just the equations that could possibly be true. In the above example, the sum of the test values for the three equations listed above is 3749.
Determine which equations could possibly be true. What is their total calibration result?
Your puzzle answer was 2299996598890.
The first half of this puzzle is complete! It provides one gold star: *
--- Part Two ---
The engineers seem concerned; the total calibration result you gave them is nowhere close to being within safety tolerances. Just then, you spot your mistake: some well-hidden elephants are holding a third type of operator.
The concatenation operator (||) combines the digits from its left and right inputs into a single number. For example, 12 || 345 would become 12345. All operators are still evaluated left-to-right.
Now, apart from the three equations that could be made true using only addition and multiplication, the above example has three more equations that can be made true by inserting operators:
156: 15 6 can be made true through a single concatenation: 15 || 6 = 156.
7290: 6 8 6 15 can be made true using 6 * 8 || 6 * 15.
192: 17 8 14 can be made true using 17 || 8 + 14.
Adding up all six test values (the three that could be made before using only + and * plus the new three that can now be made by also using ||) produces the new total calibration result of 11387.
Using your new knowledge of elephant hiding spots, determine which equations could possibly be true. What is their total calibration result?
| 429
|
--- Day 23: Unstable Diffusion ---
You enter a large crater of gray dirt where the grove is supposed to be. All around you, plants you imagine were expected to be full of fruit are instead withered and broken. A large group of Elves has formed in the middle of the grove.
"...but this volcano has been dormant for months. Without ash, the fruit can't grow!"
You look up to see a massive, snow-capped mountain towering above you.
"It's not like there are other active volcanoes here; we've looked everywhere."
"But our scanners show active magma flows; clearly it's going somewhere."
They finally notice you at the edge of the grove, your pack almost overflowing from the random star fruit you've been collecting. Behind you, elephants and monkeys explore the grove, looking concerned. Then, the Elves recognize the ash cloud slowly spreading above your recent detour.
"Why do you--" "How is--" "Did you just--"
Before any of them can form a complete question, another Elf speaks up: "Okay, new plan. We have almost enough fruit already, and ash from the plume should spread here eventually. If we quickly plant new seedlings now, we can still make it to the extraction point. Spread out!"
The Elves each reach into their pack and pull out a tiny plant. The plants rely on important nutrients from the ash, so they can't be planted too close together.
There isn't enough time to let the Elves figure out where to plant the seedlings themselves; you quickly scan the grove (your puzzle input) and note their positions.
For example:
....#..
..###.#
#...#.#
.#...##
#.###..
##.#.##
.#..#..
The scan shows Elves # and empty ground .; outside your scan, more empty ground extends a long way in every direction. The scan is oriented so that north is up; orthogonal directions are written N (north), S (south), W (west), and E (east), while diagonal directions are written NE, NW, SE, SW.
The Elves follow a time-consuming process to figure out where they should each go; you can speed up this process considerably. The process consists of some number of rounds during which Elves alternate between considering where to move and actually moving.
During the first half of each round, each Elf considers the eight positions adjacent to themself. If no other Elves are in one of those eight positions, the Elf does not do anything during this round. Otherwise, the Elf looks in each of four directions in the following order and proposes moving one step in the first valid direction:
If there is no Elf in the N, NE, or NW adjacent positions, the Elf proposes moving north one step.
If there is no Elf in the S, SE, or SW adjacent positions, the Elf proposes moving south one step.
If there is no Elf in the W, NW, or SW adjacent positions, the Elf proposes moving west one step.
If there is no Elf in the E, NE, or SE adjacent positions, the Elf proposes moving east one step.
After each Elf has had a chance to propose a move, the second half of the round can begin. Simultaneously, each Elf moves to their proposed destination tile if they were the only Elf to propose moving to that position. If two or more Elves propose moving to the same position, none of those Elves move.
Finally, at the end of the round, the first direction the Elves considered is moved to the end of the list of directions. For example, during the second round, the Elves would try proposing a move to the south first, then west, then east, then north. On the third round, the Elves would first consider west, then east, then north, then south.
As a smaller example, consider just these five Elves:
.....
..##.
..#..
.....
..##.
.....
The northernmost two Elves and southernmost two Elves all propose moving north, while the middle Elf cannot move north and proposes moving south. The middle Elf proposes the same destination as the southwest Elf, so neither of them move, but the other three do:
..##.
.....
..#..
...#.
..#..
.....
Next, the northernmost two Elves and the southernmost Elf all propose moving south. Of the remaining middle two Elves, the west one cannot move south and proposes moving west, while the east one cannot move south or west and proposes moving east. All five Elves succeed in moving to their proposed positions:
.....
..##.
.#...
....#
.....
..#..
Finally, the southernmost two Elves choose not to move at all. Of the remaining three Elves, the west one proposes moving west, the east one proposes moving east, and the middle one proposes moving north; all three succeed in moving:
..#..
....#
#....
....#
.....
..#..
At this point, no Elves need to move, and so the process ends.
The larger example above proceeds as follows:
== Initial State ==
..............
..............
.......#......
.....###.#....
...#...#.#....
....#...##....
...#.###......
...##.#.##....
....#..#......
..............
..............
..............
== End of Round 1 ==
..............
.......#......
.....#...#....
...#..#.#.....
.......#..#...
....#.#.##....
..#..#.#......
..#.#.#.##....
..............
....#..#......
..............
..............
== End of Round 2 ==
..............
.......#......
....#.....#...
...#..#.#.....
.......#...#..
...#..#.#.....
.#...#.#.#....
..............
..#.#.#.##....
....#..#......
..............
..............
== End of Round 3 ==
..............
.......#......
.....#....#...
..#..#...#....
.......#...#..
...#..#.#.....
.#..#.....#...
.......##.....
..##.#....#...
...#..........
.......#......
..............
== End of Round 4 ==
..............
.......#......
......#....#..
..#...##......
...#.....#.#..
.........#....
.#...###..#...
..#......#....
....##....#...
....#.........
.......#......
..............
== End of Round 5 ==
.......#......
..............
..#..#.....#..
.........#....
......##...#..
.#.#.####.....
...........#..
....##..#.....
..#...........
..........#...
....#..#......
..............
After a few more rounds...
== End of Round 10 ==
.......#......
...........#..
..#.#..#......
......#.......
...#.....#..#.
.#......##....
.....##.......
..#........#..
....#.#..#....
..............
....#..#..#...
..............
To make sure they're on the right track, the Elves like to check after round 10 that they're making good progress toward covering enough ground. To do this, count the number of empty ground tiles contained by the smallest rectangle that contains every Elf. (The edges of the rectangle should be aligned to the N/S/E/W directions; the Elves do not have the patience to calculate arbitrary rectangles.) In the above example, that rectangle is:
......#.....
..........#.
.#.#..#.....
.....#......
..#.....#..#
#......##...
....##......
.#........#.
...#.#..#...
............
...#..#..#..
In this region, the number of empty ground tiles is 110.
Simulate the Elves' process and find the smallest rectangle that contains the Elves after 10 rounds. How many empty ground tiles does that rectangle contain?
Your puzzle answer was 4005.
--- Part Two ---
It seems you're on the right track. Finish simulating the process and figure out where the Elves need to go. How many rounds did you save them?
In the example above, the first round where no Elf moved was round 20:
.......#......
....#......#..
..#.....#.....
......#.......
...#....#.#..#
#.............
....#.....#...
..#.....#.....
....#.#....#..
.........#....
....#......#..
.......#......
Figure out where the Elves need to go. What is the number of the first round where no Elf moves?
| 430
|
--- Day 22: Monkey Market ---
As you're all teleported deep into the jungle, a monkey steals The Historians' device! You'll need get it back while The Historians are looking for the Chief.
The monkey that stole the device seems willing to trade it, but only in exchange for an absurd number of bananas. Your only option is to buy bananas on the Monkey Exchange Market.
You aren't sure how the Monkey Exchange Market works, but one of The Historians senses trouble and comes over to help. Apparently, they've been studying these monkeys for a while and have deciphered their secrets.
Today, the Market is full of monkeys buying good hiding spots. Fortunately, because of the time you recently spent in this jungle, you know lots of good hiding spots you can sell! If you sell enough hiding spots, you should be able to get enough bananas to buy the device back.
On the Market, the buyers seem to use random prices, but their prices are actually only pseudorandom! If you know the secret of how they pick their prices, you can wait for the perfect time to sell.
The part about secrets is literal, the Historian explains. Each buyer produces a pseudorandom sequence of secret numbers where each secret is derived from the previous.
In particular, each buyer's secret number evolves into the next secret number in the sequence via the following process:
Calculate the result of multiplying the secret number by 64. Then, mix this result into the secret number. Finally, prune the secret number.
Calculate the result of dividing the secret number by 32. Round the result down to the nearest integer. Then, mix this result into the secret number. Finally, prune the secret number.
Calculate the result of multiplying the secret number by 2048. Then, mix this result into the secret number. Finally, prune the secret number.
Each step of the above process involves mixing and pruning:
To mix a value into the secret number, calculate the bitwise XOR of the given value and the secret number. Then, the secret number becomes the result of that operation. (If the secret number is 42 and you were to mix 15 into the secret number, the secret number would become 37.)
To prune the secret number, calculate the value of the secret number modulo 16777216. Then, the secret number becomes the result of that operation. (If the secret number is 100000000 and you were to prune the secret number, the secret number would become 16113920.)
After this process completes, the buyer is left with the next secret number in the sequence. The buyer can repeat this process as many times as necessary to produce more secret numbers.
So, if a buyer had a secret number of 123, that buyer's next ten secret numbers would be:
15887950
16495136
527345
704524
1553684
12683156
11100544
12249484
7753432
5908254
Each buyer uses their own secret number when choosing their price, so it's important to be able to predict the sequence of secret numbers for each buyer. Fortunately, the Historian's research has uncovered the initial secret number of each buyer (your puzzle input). For example:
1
10
100
2024
This list describes the initial secret number of four different secret-hiding-spot-buyers on the Monkey Exchange Market. If you can simulate secret numbers from each buyer, you'll be able to predict all of their future prices.
In a single day, buyers each have time to generate 2000 new secret numbers. In this example, for each buyer, their initial secret number and the 2000th new secret number they would generate are:
1: 8685429
10: 4700978
100: 15273692
2024: 8667524
Adding up the 2000th new secret number for each buyer produces 37327623.
For each buyer, simulate the creation of 2000 new secret numbers. What is the sum of the 2000th secret number generated by each buyer?
Your puzzle answer was 17724064040.
The first half of this puzzle is complete! It provides one gold star: *
--- Part Two ---
Of course, the secret numbers aren't the prices each buyer is offering! That would be ridiculous. Instead, the prices the buyer offers are just the ones digit of each of their secret numbers.
So, if a buyer starts with a secret number of 123, that buyer's first ten prices would be:
3 (from 123)
0 (from 15887950)
6 (from 16495136)
5 (etc.)
4
4
6
4
4
2
This price is the number of bananas that buyer is offering in exchange for your information about a new hiding spot. However, you still don't speak monkey, so you can't negotiate with the buyers directly. The Historian speaks a little, but not enough to negotiate; instead, he can ask another monkey to negotiate on your behalf.
Unfortunately, the monkey only knows how to decide when to sell by looking at the changes in price. Specifically, the monkey will only look for a specific sequence of four consecutive changes in price, then immediately sell when it sees that sequence.
So, if a buyer starts with a secret number of 123, that buyer's first ten secret numbers, prices, and the associated changes would be:
123: 3
15887950: 0 (-3)
16495136: 6 (6)
527345: 5 (-1)
704524: 4 (-1)
1553684: 4 (0)
12683156: 6 (2)
11100544: 4 (-2)
12249484: 4 (0)
7753432: 2 (-2)
Note that the first price has no associated change because there was no previous price to compare it with.
In this short example, within just these first few prices, the highest price will be 6, so it would be nice to give the monkey instructions that would make it sell at that time. The first 6 occurs after only two changes, so there's no way to instruct the monkey to sell then, but the second 6 occurs after the changes -1,-1,0,2. So, if you gave the monkey that sequence of changes, it would wait until the first time it sees that sequence and then immediately sell your hiding spot information at the current price, winning you 6 bananas.
Each buyer only wants to buy one hiding spot, so after the hiding spot is sold, the monkey will move on to the next buyer. If the monkey never hears that sequence of price changes from a buyer, the monkey will never sell, and will instead just move on to the next buyer.
Worse, you can only give the monkey a single sequence of four price changes to look for. You can't change the sequence between buyers.
You're going to need as many bananas as possible, so you'll need to determine which sequence of four price changes will cause the monkey to get you the most bananas overall. Each buyer is going to generate 2000 secret numbers after their initial secret number, so, for each buyer, you'll have 2000 price changes in which your sequence can occur.
Suppose the initial secret number of each buyer is:
1
2
3
2024
There are many sequences of four price changes you could tell the monkey, but for these four buyers, the sequence that will get you the most bananas is -2,1,-1,3. Using that sequence, the monkey will make the following sales:
For the buyer with an initial secret number of 1, changes -2,1,-1,3 first occur when the price is 7.
For the buyer with initial secret 2, changes -2,1,-1,3 first occur when the price is 7.
For the buyer with initial secret 3, the change sequence -2,1,-1,3 does not occur in the first 2000 changes.
For the buyer starting with 2024, changes -2,1,-1,3 first occur when the price is 9.
So, by asking the monkey to sell the first time each buyer's prices go down 2, then up 1, then down 1, then up 3, you would get 23 (7 + 7 + 9) bananas!
Figure out the best sequence to tell the monkey so that by looking for that same sequence of changes in every buyer's future prices, you get the most bananas in total. What is the most bananas you can get?
| 431
|
--- Day 15: Beverage Bandits ---
Having perfected their hot chocolate, the Elves have a new problem: the Goblins that live in these caves will do anything to steal it. Looks like they're here for a fight.
You scan the area, generating a map of the walls (#), open cavern (.), and starting position of every Goblin (G) and Elf (E) (your puzzle input).
Combat proceeds in rounds; in each round, each unit that is still alive takes a turn, resolving all of its actions before the next unit's turn begins. On each unit's turn, it tries to move into range of an enemy (if it isn't already) and then attack (if it is in range).
All units are very disciplined and always follow very strict combat rules. Units never move or attack diagonally, as doing so would be dishonorable. When multiple choices are equally valid, ties are broken in reading order: top-to-bottom, then left-to-right. For instance, the order in which units take their turns within a round is the reading order of their starting positions in that round, regardless of the type of unit or whether other units have moved after the round started. For example:
would take their
These units: turns in this order:
####### #######
#.G.E.# #.1.2.#
#E.G.E# #3.4.5#
#.G.E.# #.6.7.#
####### #######
Each unit begins its turn by identifying all possible targets (enemy units). If no targets remain, combat ends.
Then, the unit identifies all of the open squares (.) that are in range of each target; these are the squares which are adjacent (immediately up, down, left, or right) to any target and which aren't already occupied by a wall or another unit. Alternatively, the unit might already be in range of a target. If the unit is not already in range of a target, and there are no open squares which are in range of a target, the unit ends its turn.
If the unit is already in range of a target, it does not move, but continues its turn with an attack. Otherwise, since it is not in range of a target, it moves.
To move, the unit first considers the squares that are in range and determines which of those squares it could reach in the fewest steps. A step is a single movement to any adjacent (immediately up, down, left, or right) open (.) square. Units cannot move into walls or other units. The unit does this while considering the current positions of units and does not do any prediction about where units will be later. If the unit cannot reach (find an open path to) any of the squares that are in range, it ends its turn. If multiple squares are in range and tied for being reachable in the fewest steps, the square which is first in reading order is chosen. For example:
Targets: In range: Reachable: Nearest: Chosen:
####### ####### ####### ####### #######
#E..G.# #E.?G?# #E.@G.# #E.!G.# #E.+G.#
#...#.# --> #.?.#?# --> #.@.#.# --> #.!.#.# --> #...#.#
#.G.#G# #?G?#G# #@G@#G# #!G.#G# #.G.#G#
####### ####### ####### ####### #######
In the above scenario, the Elf has three targets (the three Goblins):
Each of the Goblins has open, adjacent squares which are in range (marked with a ? on the map).
Of those squares, four are reachable (marked @); the other two (on the right) would require moving through a wall or unit to reach.
Three of these reachable squares are nearest, requiring the fewest steps (only 2) to reach (marked !).
Of those, the square which is first in reading order is chosen (+).
The unit then takes a single step toward the chosen square along the shortest path to that square. If multiple steps would put the unit equally closer to its destination, the unit chooses the step which is first in reading order. (This requires knowing when there is more than one shortest path so that you can consider the first step of each such path.) For example:
In range: Nearest: Chosen: Distance: Step:
####### ####### ####### ####### #######
#.E...# #.E...# #.E...# #4E212# #..E..#
#...?.# --> #...!.# --> #...+.# --> #32101# --> #.....#
#..?G?# #..!G.# #...G.# #432G2# #...G.#
####### ####### ####### ####### #######
The Elf sees three squares in range of a target (?), two of which are nearest (!), and so the first in reading order is chosen (+). Under "Distance", each open square is marked with its distance from the destination square; the two squares to which the Elf could move on this turn (down and to the right) are both equally good moves and would leave the Elf 2 steps from being in range of the Goblin. Because the step which is first in reading order is chosen, the Elf moves right one square.
Here's a larger example of movement:
Initially:
#########
#G..G..G#
#.......#
#.......#
#G..E..G#
#.......#
#.......#
#G..G..G#
#########
After 1 round:
#########
#.G...G.#
#...G...#
#...E..G#
#.G.....#
#.......#
#G..G..G#
#.......#
#########
After 2 rounds:
#########
#..G.G..#
#...G...#
#.G.E.G.#
#.......#
#G..G..G#
#.......#
#.......#
#########
After 3 rounds:
#########
#.......#
#..GGG..#
#..GEG..#
#G..G...#
#......G#
#.......#
#.......#
#########
Once the Goblins and Elf reach the positions above, they all are either in range of a target or cannot find any square in range of a target, and so none of the units can move until a unit dies.
After moving (or if the unit began its turn in range of a target), the unit attacks.
To attack, the unit first determines all of the targets that are in range of it by being immediately adjacent to it. If there are no such targets, the unit ends its turn. Otherwise, the adjacent target with the fewest hit points is selected; in a tie, the adjacent target with the fewest hit points which is first in reading order is selected.
The unit deals damage equal to its attack power to the selected target, reducing its hit points by that amount. If this reduces its hit points to 0 or fewer, the selected target dies: its square becomes . and it takes no further turns.
Each unit, either Goblin or Elf, has 3 attack power and starts with 200 hit points.
For example, suppose the only Elf is about to attack:
HP: HP:
G.... 9 G.... 9
..G.. 4 ..G.. 4
..EG. 2 --> ..E..
..G.. 2 ..G.. 2
...G. 1 ...G. 1
The "HP" column shows the hit points of the Goblin to the left in the corresponding row. The Elf is in range of three targets: the Goblin above it (with 4 hit points), the Goblin to its right (with 2 hit points), and the Goblin below it (also with 2 hit points). Because three targets are in range, the ones with the lowest hit points are selected: the two Goblins with 2 hit points each (one to the right of the Elf and one below the Elf). Of those, the Goblin first in reading order (the one to the right of the Elf) is selected. The selected Goblin's hit points (2) are reduced by the Elf's attack power (3), reducing its hit points to -1, killing it.
After attacking, the unit's turn ends. Regardless of how the unit's turn ends, the next unit in the round takes its turn. If all units have taken turns in this round, the round ends, and a new round begins.
The Elves look quite outnumbered. You need to determine the outcome of the battle: the number of full rounds that were completed (not counting the round in which combat ends) multiplied by the sum of the hit points of all remaining units at the moment combat ends. (Combat only ends when a unit finds no targets during its turn.)
Below is an entire sample combat. Next to each map, each row's units' hit points are listed from left to right.
Initially:
#######
#.G...# G(200)
#...EG# E(200), G(200)
#.#.#G# G(200)
#..G#E# G(200), E(200)
#.....#
#######
After 1 round:
#######
#..G..# G(200)
#...EG# E(197), G(197)
#.#G#G# G(200), G(197)
#...#E# E(197)
#.....#
#######
After 2 rounds:
#######
#...G.# G(200)
#..GEG# G(200), E(188), G(194)
#.#.#G# G(194)
#...#E# E(194)
#.....#
#######
Combat ensues; eventually, the top Elf dies:
After 23 rounds:
#######
#...G.# G(200)
#..G.G# G(200), G(131)
#.#.#G# G(131)
#...#E# E(131)
#.....#
#######
After 24 rounds:
#######
#..G..# G(200)
#...G.# G(131)
#.#G#G# G(200), G(128)
#...#E# E(128)
#.....#
#######
After 25 rounds:
#######
#.G...# G(200)
#..G..# G(131)
#.#.#G# G(125)
#..G#E# G(200), E(125)
#.....#
#######
After 26 rounds:
#######
#G....# G(200)
#.G...# G(131)
#.#.#G# G(122)
#...#E# E(122)
#..G..# G(200)
#######
After 27 rounds:
#######
#G....# G(200)
#.G...# G(131)
#.#.#G# G(119)
#...#E# E(119)
#...G.# G(200)
#######
After 28 rounds:
#######
#G....# G(200)
#.G...# G(131)
#.#.#G# G(116)
#...#E# E(113)
#....G# G(200)
#######
More combat ensues; eventually, the bottom Elf dies:
After 47 rounds:
#######
#G....# G(200)
#.G...# G(131)
#.#.#G# G(59)
#...#.#
#....G# G(200)
#######
Before the 48th round can finish, the top-left Goblin finds that there are no targets remaining, and so combat ends. So, the number of full rounds that were completed is 47, and the sum of the hit points of all remaining units is 200+131+59+200 = 590. From these, the outcome of the battle is 47 * 590 = 27730.
Here are a few example summarized combats:
####### #######
#G..#E# #...#E# E(200)
#E#E.E# #E#...# E(197)
#G.##.# --> #.E##.# E(185)
#...#E# #E..#E# E(200), E(200)
#...E.# #.....#
####### #######
Combat ends after 37 full rounds
Elves win with 982 total hit points left
Outcome: 37 * 982 = 36334
####### #######
#E..EG# #.E.E.# E(164), E(197)
#.#G.E# #.#E..# E(200)
#E.##E# --> #E.##.# E(98)
#G..#.# #.E.#.# E(200)
#..E#.# #...#.#
####### #######
Combat ends after 46 full rounds
Elves win with 859 total hit points left
Outcome: 46 * 859 = 39514
####### #######
#E.G#.# #G.G#.# G(200), G(98)
#.#G..# #.#G..# G(200)
#G.#.G# --> #..#..#
#G..#.# #...#G# G(95)
#...E.# #...G.# G(200)
####### #######
Combat ends after 35 full rounds
Goblins win with 793 total hit points left
Outcome: 35 * 793 = 27755
####### #######
#.E...# #.....#
#.#..G# #.#G..# G(200)
#.###.# --> #.###.#
#E#G#G# #.#.#.#
#...#G# #G.G#G# G(98), G(38), G(200)
####### #######
Combat ends after 54 full rounds
Goblins win with 536 total hit points left
Outcome: 54 * 536 = 28944
######### #########
#G......# #.G.....# G(137)
#.E.#...# #G.G#...# G(200), G(200)
#..##..G# #.G##...# G(200)
#...##..# --> #...##..#
#...#...# #.G.#...# G(200)
#.G...G.# #.......#
#.....G.# #.......#
######### #########
Combat ends after 20 full rounds
Goblins win with 937 total hit points left
Outcome: 20 * 937 = 18740
What is the outcome of the combat described in your puzzle input?
Your puzzle answer was 269430.
--- Part Two ---
According to your calculations, the Elves are going to lose badly. Surely, you won't mess up the timeline too much if you give them just a little advanced technology, right?
You need to make sure the Elves not only win, but also suffer no losses: even the death of a single Elf is unacceptable.
However, you can't go too far: larger changes will be more likely to permanently alter spacetime.
So, you need to find the outcome of the battle in which the Elves have the lowest integer attack power (at least 4) that allows them to win without a single death. The Goblins always have an attack power of 3.
In the first summarized example above, the lowest attack power the Elves need to win without losses is 15:
####### #######
#.G...# #..E..# E(158)
#...EG# #...E.# E(14)
#.#.#G# --> #.#.#.#
#..G#E# #...#.#
#.....# #.....#
####### #######
Combat ends after 29 full rounds
Elves win with 172 total hit points left
Outcome: 29 * 172 = 4988
In the second example above, the Elves need only 4 attack power:
####### #######
#E..EG# #.E.E.# E(200), E(23)
#.#G.E# #.#E..# E(200)
#E.##E# --> #E.##E# E(125), E(200)
#G..#.# #.E.#.# E(200)
#..E#.# #...#.#
####### #######
Combat ends after 33 full rounds
Elves win with 948 total hit points left
Outcome: 33 * 948 = 31284
In the third example above, the Elves need 15 attack power:
####### #######
#E.G#.# #.E.#.# E(8)
#.#G..# #.#E..# E(86)
#G.#.G# --> #..#..#
#G..#.# #...#.#
#...E.# #.....#
####### #######
Combat ends after 37 full rounds
Elves win with 94 total hit points left
Outcome: 37 * 94 = 3478
In the fourth example above, the Elves need 12 attack power:
####### #######
#.E...# #...E.# E(14)
#.#..G# #.#..E# E(152)
#.###.# --> #.###.#
#E#G#G# #.#.#.#
#...#G# #...#.#
####### #######
Combat ends after 39 full rounds
Elves win with 166 total hit points left
Outcome: 39 * 166 = 6474
In the last example above, the lone Elf needs 34 attack power:
######### #########
#G......# #.......#
#.E.#...# #.E.#...# E(38)
#..##..G# #..##...#
#...##..# --> #...##..#
#...#...# #...#...#
#.G...G.# #.......#
#.....G.# #.......#
######### #########
Combat ends after 30 full rounds
Elves win with 38 total hit points left
Outcome: 30 * 38 = 1140
After increasing the Elves' attack power until it is just barely enough for them to win without any Elves dying, what is the outcome of the combat described in your puzzle input?
| 432
|
--- Day 5: Hydrothermal Venture ---
You come across a field of hydrothermal vents on the ocean floor! These vents constantly produce large, opaque clouds, so it would be best to avoid them if possible.
They tend to form in lines; the submarine helpfully produces a list of nearby lines of vents (your puzzle input) for you to review. For example:
0,9 -> 5,9
8,0 -> 0,8
9,4 -> 3,4
2,2 -> 2,1
7,0 -> 7,4
6,4 -> 2,0
0,9 -> 2,9
3,4 -> 1,4
0,0 -> 8,8
5,5 -> 8,2
Each line of vents is given as a line segment in the format x1,y1 -> x2,y2 where x1,y1 are the coordinates of one end the line segment and x2,y2 are the coordinates of the other end. These line segments include the points at both ends. In other words:
An entry like 1,1 -> 1,3 covers points 1,1, 1,2, and 1,3.
An entry like 9,7 -> 7,7 covers points 9,7, 8,7, and 7,7.
For now, only consider horizontal and vertical lines: lines where either x1 = x2 or y1 = y2.
So, the horizontal and vertical lines from the above list would produce the following diagram:
.......1..
..1....1..
..1....1..
.......1..
.112111211
..........
..........
..........
..........
222111....
In this diagram, the top left corner is 0,0 and the bottom right corner is 9,9. Each position is shown as the number of lines which cover that point or . if no line covers that point. The top-left pair of 1s, for example, comes from 2,2 -> 2,1; the very bottom row is formed by the overlapping lines 0,9 -> 5,9 and 0,9 -> 2,9.
To avoid the most dangerous areas, you need to determine the number of points where at least two lines overlap. In the above example, this is anywhere in the diagram with a 2 or larger - a total of 5 points.
Consider only horizontal and vertical lines. At how many points do at least two lines overlap?
Your puzzle answer was 7380.
--- Part Two ---
Unfortunately, considering only horizontal and vertical lines doesn't give you the full picture; you need to also consider diagonal lines.
Because of the limits of the hydrothermal vent mapping system, the lines in your list will only ever be horizontal, vertical, or a diagonal line at exactly 45 degrees. In other words:
An entry like 1,1 -> 3,3 covers points 1,1, 2,2, and 3,3.
An entry like 9,7 -> 7,9 covers points 9,7, 8,8, and 7,9.
Considering all lines from the above example would now produce the following diagram:
1.1....11.
.111...2..
..2.1.111.
...1.2.2..
.112313211
...1.2....
..1...1...
.1.....1..
1.......1.
222111....
You still need to determine the number of points where at least two lines overlap. In the above example, this is still anywhere in the diagram with a 2 or larger - now a total of 12 points.
Consider all of the lines. At how many points do at least two lines overlap?
| 433
|
--- Day 18: Settlers of The North Pole ---
On the outskirts of the North Pole base construction project, many Elves are collecting lumber.
The lumber collection area is 50 acres by 50 acres; each acre can be either open ground (.), trees (|), or a lumberyard (#). You take a scan of the area (your puzzle input).
Strange magic is at work here: each minute, the landscape looks entirely different. In exactly one minute, an open acre can fill with trees, a wooded acre can be converted to a lumberyard, or a lumberyard can be cleared to open ground (the lumber having been sent to other projects).
The change to each acre is based entirely on the contents of that acre as well as the number of open, wooded, or lumberyard acres adjacent to it at the start of each minute. Here, "adjacent" means any of the eight acres surrounding that acre. (Acres on the edges of the lumber collection area might have fewer than eight adjacent acres; the missing acres aren't counted.)
In particular:
An open acre will become filled with trees if three or more adjacent acres contained trees. Otherwise, nothing happens.
An acre filled with trees will become a lumberyard if three or more adjacent acres were lumberyards. Otherwise, nothing happens.
An acre containing a lumberyard will remain a lumberyard if it was adjacent to at least one other lumberyard and at least one acre containing trees. Otherwise, it becomes open.
These changes happen across all acres simultaneously, each of them using the state of all acres at the beginning of the minute and changing to their new form by the end of that same minute. Changes that happen during the minute don't affect each other.
For example, suppose the lumber collection area is instead only 10 by 10 acres with this initial configuration:
Initial state:
.#.#...|#.
.....#|##|
.|..|...#.
..|#.....#
#.#|||#|#|
...#.||...
.|....|...
||...#|.#|
|.||||..|.
...#.|..|.
After 1 minute:
.......##.
......|###
.|..|...#.
..|#||...#
..##||.|#|
...#||||..
||...|||..
|||||.||.|
||||||||||
....||..|.
After 2 minutes:
.......#..
......|#..
.|.|||....
..##|||..#
..###|||#|
...#|||||.
|||||||||.
||||||||||
||||||||||
.|||||||||
After 3 minutes:
.......#..
....|||#..
.|.||||...
..###|||.#
...##|||#|
.||##|||||
||||||||||
||||||||||
||||||||||
||||||||||
After 4 minutes:
.....|.#..
...||||#..
.|.#||||..
..###||||#
...###||#|
|||##|||||
||||||||||
||||||||||
||||||||||
||||||||||
After 5 minutes:
....|||#..
...||||#..
.|.##||||.
..####|||#
.|.###||#|
|||###||||
||||||||||
||||||||||
||||||||||
||||||||||
After 6 minutes:
...||||#..
...||||#..
.|.###|||.
..#.##|||#
|||#.##|#|
|||###||||
||||#|||||
||||||||||
||||||||||
||||||||||
After 7 minutes:
...||||#..
..||#|##..
.|.####||.
||#..##||#
||##.##|#|
|||####|||
|||###||||
||||||||||
||||||||||
||||||||||
After 8 minutes:
..||||##..
..|#####..
|||#####|.
||#...##|#
||##..###|
||##.###||
|||####|||
||||#|||||
||||||||||
||||||||||
After 9 minutes:
..||###...
.||#####..
||##...##.
||#....###
|##....##|
||##..###|
||######||
|||###||||
||||||||||
||||||||||
After 10 minutes:
.||##.....
||###.....
||##......
|##.....##
|##.....##
|##....##|
||##.####|
||#####|||
||||#|||||
||||||||||
After 10 minutes, there are 37 wooded acres and 31 lumberyards. Multiplying the number of wooded acres by the number of lumberyards gives the total resource value after ten minutes: 37 * 31 = 1147.
What will the total resource value of the lumber collection area be after 10 minutes?
Your puzzle answer was 466312.
--- Part Two ---
This important natural resource will need to last for at least thousands of years. Are the Elves collecting this lumber sustainably?
What will the total resource value of the lumber collection area be after 1000000000 minutes?
| 434
|
--- Day 2: Red-Nosed Reports ---
Fortunately, the first location The Historians want to search isn't a long walk from the Chief Historian's office.
While the Red-Nosed Reindeer nuclear fusion/fission plant appears to contain no sign of the Chief Historian, the engineers there run up to you as soon as they see you. Apparently, they still talk about the time Rudolph was saved through molecular synthesis from a single electron.
They're quick to add that - since you're already here - they'd really appreciate your help analyzing some unusual data from the Red-Nosed reactor. You turn to check if The Historians are waiting for you, but they seem to have already divided into groups that are currently searching every corner of the facility. You offer to help with the unusual data.
The unusual data (your puzzle input) consists of many reports, one report per line. Each report is a list of numbers called levels that are separated by spaces. For example:
7 6 4 2 1
1 2 7 8 9
9 7 6 2 1
1 3 2 4 5
8 6 4 4 1
1 3 6 7 9
This example data contains six reports each containing five levels.
The engineers are trying to figure out which reports are safe. The Red-Nosed reactor safety systems can only tolerate levels that are either gradually increasing or gradually decreasing. So, a report only counts as safe if both of the following are true:
The levels are either all increasing or all decreasing.
Any two adjacent levels differ by at least one and at most three.
In the example above, the reports can be found safe or unsafe by checking those rules:
7 6 4 2 1: Safe because the levels are all decreasing by 1 or 2.
1 2 7 8 9: Unsafe because 2 7 is an increase of 5.
9 7 6 2 1: Unsafe because 6 2 is a decrease of 4.
1 3 2 4 5: Unsafe because 1 3 is increasing but 3 2 is decreasing.
8 6 4 4 1: Unsafe because 4 4 is neither an increase or a decrease.
1 3 6 7 9: Safe because the levels are all increasing by 1, 2, or 3.
So, in this example, 2 reports are safe.
Analyze the unusual data from the engineers. How many reports are safe?
Your puzzle answer was 383.
The first half of this puzzle is complete! It provides one gold star: *
--- Part Two ---
The engineers are surprised by the low number of safe reports until they realize they forgot to tell you about the Problem Dampener.
The Problem Dampener is a reactor-mounted module that lets the reactor safety systems tolerate a single bad level in what would otherwise be a safe report. It's like the bad level never happened!
Now, the same rules apply as before, except if removing a single level from an unsafe report would make it safe, the report instead counts as safe.
More of the above example's reports are now safe:
7 6 4 2 1: Safe without removing any level.
1 2 7 8 9: Unsafe regardless of which level is removed.
9 7 6 2 1: Unsafe regardless of which level is removed.
1 3 2 4 5: Safe by removing the second level, 3.
8 6 4 4 1: Safe by removing the third level, 4.
1 3 6 7 9: Safe without removing any level.
Thanks to the Problem Dampener, 4 reports are actually safe!
Update your analysis by handling situations where the Problem Dampener can remove a single level from unsafe reports. How many reports are now safe?
| 435
|
--- Day 3: Squares With Three Sides ---
Now that you can think clearly, you move deeper into the labyrinth of hallways and office furniture that makes up this part of Easter Bunny HQ. This must be a graphic design department; the walls are covered in specifications for triangles.
Or are they?
The design document gives the side lengths of each triangle it describes, but... 5 10 25? Some of these aren't triangles. You can't help but mark the impossible ones.
In a valid triangle, the sum of any two sides must be larger than the remaining side. For example, the "triangle" given above is impossible, because 5 + 10 is not larger than 25.
In your puzzle input, how many of the listed triangles are possible?
Your puzzle answer was 862.
--- Part Two ---
Now that you've helpfully marked up their design documents, it occurs to you that triangles are specified in groups of three vertically. Each set of three numbers in a column specifies a triangle. Rows are unrelated.
For example, given the following specification, numbers with the same hundreds digit would be part of the same triangle:
101 301 501
102 302 502
103 303 503
201 401 601
202 402 602
203 403 603
In your puzzle input, and instead reading by columns, how many of the listed triangles are possible?
| 436
|
--- Day 12: Subterranean Sustainability ---
The year 518 is significantly more underground than your history books implied. Either that, or you've arrived in a vast cavern network under the North Pole.
After exploring a little, you discover a long tunnel that contains a row of small pots as far as you can see to your left and right. A few of them contain plants - someone is trying to grow things in these geothermally-heated caves.
The pots are numbered, with 0 in front of you. To the left, the pots are numbered -1, -2, -3, and so on; to the right, 1, 2, 3.... Your puzzle input contains a list of pots from 0 to the right and whether they do (#) or do not (.) currently contain a plant, the initial state. (No other pots currently contain plants.) For example, an initial state of #..##.... indicates that pots 0, 3, and 4 currently contain plants.
Your puzzle input also contains some notes you find on a nearby table: someone has been trying to figure out how these plants spread to nearby pots. Based on the notes, for each generation of plants, a given pot has or does not have a plant based on whether that pot (and the two pots on either side of it) had a plant in the last generation. These are written as LLCRR => N, where L are pots to the left, C is the current pot being considered, R are the pots to the right, and N is whether the current pot will have a plant in the next generation. For example:
A note like ..#.. => . means that a pot that contains a plant but with no plants within two pots of it will not have a plant in it during the next generation.
A note like ##.## => . means that an empty pot with two plants on each side of it will remain empty in the next generation.
A note like .##.# => # means that a pot has a plant in a given generation if, in the previous generation, there were plants in that pot, the one immediately to the left, and the one two pots to the right, but not in the ones immediately to the right and two to the left.
It's not clear what these plants are for, but you're sure it's important, so you'd like to make sure the current configuration of plants is sustainable by determining what will happen after 20 generations.
For example, given the following input:
initial state: #..#.#..##......###...###
...## => #
..#.. => #
.#... => #
.#.#. => #
.#.## => #
.##.. => #
.#### => #
#.#.# => #
#.### => #
##.#. => #
##.## => #
###.. => #
###.# => #
####. => #
For brevity, in this example, only the combinations which do produce a plant are listed. (Your input includes all possible combinations.) Then, the next 20 generations will look like this:
1 2 3
0 0 0 0
0: ...#..#.#..##......###...###...........
1: ...#...#....#.....#..#..#..#...........
2: ...##..##...##....#..#..#..##..........
3: ..#.#...#..#.#....#..#..#...#..........
4: ...#.#..#...#.#...#..#..##..##.........
5: ....#...##...#.#..#..#...#...#.........
6: ....##.#.#....#...#..##..##..##........
7: ...#..###.#...##..#...#...#...#........
8: ...#....##.#.#.#..##..##..##..##.......
9: ...##..#..#####....#...#...#...#.......
10: ..#.#..#...#.##....##..##..##..##......
11: ...#...##...#.#...#.#...#...#...#......
12: ...##.#.#....#.#...#.#..##..##..##.....
13: ..#..###.#....#.#...#....#...#...#.....
14: ..#....##.#....#.#..##...##..##..##....
15: ..##..#..#.#....#....#..#.#...#...#....
16: .#.#..#...#.#...##...#...#.#..##..##...
17: ..#...##...#.#.#.#...##...#....#...#...
18: ..##.#.#....#####.#.#.#...##...##..##..
19: .#..###.#..#.#.#######.#.#.#..#.#...#..
20: .#....##....#####...#######....#.#..##.
The generation is shown along the left, where 0 is the initial state. The pot numbers are shown along the top, where 0 labels the center pot, negative-numbered pots extend to the left, and positive pots extend toward the right. Remember, the initial state begins at pot 0, which is not the leftmost pot used in this example.
After one generation, only seven plants remain. The one in pot 0 matched the rule looking for ..#.., the one in pot 4 matched the rule looking for .#.#., pot 9 matched .##.., and so on.
In this example, after 20 generations, the pots shown as # contain plants, the furthest left of which is pot -2, and the furthest right of which is pot 34. Adding up all the numbers of plant-containing pots after the 20th generation produces 325.
After 20 generations, what is the sum of the numbers of all pots which contain a plant?
| 437
|
--- Day 6: Custom Customs ---
As your flight approaches the regional airport where you'll switch to a much larger plane, customs declaration forms are distributed to the passengers.
The form asks a series of 26 yes-or-no questions marked a through z. All you need to do is identify the questions for which anyone in your group answers "yes". Since your group is just you, this doesn't take very long.
However, the person sitting next to you seems to be experiencing a language barrier and asks if you can help. For each of the people in their group, you write down the questions for which they answer "yes", one per line. For example:
abcx
abcy
abcz
In this group, there are 6 questions to which anyone answered "yes": a, b, c, x, y, and z. (Duplicate answers to the same question don't count extra; each question counts at most once.)
Another group asks for your help, then another, and eventually you've collected answers from every group on the plane (your puzzle input). Each group's answers are separated by a blank line, and within each group, each person's answers are on a single line. For example:
abc
a
b
c
ab
ac
a
a
a
a
b
This list represents answers from five groups:
The first group contains one person who answered "yes" to 3 questions: a, b, and c.
The second group contains three people; combined, they answered "yes" to 3 questions: a, b, and c.
The third group contains two people; combined, they answered "yes" to 3 questions: a, b, and c.
The fourth group contains four people; combined, they answered "yes" to only 1 question, a.
The last group contains one person who answered "yes" to only 1 question, b.
In this example, the sum of these counts is 3 + 3 + 3 + 1 + 1 = 11.
For each group, count the number of questions to which anyone answered "yes". What is the sum of those counts?
| 438
|
--- Day 5: If You Give A Seed A Fertilizer ---
You take the boat and find the gardener right where you were told he would be: managing a giant "garden" that looks more to you like a farm.
"A water source? Island Island is the water source!" You point out that Snow Island isn't receiving any water.
"Oh, we had to stop the water because we ran out of sand to filter it with! Can't make snow with dirty water. Don't worry, I'm sure we'll get more sand soon; we only turned off the water a few days... weeks... oh no." His face sinks into a look of horrified realization.
"I've been so busy making sure everyone here has food that I completely forgot to check why we stopped getting more sand! There's a ferry leaving soon that is headed over in that direction - it's much faster than your boat. Could you please go check it out?"
You barely have time to agree to this request when he brings up another. "While you wait for the ferry, maybe you can help us with our food production problem. The latest Island Island Almanac just arrived and we're having trouble making sense of it."
The almanac (your puzzle input) lists all of the seeds that need to be planted. It also lists what type of soil to use with each kind of seed, what type of fertilizer to use with each kind of soil, what type of water to use with each kind of fertilizer, and so on. Every type of seed, soil, fertilizer and so on is identified with a number, but numbers are reused by each category - that is, soil 123 and fertilizer 123 aren't necessarily related to each other.
For example:
seeds: 79 14 55 13
seed-to-soil map:
50 98 2
52 50 48
soil-to-fertilizer map:
0 15 37
37 52 2
39 0 15
fertilizer-to-water map:
49 53 8
0 11 42
42 0 7
57 7 4
water-to-light map:
88 18 7
18 25 70
light-to-temperature map:
45 77 23
81 45 19
68 64 13
temperature-to-humidity map:
0 69 1
1 0 69
humidity-to-location map:
60 56 37
56 93 4
The almanac starts by listing which seeds need to be planted: seeds 79, 14, 55, and 13.
The rest of the almanac contains a list of maps which describe how to convert numbers from a source category into numbers in a destination category. That is, the section that starts with seed-to-soil map: describes how to convert a seed number (the source) to a soil number (the destination). This lets the gardener and his team know which soil to use with which seeds, which water to use with which fertilizer, and so on.
Rather than list every source number and its corresponding destination number one by one, the maps describe entire ranges of numbers that can be converted. Each line within a map contains three numbers: the destination range start, the source range start, and the range length.
Consider again the example seed-to-soil map:
50 98 2
52 50 48
The first line has a destination range start of 50, a source range start of 98, and a range length of 2. This line means that the source range starts at 98 and contains two values: 98 and 99. The destination range is the same length, but it starts at 50, so its two values are 50 and 51. With this information, you know that seed number 98 corresponds to soil number 50 and that seed number 99 corresponds to soil number 51.
The second line means that the source range starts at 50 and contains 48 values: 50, 51, ..., 96, 97. This corresponds to a destination range starting at 52 and also containing 48 values: 52, 53, ..., 98, 99. So, seed number 53 corresponds to soil number 55.
Any source numbers that aren't mapped correspond to the same destination number. So, seed number 10 corresponds to soil number 10.
So, the entire list of seed numbers and their corresponding soil numbers looks like this:
seed soil
0 0
1 1
... ...
48 48
49 49
50 52
51 53
... ...
96 98
97 99
98 50
99 51
With this map, you can look up the soil number required for each initial seed number:
Seed number 79 corresponds to soil number 81.
Seed number 14 corresponds to soil number 14.
Seed number 55 corresponds to soil number 57.
Seed number 13 corresponds to soil number 13.
The gardener and his team want to get started as soon as possible, so they'd like to know the closest location that needs a seed. Using these maps, find the lowest location number that corresponds to any of the initial seeds. To do this, you'll need to convert each seed number through other categories until you can find its corresponding location number. In this example, the corresponding types are:
Seed 79, soil 81, fertilizer 81, water 81, light 74, temperature 78, humidity 78, location 82.
Seed 14, soil 14, fertilizer 53, water 49, light 42, temperature 42, humidity 43, location 43.
Seed 55, soil 57, fertilizer 57, water 53, light 46, temperature 82, humidity 82, location 86.
Seed 13, soil 13, fertilizer 52, water 41, light 34, temperature 34, humidity 35, location 35.
So, the lowest location number in this example is 35.
What is the lowest location number that corresponds to any of the initial seed numbers?
| 439
|
--- Day 13: Transparent Origami ---
You reach another volcanically active part of the cave. It would be nice if you could do some kind of thermal imaging so you could tell ahead of time which caves are too hot to safely enter.
Fortunately, the submarine seems to be equipped with a thermal camera! When you activate it, you are greeted with:
Congratulations on your purchase! To activate this infrared thermal imaging
camera system, please enter the code found on page 1 of the manual.
Apparently, the Elves have never used this feature. To your surprise, you manage to find the manual; as you go to open it, page 1 falls out. It's a large sheet of transparent paper! The transparent paper is marked with random dots and includes instructions on how to fold it up (your puzzle input). For example:
6,10
0,14
9,10
0,3
10,4
4,11
6,0
6,12
4,1
0,13
10,12
3,4
3,0
8,4
1,10
2,14
8,10
9,0
fold along y=7
fold along x=5
The first section is a list of dots on the transparent paper. 0,0 represents the top-left coordinate. The first value, x, increases to the right. The second value, y, increases downward. So, the coordinate 3,0 is to the right of 0,0, and the coordinate 0,7 is below 0,0. The coordinates in this example form the following pattern, where # is a dot on the paper and . is an empty, unmarked position:
...#..#..#.
....#......
...........
#..........
...#....#.#
...........
...........
...........
...........
...........
.#....#.##.
....#......
......#...#
#..........
#.#........
Then, there is a list of fold instructions. Each instruction indicates a line on the transparent paper and wants you to fold the paper up (for horizontal y=... lines) or left (for vertical x=... lines). In this example, the first fold instruction is fold along y=7, which designates the line formed by all of the positions where y is 7 (marked here with -):
...#..#..#.
....#......
...........
#..........
...#....#.#
...........
...........
-----------
...........
...........
.#....#.##.
....#......
......#...#
#..........
#.#........
Because this is a horizontal line, fold the bottom half up. Some of the dots might end up overlapping after the fold is complete, but dots will never appear exactly on a fold line. The result of doing this fold looks like this:
#.##..#..#.
#...#......
......#...#
#...#......
.#.#..#.###
...........
...........
Now, only 17 dots are visible.
Notice, for example, the two dots in the bottom left corner before the transparent paper is folded; after the fold is complete, those dots appear in the top left corner (at 0,0 and 0,1). Because the paper is transparent, the dot just below them in the result (at 0,3) remains visible, as it can be seen through the transparent paper.
Also notice that some dots can end up overlapping; in this case, the dots merge together and become a single dot.
The second fold instruction is fold along x=5, which indicates this line:
#.##.|#..#.
#...#|.....
.....|#...#
#...#|.....
.#.#.|#.###
.....|.....
.....|.....
Because this is a vertical line, fold left:
#####
#...#
#...#
#...#
#####
.....
.....
The instructions made a square!
The transparent paper is pretty big, so for now, focus on just completing the first fold. After the first fold in the example above, 17 dots are visible - dots that end up overlapping after the fold is completed count as a single dot.
How many dots are visible after completing just the first fold instruction on your transparent paper?
Your puzzle answer was 720.
--- Part Two ---
Finish folding the transparent paper according to the instructions. The manual says the code is always eight capital letters.
What code do you use to activate the infrared thermal imaging camera system?
| 440
|
--- Day 3: Binary Diagnostic ---
The submarine has been making some odd creaking noises, so you ask it to produce a diagnostic report just in case.
The diagnostic report (your puzzle input) consists of a list of binary numbers which, when decoded properly, can tell you many useful things about the conditions of the submarine. The first parameter to check is the power consumption.
You need to use the binary numbers in the diagnostic report to generate two new binary numbers (called the gamma rate and the epsilon rate). The power consumption can then be found by multiplying the gamma rate by the epsilon rate.
Each bit in the gamma rate can be determined by finding the most common bit in the corresponding position of all numbers in the diagnostic report. For example, given the following diagnostic report:
00100
11110
10110
10111
10101
01111
00111
11100
10000
11001
00010
01010
Considering only the first bit of each number, there are five 0 bits and seven 1 bits. Since the most common bit is 1, the first bit of the gamma rate is 1.
The most common second bit of the numbers in the diagnostic report is 0, so the second bit of the gamma rate is 0.
The most common value of the third, fourth, and fifth bits are 1, 1, and 0, respectively, and so the final three bits of the gamma rate are 110.
So, the gamma rate is the binary number 10110, or 22 in decimal.
The epsilon rate is calculated in a similar way; rather than use the most common bit, the least common bit from each position is used. So, the epsilon rate is 01001, or 9 in decimal. Multiplying the gamma rate (22) by the epsilon rate (9) produces the power consumption, 198.
Use the binary numbers in your diagnostic report to calculate the gamma rate and epsilon rate, then multiply them together. What is the power consumption of the submarine? (Be sure to represent your answer in decimal, not binary.)
| 441
|
--- Day 1: Sonar Sweep ---
You're minding your own business on a ship at sea when the overboard alarm goes off! You rush to see if you can help. Apparently, one of the Elves tripped and accidentally sent the sleigh keys flying into the ocean!
Before you know it, you're inside a submarine the Elves keep ready for situations like this. It's covered in Christmas lights (because of course it is), and it even has an experimental antenna that should be able to track the keys if you can boost its signal strength high enough; there's a little meter that indicates the antenna's signal strength by displaying 0-50 stars.
Your instincts tell you that in order to save Christmas, you'll need to get all fifty stars by December 25th.
Collect stars by solving puzzles. Two puzzles will be made available on each day in the Advent calendar; the second puzzle is unlocked when you complete the first. Each puzzle grants one star. Good luck!
As the submarine drops below the surface of the ocean, it automatically performs a sonar sweep of the nearby sea floor. On a small screen, the sonar sweep report (your puzzle input) appears: each line is a measurement of the sea floor depth as the sweep looks further and further away from the submarine.
For example, suppose you had the following report:
199
200
208
210
200
207
240
269
260
263
This report indicates that, scanning outward from the submarine, the sonar sweep found depths of 199, 200, 208, 210, and so on.
The first order of business is to figure out how quickly the depth increases, just so you know what you're dealing with - you never know if the keys will get carried into deeper water by an ocean current or a fish or something.
To do this, count the number of times a depth measurement increases from the previous measurement. (There is no measurement before the first measurement.) In the example above, the changes are as follows:
199 (N/A - no previous measurement)
200 (increased)
208 (increased)
210 (increased)
200 (decreased)
207 (increased)
240 (increased)
269 (increased)
260 (decreased)
263 (increased)
In this example, there are 7 measurements that are larger than the previous measurement.
How many measurements are larger than the previous measurement?
| 442
|
--- Day 23: Coprocessor Conflagration ---
You decide to head directly to the CPU and fix the printer from there. As you get close, you find an experimental coprocessor doing so much work that the local programs are afraid it will halt and catch fire. This would cause serious issues for the rest of the computer, so you head in and see what you can do.
The code it's running seems to be a variant of the kind you saw recently on that tablet. The general functionality seems very similar, but some of the instructions are different:
set X Y sets register X to the value of Y.
sub X Y decreases register X by the value of Y.
mul X Y sets register X to the result of multiplying the value contained in register X by the value of Y.
jnz X Y jumps with an offset of the value of Y, but only if the value of X is not zero. (An offset of 2 skips the next instruction, an offset of -1 jumps to the previous instruction, and so on.)
Only the instructions listed above are used. The eight registers here, named a through h, all start at 0.
The coprocessor is currently set to some kind of debug mode, which allows for testing, but prevents it from doing any meaningful work.
If you run the program (your puzzle input), how many times is the mul instruction invoked?
| 443
|
--- Day 3: No Matter How You Slice It ---
The Elves managed to locate the chimney-squeeze prototype fabric for Santa's suit (thanks to someone who helpfully wrote its box IDs on the wall of the warehouse in the middle of the night). Unfortunately, anomalies are still affecting them - nobody can even agree on how to cut the fabric.
The whole piece of fabric they're working on is a very large square - at least 1000 inches on each side.
Each Elf has made a claim about which area of fabric would be ideal for Santa's suit. All claims have an ID and consist of a single rectangle with edges parallel to the edges of the fabric. Each claim's rectangle is defined as follows:
The number of inches between the left edge of the fabric and the left edge of the rectangle.
The number of inches between the top edge of the fabric and the top edge of the rectangle.
The width of the rectangle in inches.
The height of the rectangle in inches.
A claim like #123 @ 3,2: 5x4 means that claim ID 123 specifies a rectangle 3 inches from the left edge, 2 inches from the top edge, 5 inches wide, and 4 inches tall. Visually, it claims the square inches of fabric represented by # (and ignores the square inches of fabric represented by .) in the diagram below:
...........
...........
...#####...
...#####...
...#####...
...#####...
...........
...........
...........
The problem is that many of the claims overlap, causing two or more claims to cover part of the same areas. For example, consider the following claims:
#1 @ 1,3: 4x4
#2 @ 3,1: 4x4
#3 @ 5,5: 2x2
Visually, these claim the following areas:
........
...2222.
...2222.
.11XX22.
.11XX22.
.111133.
.111133.
........
The four square inches marked with X are claimed by both 1 and 2. (Claim 3, while adjacent to the others, does not overlap either of them.)
If the Elves all proceed with their own plans, none of them will have enough fabric. How many square inches of fabric are within two or more claims?
| 444
|
--- Day 10: The Stars Align ---
It's no use; your navigation system simply isn't capable of providing walking directions in the arctic circle, and certainly not in 1018.
The Elves suggest an alternative. In times like these, North Pole rescue operations will arrange points of light in the sky to guide missing Elves back to base. Unfortunately, the message is easy to miss: the points move slowly enough that it takes hours to align them, but have so much momentum that they only stay aligned for a second. If you blink at the wrong time, it might be hours before another message appears.
You can see these points of light floating in the distance, and record their position in the sky and their velocity, the relative change in position per second (your puzzle input). The coordinates are all given from your perspective; given enough time, those positions and velocities will move the points into a cohesive message!
Rather than wait, you decide to fast-forward the process and calculate what the points will eventually spell.
For example, suppose you note the following points:
position=< 9, 1> velocity=< 0, 2>
position=< 7, 0> velocity=<-1, 0>
position=< 3, -2> velocity=<-1, 1>
position=< 6, 10> velocity=<-2, -1>
position=< 2, -4> velocity=< 2, 2>
position=<-6, 10> velocity=< 2, -2>
position=< 1, 8> velocity=< 1, -1>
position=< 1, 7> velocity=< 1, 0>
position=<-3, 11> velocity=< 1, -2>
position=< 7, 6> velocity=<-1, -1>
position=<-2, 3> velocity=< 1, 0>
position=<-4, 3> velocity=< 2, 0>
position=<10, -3> velocity=<-1, 1>
position=< 5, 11> velocity=< 1, -2>
position=< 4, 7> velocity=< 0, -1>
position=< 8, -2> velocity=< 0, 1>
position=<15, 0> velocity=<-2, 0>
position=< 1, 6> velocity=< 1, 0>
position=< 8, 9> velocity=< 0, -1>
position=< 3, 3> velocity=<-1, 1>
position=< 0, 5> velocity=< 0, -1>
position=<-2, 2> velocity=< 2, 0>
position=< 5, -2> velocity=< 1, 2>
position=< 1, 4> velocity=< 2, 1>
position=<-2, 7> velocity=< 2, -2>
position=< 3, 6> velocity=<-1, -1>
position=< 5, 0> velocity=< 1, 0>
position=<-6, 0> velocity=< 2, 0>
position=< 5, 9> velocity=< 1, -2>
position=<14, 7> velocity=<-2, 0>
position=<-3, 6> velocity=< 2, -1>
Each line represents one point. Positions are given as <X, Y> pairs: X represents how far left (negative) or right (positive) the point appears, while Y represents how far up (negative) or down (positive) the point appears.
At 0 seconds, each point has the position given. Each second, each point's velocity is added to its position. So, a point with velocity <1, -2> is moving to the right, but is moving upward twice as quickly. If this point's initial position were <3, 9>, after 3 seconds, its position would become <6, 3>.
Over time, the points listed above would move like this:
Initially:
........#.............
................#.....
.........#.#..#.......
......................
#..........#.#.......#
...............#......
....#.................
..#.#....#............
.......#..............
......#...............
...#...#.#...#........
....#..#..#.........#.
.......#..............
...........#..#.......
#...........#.........
...#.......#..........
After 1 second:
......................
......................
..........#....#......
........#.....#.......
..#.........#......#..
......................
......#...............
....##.........#......
......#.#.............
.....##.##..#.........
........#.#...........
........#...#.....#...
..#...........#.......
....#.....#.#.........
......................
......................
After 2 seconds:
......................
......................
......................
..............#.......
....#..#...####..#....
......................
........#....#........
......#.#.............
.......#...#..........
.......#..#..#.#......
....#....#.#..........
.....#...#...##.#.....
........#.............
......................
......................
......................
After 3 seconds:
......................
......................
......................
......................
......#...#..###......
......#...#...#.......
......#...#...#.......
......#####...#.......
......#...#...#.......
......#...#...#.......
......#...#...#.......
......#...#..###......
......................
......................
......................
......................
After 4 seconds:
......................
......................
......................
............#.........
........##...#.#......
......#.....#..#......
.....#..##.##.#.......
.......##.#....#......
...........#....#.....
..............#.......
....#......#...#......
.....#.....##.........
...............#......
...............#......
......................
......................
After 3 seconds, the message appeared briefly: HI. Of course, your message will be much longer and will take many more seconds to appear.
What message will eventually appear in the sky?
Your puzzle answer was HKJFAKAF.
--- Part Two ---
Good thing you didn't have to wait, because that would have taken a long time - much longer than the 3 seconds in the example above.
Impressed by your sub-hour communication capabilities, the Elves are curious: exactly how many seconds would they have needed to wait for that message to appear?
| 445
|
--- Day 6: Guard Gallivant ---
The Historians use their fancy device again, this time to whisk you all away to the North Pole prototype suit manufacturing lab... in the year 1518! It turns out that having direct access to history is very convenient for a group of historians.
You still have to be careful of time paradoxes, and so it will be important to avoid anyone from 1518 while The Historians search for the Chief. Unfortunately, a single guard is patrolling this part of the lab.
Maybe you can work out where the guard will go ahead of time so that The Historians can search safely?
You start by making a map (your puzzle input) of the situation. For example:
....#.....
.........#
..........
..#.......
.......#..
..........
.#..^.....
........#.
#.........
......#...
The map shows the current position of the guard with ^ (to indicate the guard is currently facing up from the perspective of the map). Any obstructions - crates, desks, alchemical reactors, etc. - are shown as #.
Lab guards in 1518 follow a very strict patrol protocol which involves repeatedly following these steps:
If there is something directly in front of you, turn right 90 degrees.
Otherwise, take a step forward.
Following the above protocol, the guard moves up several times until she reaches an obstacle (in this case, a pile of failed suit prototypes):
....#.....
....^....#
..........
..#.......
.......#..
..........
.#........
........#.
#.........
......#...
Because there is now an obstacle in front of the guard, she turns right before continuing straight in her new facing direction:
....#.....
........>#
..........
..#.......
.......#..
..........
.#........
........#.
#.........
......#...
Reaching another obstacle (a spool of several very long polymers), she turns right again and continues downward:
....#.....
.........#
..........
..#.......
.......#..
..........
.#......v.
........#.
#.........
......#...
This process continues for a while, but the guard eventually leaves the mapped area (after walking past a tank of universal solvent):
....#.....
.........#
..........
..#.......
.......#..
..........
.#........
........#.
#.........
......#v..
By predicting the guard's route, you can determine which specific positions in the lab will be in the patrol path. Including the guard's starting position, the positions visited by the guard before leaving the area are marked with an X:
....#.....
....XXXXX#
....X...X.
..#.X...X.
..XXXXX#X.
..X.X.X.X.
.#XXXXXXX.
.XXXXXXX#.
#XXXXXXX..
......#X..
In this example, the guard will visit 41 distinct positions on your map.
Predict the path of the guard. How many distinct positions will the guard visit before leaving the mapped area?
| 446
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--- Day 7: Camel Cards ---
Your all-expenses-paid trip turns out to be a one-way, five-minute ride in an airship. (At least it's a cool airship!) It drops you off at the edge of a vast desert and descends back to Island Island.
"Did you bring the parts?"
You turn around to see an Elf completely covered in white clothing, wearing goggles, and riding a large camel.
"Did you bring the parts?" she asks again, louder this time. You aren't sure what parts she's looking for; you're here to figure out why the sand stopped.
"The parts! For the sand, yes! Come with me; I will show you." She beckons you onto the camel.
After riding a bit across the sands of Desert Island, you can see what look like very large rocks covering half of the horizon. The Elf explains that the rocks are all along the part of Desert Island that is directly above Island Island, making it hard to even get there. Normally, they use big machines to move the rocks and filter the sand, but the machines have broken down because Desert Island recently stopped receiving the parts they need to fix the machines.
You've already assumed it'll be your job to figure out why the parts stopped when she asks if you can help. You agree automatically.
Because the journey will take a few days, she offers to teach you the game of Camel Cards. Camel Cards is sort of similar to poker except it's designed to be easier to play while riding a camel.
In Camel Cards, you get a list of hands, and your goal is to order them based on the strength of each hand. A hand consists of five cards labeled one of A, K, Q, J, T, 9, 8, 7, 6, 5, 4, 3, or 2. The relative strength of each card follows this order, where A is the highest and 2 is the lowest.
Every hand is exactly one type. From strongest to weakest, they are:
Five of a kind, where all five cards have the same label: AAAAA
Four of a kind, where four cards have the same label and one card has a different label: AA8AA
Full house, where three cards have the same label, and the remaining two cards share a different label: 23332
Three of a kind, where three cards have the same label, and the remaining two cards are each different from any other card in the hand: TTT98
Two pair, where two cards share one label, two other cards share a second label, and the remaining card has a third label: 23432
One pair, where two cards share one label, and the other three cards have a different label from the pair and each other: A23A4
High card, where all cards' labels are distinct: 23456
Hands are primarily ordered based on type; for example, every full house is stronger than any three of a kind.
If two hands have the same type, a second ordering rule takes effect. Start by comparing the first card in each hand. If these cards are different, the hand with the stronger first card is considered stronger. If the first card in each hand have the same label, however, then move on to considering the second card in each hand. If they differ, the hand with the higher second card wins; otherwise, continue with the third card in each hand, then the fourth, then the fifth.
So, 33332 and 2AAAA are both four of a kind hands, but 33332 is stronger because its first card is stronger. Similarly, 77888 and 77788 are both a full house, but 77888 is stronger because its third card is stronger (and both hands have the same first and second card).
To play Camel Cards, you are given a list of hands and their corresponding bid (your puzzle input). For example:
32T3K 765
T55J5 684
KK677 28
KTJJT 220
QQQJA 483
This example shows five hands; each hand is followed by its bid amount. Each hand wins an amount equal to its bid multiplied by its rank, where the weakest hand gets rank 1, the second-weakest hand gets rank 2, and so on up to the strongest hand. Because there are five hands in this example, the strongest hand will have rank 5 and its bid will be multiplied by 5.
So, the first step is to put the hands in order of strength:
32T3K is the only one pair and the other hands are all a stronger type, so it gets rank 1.
KK677 and KTJJT are both two pair. Their first cards both have the same label, but the second card of KK677 is stronger (K vs T), so KTJJT gets rank 2 and KK677 gets rank 3.
T55J5 and QQQJA are both three of a kind. QQQJA has a stronger first card, so it gets rank 5 and T55J5 gets rank 4.
Now, you can determine the total winnings of this set of hands by adding up the result of multiplying each hand's bid with its rank (765 * 1 + 220 * 2 + 28 * 3 + 684 * 4 + 483 * 5). So the total winnings in this example are 6440.
Find the rank of every hand in your set. What are the total winnings?
Your puzzle answer was 250957639.
--- Part Two ---
To make things a little more interesting, the Elf introduces one additional rule. Now, J cards are jokers - wildcards that can act like whatever card would make the hand the strongest type possible.
To balance this, J cards are now the weakest individual cards, weaker even than 2. The other cards stay in the same order: A, K, Q, T, 9, 8, 7, 6, 5, 4, 3, 2, J.
J cards can pretend to be whatever card is best for the purpose of determining hand type; for example, QJJQ2 is now considered four of a kind. However, for the purpose of breaking ties between two hands of the same type, J is always treated as J, not the card it's pretending to be: JKKK2 is weaker than QQQQ2 because J is weaker than Q.
Now, the above example goes very differently:
32T3K 765
T55J5 684
KK677 28
KTJJT 220
QQQJA 483
32T3K is still the only one pair; it doesn't contain any jokers, so its strength doesn't increase.
KK677 is now the only two pair, making it the second-weakest hand.
T55J5, KTJJT, and QQQJA are now all four of a kind! T55J5 gets rank 3, QQQJA gets rank 4, and KTJJT gets rank 5.
With the new joker rule, the total winnings in this example are 5905.
Using the new joker rule, find the rank of every hand in your set. What are the new total winnings?
| 447
|
--- Day 7: The Treachery of Whales ---
A giant whale has decided your submarine is its next meal, and it's much faster than you are. There's nowhere to run!
Suddenly, a swarm of crabs (each in its own tiny submarine - it's too deep for them otherwise) zooms in to rescue you! They seem to be preparing to blast a hole in the ocean floor; sensors indicate a massive underground cave system just beyond where they're aiming!
The crab submarines all need to be aligned before they'll have enough power to blast a large enough hole for your submarine to get through. However, it doesn't look like they'll be aligned before the whale catches you! Maybe you can help?
There's one major catch - crab submarines can only move horizontally.
You quickly make a list of the horizontal position of each crab (your puzzle input). Crab submarines have limited fuel, so you need to find a way to make all of their horizontal positions match while requiring them to spend as little fuel as possible.
For example, consider the following horizontal positions:
16,1,2,0,4,2,7,1,2,14
This means there's a crab with horizontal position 16, a crab with horizontal position 1, and so on.
Each change of 1 step in horizontal position of a single crab costs 1 fuel. You could choose any horizontal position to align them all on, but the one that costs the least fuel is horizontal position 2:
Move from 16 to 2: 14 fuel
Move from 1 to 2: 1 fuel
Move from 2 to 2: 0 fuel
Move from 0 to 2: 2 fuel
Move from 4 to 2: 2 fuel
Move from 2 to 2: 0 fuel
Move from 7 to 2: 5 fuel
Move from 1 to 2: 1 fuel
Move from 2 to 2: 0 fuel
Move from 14 to 2: 12 fuel
This costs a total of 37 fuel. This is the cheapest possible outcome; more expensive outcomes include aligning at position 1 (41 fuel), position 3 (39 fuel), or position 10 (71 fuel).
Determine the horizontal position that the crabs can align to using the least fuel possible. How much fuel must they spend to align to that position?
| 448
|
--- Day 22: Crab Combat ---
It only takes a few hours of sailing the ocean on a raft for boredom to sink in. Fortunately, you brought a small deck of space cards! You'd like to play a game of Combat, and there's even an opponent available: a small crab that climbed aboard your raft before you left.
Fortunately, it doesn't take long to teach the crab the rules.
Before the game starts, split the cards so each player has their own deck (your puzzle input). Then, the game consists of a series of rounds: both players draw their top card, and the player with the higher-valued card wins the round. The winner keeps both cards, placing them on the bottom of their own deck so that the winner's card is above the other card. If this causes a player to have all of the cards, they win, and the game ends.
For example, consider the following starting decks:
Player 1:
9
2
6
3
1
Player 2:
5
8
4
7
10
This arrangement means that player 1's deck contains 5 cards, with 9 on top and 1 on the bottom; player 2's deck also contains 5 cards, with 5 on top and 10 on the bottom.
The first round begins with both players drawing the top card of their decks: 9 and 5. Player 1 has the higher card, so both cards move to the bottom of player 1's deck such that 9 is above 5. In total, it takes 29 rounds before a player has all of the cards:
-- Round 1 --
Player 1's deck: 9, 2, 6, 3, 1
Player 2's deck: 5, 8, 4, 7, 10
Player 1 plays: 9
Player 2 plays: 5
Player 1 wins the round!
-- Round 2 --
Player 1's deck: 2, 6, 3, 1, 9, 5
Player 2's deck: 8, 4, 7, 10
Player 1 plays: 2
Player 2 plays: 8
Player 2 wins the round!
-- Round 3 --
Player 1's deck: 6, 3, 1, 9, 5
Player 2's deck: 4, 7, 10, 8, 2
Player 1 plays: 6
Player 2 plays: 4
Player 1 wins the round!
-- Round 4 --
Player 1's deck: 3, 1, 9, 5, 6, 4
Player 2's deck: 7, 10, 8, 2
Player 1 plays: 3
Player 2 plays: 7
Player 2 wins the round!
-- Round 5 --
Player 1's deck: 1, 9, 5, 6, 4
Player 2's deck: 10, 8, 2, 7, 3
Player 1 plays: 1
Player 2 plays: 10
Player 2 wins the round!
...several more rounds pass...
-- Round 27 --
Player 1's deck: 5, 4, 1
Player 2's deck: 8, 9, 7, 3, 2, 10, 6
Player 1 plays: 5
Player 2 plays: 8
Player 2 wins the round!
-- Round 28 --
Player 1's deck: 4, 1
Player 2's deck: 9, 7, 3, 2, 10, 6, 8, 5
Player 1 plays: 4
Player 2 plays: 9
Player 2 wins the round!
-- Round 29 --
Player 1's deck: 1
Player 2's deck: 7, 3, 2, 10, 6, 8, 5, 9, 4
Player 1 plays: 1
Player 2 plays: 7
Player 2 wins the round!
== Post-game results ==
Player 1's deck:
Player 2's deck: 3, 2, 10, 6, 8, 5, 9, 4, 7, 1
Once the game ends, you can calculate the winning player's score. The bottom card in their deck is worth the value of the card multiplied by 1, the second-from-the-bottom card is worth the value of the card multiplied by 2, and so on. With 10 cards, the top card is worth the value on the card multiplied by 10. In this example, the winning player's score is:
3 * 10
+ 2 * 9
+ 10 * 8
+ 6 * 7
+ 8 * 6
+ 5 * 5
+ 9 * 4
+ 4 * 3
+ 7 * 2
+ 1 * 1
= 306
So, once the game ends, the winning player's score is 306.
Play the small crab in a game of Combat using the two decks you just dealt. What is the winning player's score?
Your puzzle answer was 31269.
--- Part Two ---
You lost to the small crab! Fortunately, crabs aren't very good at recursion. To defend your honor as a Raft Captain, you challenge the small crab to a game of Recursive Combat.
Recursive Combat still starts by splitting the cards into two decks (you offer to play with the same starting decks as before - it's only fair). Then, the game consists of a series of rounds with a few changes:
Before either player deals a card, if there was a previous round in this game that had exactly the same cards in the same order in the same players' decks, the game instantly ends in a win for player 1. Previous rounds from other games are not considered. (This prevents infinite games of Recursive Combat, which everyone agrees is a bad idea.)
Otherwise, this round's cards must be in a new configuration; the players begin the round by each drawing the top card of their deck as normal.
If both players have at least as many cards remaining in their deck as the value of the card they just drew, the winner of the round is determined by playing a new game of Recursive Combat (see below).
Otherwise, at least one player must not have enough cards left in their deck to recurse; the winner of the round is the player with the higher-value card.
As in regular Combat, the winner of the round (even if they won the round by winning a sub-game) takes the two cards dealt at the beginning of the round and places them on the bottom of their own deck (again so that the winner's card is above the other card). Note that the winner's card might be the lower-valued of the two cards if they won the round due to winning a sub-game. If collecting cards by winning the round causes a player to have all of the cards, they win, and the game ends.
Here is an example of a small game that would loop forever without the infinite game prevention rule:
Player 1:
43
19
Player 2:
2
29
14
During a round of Recursive Combat, if both players have at least as many cards in their own decks as the number on the card they just dealt, the winner of the round is determined by recursing into a sub-game of Recursive Combat. (For example, if player 1 draws the 3 card, and player 2 draws the 7 card, this would occur if player 1 has at least 3 cards left and player 2 has at least 7 cards left, not counting the 3 and 7 cards that were drawn.)
To play a sub-game of Recursive Combat, each player creates a new deck by making a copy of the next cards in their deck (the quantity of cards copied is equal to the number on the card they drew to trigger the sub-game). During this sub-game, the game that triggered it is on hold and completely unaffected; no cards are removed from players' decks to form the sub-game. (For example, if player 1 drew the 3 card, their deck in the sub-game would be copies of the next three cards in their deck.)
Here is a complete example of gameplay, where Game 1 is the primary game of Recursive Combat:
=== Game 1 ===
-- Round 1 (Game 1) --
Player 1's deck: 9, 2, 6, 3, 1
Player 2's deck: 5, 8, 4, 7, 10
Player 1 plays: 9
Player 2 plays: 5
Player 1 wins round 1 of game 1!
-- Round 2 (Game 1) --
Player 1's deck: 2, 6, 3, 1, 9, 5
Player 2's deck: 8, 4, 7, 10
Player 1 plays: 2
Player 2 plays: 8
Player 2 wins round 2 of game 1!
-- Round 3 (Game 1) --
Player 1's deck: 6, 3, 1, 9, 5
Player 2's deck: 4, 7, 10, 8, 2
Player 1 plays: 6
Player 2 plays: 4
Player 1 wins round 3 of game 1!
-- Round 4 (Game 1) --
Player 1's deck: 3, 1, 9, 5, 6, 4
Player 2's deck: 7, 10, 8, 2
Player 1 plays: 3
Player 2 plays: 7
Player 2 wins round 4 of game 1!
-- Round 5 (Game 1) --
Player 1's deck: 1, 9, 5, 6, 4
Player 2's deck: 10, 8, 2, 7, 3
Player 1 plays: 1
Player 2 plays: 10
Player 2 wins round 5 of game 1!
-- Round 6 (Game 1) --
Player 1's deck: 9, 5, 6, 4
Player 2's deck: 8, 2, 7, 3, 10, 1
Player 1 plays: 9
Player 2 plays: 8
Player 1 wins round 6 of game 1!
-- Round 7 (Game 1) --
Player 1's deck: 5, 6, 4, 9, 8
Player 2's deck: 2, 7, 3, 10, 1
Player 1 plays: 5
Player 2 plays: 2
Player 1 wins round 7 of game 1!
-- Round 8 (Game 1) --
Player 1's deck: 6, 4, 9, 8, 5, 2
Player 2's deck: 7, 3, 10, 1
Player 1 plays: 6
Player 2 plays: 7
Player 2 wins round 8 of game 1!
-- Round 9 (Game 1) --
Player 1's deck: 4, 9, 8, 5, 2
Player 2's deck: 3, 10, 1, 7, 6
Player 1 plays: 4
Player 2 plays: 3
Playing a sub-game to determine the winner...
=== Game 2 ===
-- Round 1 (Game 2) --
Player 1's deck: 9, 8, 5, 2
Player 2's deck: 10, 1, 7
Player 1 plays: 9
Player 2 plays: 10
Player 2 wins round 1 of game 2!
-- Round 2 (Game 2) --
Player 1's deck: 8, 5, 2
Player 2's deck: 1, 7, 10, 9
Player 1 plays: 8
Player 2 plays: 1
Player 1 wins round 2 of game 2!
-- Round 3 (Game 2) --
Player 1's deck: 5, 2, 8, 1
Player 2's deck: 7, 10, 9
Player 1 plays: 5
Player 2 plays: 7
Player 2 wins round 3 of game 2!
-- Round 4 (Game 2) --
Player 1's deck: 2, 8, 1
Player 2's deck: 10, 9, 7, 5
Player 1 plays: 2
Player 2 plays: 10
Player 2 wins round 4 of game 2!
-- Round 5 (Game 2) --
Player 1's deck: 8, 1
Player 2's deck: 9, 7, 5, 10, 2
Player 1 plays: 8
Player 2 plays: 9
Player 2 wins round 5 of game 2!
-- Round 6 (Game 2) --
Player 1's deck: 1
Player 2's deck: 7, 5, 10, 2, 9, 8
Player 1 plays: 1
Player 2 plays: 7
Player 2 wins round 6 of game 2!
The winner of game 2 is player 2!
...anyway, back to game 1.
Player 2 wins round 9 of game 1!
-- Round 10 (Game 1) --
Player 1's deck: 9, 8, 5, 2
Player 2's deck: 10, 1, 7, 6, 3, 4
Player 1 plays: 9
Player 2 plays: 10
Player 2 wins round 10 of game 1!
-- Round 11 (Game 1) --
Player 1's deck: 8, 5, 2
Player 2's deck: 1, 7, 6, 3, 4, 10, 9
Player 1 plays: 8
Player 2 plays: 1
Player 1 wins round 11 of game 1!
-- Round 12 (Game 1) --
Player 1's deck: 5, 2, 8, 1
Player 2's deck: 7, 6, 3, 4, 10, 9
Player 1 plays: 5
Player 2 plays: 7
Player 2 wins round 12 of game 1!
-- Round 13 (Game 1) --
Player 1's deck: 2, 8, 1
Player 2's deck: 6, 3, 4, 10, 9, 7, 5
Player 1 plays: 2
Player 2 plays: 6
Playing a sub-game to determine the winner...
=== Game 3 ===
-- Round 1 (Game 3) --
Player 1's deck: 8, 1
Player 2's deck: 3, 4, 10, 9, 7, 5
Player 1 plays: 8
Player 2 plays: 3
Player 1 wins round 1 of game 3!
-- Round 2 (Game 3) --
Player 1's deck: 1, 8, 3
Player 2's deck: 4, 10, 9, 7, 5
Player 1 plays: 1
Player 2 plays: 4
Playing a sub-game to determine the winner...
=== Game 4 ===
-- Round 1 (Game 4) --
Player 1's deck: 8
Player 2's deck: 10, 9, 7, 5
Player 1 plays: 8
Player 2 plays: 10
Player 2 wins round 1 of game 4!
The winner of game 4 is player 2!
...anyway, back to game 3.
Player 2 wins round 2 of game 3!
-- Round 3 (Game 3) --
Player 1's deck: 8, 3
Player 2's deck: 10, 9, 7, 5, 4, 1
Player 1 plays: 8
Player 2 plays: 10
Player 2 wins round 3 of game 3!
-- Round 4 (Game 3) --
Player 1's deck: 3
Player 2's deck: 9, 7, 5, 4, 1, 10, 8
Player 1 plays: 3
Player 2 plays: 9
Player 2 wins round 4 of game 3!
The winner of game 3 is player 2!
...anyway, back to game 1.
Player 2 wins round 13 of game 1!
-- Round 14 (Game 1) --
Player 1's deck: 8, 1
Player 2's deck: 3, 4, 10, 9, 7, 5, 6, 2
Player 1 plays: 8
Player 2 plays: 3
Player 1 wins round 14 of game 1!
-- Round 15 (Game 1) --
Player 1's deck: 1, 8, 3
Player 2's deck: 4, 10, 9, 7, 5, 6, 2
Player 1 plays: 1
Player 2 plays: 4
Playing a sub-game to determine the winner...
=== Game 5 ===
-- Round 1 (Game 5) --
Player 1's deck: 8
Player 2's deck: 10, 9, 7, 5
Player 1 plays: 8
Player 2 plays: 10
Player 2 wins round 1 of game 5!
The winner of game 5 is player 2!
...anyway, back to game 1.
Player 2 wins round 15 of game 1!
-- Round 16 (Game 1) --
Player 1's deck: 8, 3
Player 2's deck: 10, 9, 7, 5, 6, 2, 4, 1
Player 1 plays: 8
Player 2 plays: 10
Player 2 wins round 16 of game 1!
-- Round 17 (Game 1) --
Player 1's deck: 3
Player 2's deck: 9, 7, 5, 6, 2, 4, 1, 10, 8
Player 1 plays: 3
Player 2 plays: 9
Player 2 wins round 17 of game 1!
The winner of game 1 is player 2!
== Post-game results ==
Player 1's deck:
Player 2's deck: 7, 5, 6, 2, 4, 1, 10, 8, 9, 3
After the game, the winning player's score is calculated from the cards they have in their original deck using the same rules as regular Combat. In the above game, the winning player's score is 291.
Defend your honor as Raft Captain by playing the small crab in a game of Recursive Combat using the same two decks as before. What is the winning player's score?
| 449
|
--- Day 3: Gear Ratios ---
You and the Elf eventually reach a gondola lift station; he says the gondola lift will take you up to the water source, but this is as far as he can bring you. You go inside.
It doesn't take long to find the gondolas, but there seems to be a problem: they're not moving.
"Aaah!"
You turn around to see a slightly-greasy Elf with a wrench and a look of surprise. "Sorry, I wasn't expecting anyone! The gondola lift isn't working right now; it'll still be a while before I can fix it." You offer to help.
The engineer explains that an engine part seems to be missing from the engine, but nobody can figure out which one. If you can add up all the part numbers in the engine schematic, it should be easy to work out which part is missing.
The engine schematic (your puzzle input) consists of a visual representation of the engine. There are lots of numbers and symbols you don't really understand, but apparently any number adjacent to a symbol, even diagonally, is a "part number" and should be included in your sum. (Periods (.) do not count as a symbol.)
Here is an example engine schematic:
467..114..
...*......
..35..633.
......#...
617*......
.....+.58.
..592.....
......755.
...$.*....
.664.598..
In this schematic, two numbers are not part numbers because they are not adjacent to a symbol: 114 (top right) and 58 (middle right). Every other number is adjacent to a symbol and so is a part number; their sum is 4361.
Of course, the actual engine schematic is much larger. What is the sum of all of the part numbers in the engine schematic?
| 450
|
--- Day 2: Dive! ---
Now, you need to figure out how to pilot this thing.
It seems like the submarine can take a series of commands like forward 1, down 2, or up 3:
forward X increases the horizontal position by X units.
down X increases the depth by X units.
up X decreases the depth by X units.
Note that since you're on a submarine, down and up affect your depth, and so they have the opposite result of what you might expect.
The submarine seems to already have a planned course (your puzzle input). You should probably figure out where it's going. For example:
forward 5
down 5
forward 8
up 3
down 8
forward 2
Your horizontal position and depth both start at 0. The steps above would then modify them as follows:
forward 5 adds 5 to your horizontal position, a total of 5.
down 5 adds 5 to your depth, resulting in a value of 5.
forward 8 adds 8 to your horizontal position, a total of 13.
up 3 decreases your depth by 3, resulting in a value of 2.
down 8 adds 8 to your depth, resulting in a value of 10.
forward 2 adds 2 to your horizontal position, a total of 15.
After following these instructions, you would have a horizontal position of 15 and a depth of 10. (Multiplying these together produces 150.)
Calculate the horizontal position and depth you would have after following the planned course. What do you get if you multiply your final horizontal position by your final depth?
| 451
|
--- Day 4: Security Through Obscurity ---
Finally, you come across an information kiosk with a list of rooms. Of course, the list is encrypted and full of decoy data, but the instructions to decode the list are barely hidden nearby. Better remove the decoy data first.
Each room consists of an encrypted name (lowercase letters separated by dashes) followed by a dash, a sector ID, and a checksum in square brackets.
A room is real (not a decoy) if the checksum is the five most common letters in the encrypted name, in order, with ties broken by alphabetization. For example:
aaaaa-bbb-z-y-x-123[abxyz] is a real room because the most common letters are a (5), b (3), and then a tie between x, y, and z, which are listed alphabetically.
a-b-c-d-e-f-g-h-987[abcde] is a real room because although the letters are all tied (1 of each), the first five are listed alphabetically.
not-a-real-room-404[oarel] is a real room.
totally-real-room-200[decoy] is not.
Of the real rooms from the list above, the sum of their sector IDs is 1514.
What is the sum of the sector IDs of the real rooms?
Your puzzle answer was 173787.
--- Part Two ---
With all the decoy data out of the way, it's time to decrypt this list and get moving.
The room names are encrypted by a state-of-the-art shift cipher, which is nearly unbreakable without the right software. However, the information kiosk designers at Easter Bunny HQ were not expecting to deal with a master cryptographer like yourself.
To decrypt a room name, rotate each letter forward through the alphabet a number of times equal to the room's sector ID. A becomes B, B becomes C, Z becomes A, and so on. Dashes become spaces.
For example, the real name for qzmt-zixmtkozy-ivhz-343 is very encrypted name.
What is the sector ID of the room where North Pole objects are stored?
| 452
|
--- Day 16: Chronal Classification ---
As you see the Elves defend their hot chocolate successfully, you go back to falling through time. This is going to become a problem.
If you're ever going to return to your own time, you need to understand how this device on your wrist works. You have a little while before you reach your next destination, and with a bit of trial and error, you manage to pull up a programming manual on the device's tiny screen.
According to the manual, the device has four registers (numbered 0 through 3) that can be manipulated by instructions containing one of 16 opcodes. The registers start with the value 0.
Every instruction consists of four values: an opcode, two inputs (named A and B), and an output (named C), in that order. The opcode specifies the behavior of the instruction and how the inputs are interpreted. The output, C, is always treated as a register.
In the opcode descriptions below, if something says "value A", it means to take the number given as A literally. (This is also called an "immediate" value.) If something says "register A", it means to use the number given as A to read from (or write to) the register with that number. So, if the opcode addi adds register A and value B, storing the result in register C, and the instruction addi 0 7 3 is encountered, it would add 7 to the value contained by register 0 and store the sum in register 3, never modifying registers 0, 1, or 2 in the process.
Many opcodes are similar except for how they interpret their arguments. The opcodes fall into seven general categories:
Addition:
addr (add register) stores into register C the result of adding register A and register B.
addi (add immediate) stores into register C the result of adding register A and value B.
Multiplication:
mulr (multiply register) stores into register C the result of multiplying register A and register B.
muli (multiply immediate) stores into register C the result of multiplying register A and value B.
Bitwise AND:
banr (bitwise AND register) stores into register C the result of the bitwise AND of register A and register B.
bani (bitwise AND immediate) stores into register C the result of the bitwise AND of register A and value B.
Bitwise OR:
borr (bitwise OR register) stores into register C the result of the bitwise OR of register A and register B.
bori (bitwise OR immediate) stores into register C the result of the bitwise OR of register A and value B.
Assignment:
setr (set register) copies the contents of register A into register C. (Input B is ignored.)
seti (set immediate) stores value A into register C. (Input B is ignored.)
Greater-than testing:
gtir (greater-than immediate/register) sets register C to 1 if value A is greater than register B. Otherwise, register C is set to 0.
gtri (greater-than register/immediate) sets register C to 1 if register A is greater than value B. Otherwise, register C is set to 0.
gtrr (greater-than register/register) sets register C to 1 if register A is greater than register B. Otherwise, register C is set to 0.
Equality testing:
eqir (equal immediate/register) sets register C to 1 if value A is equal to register B. Otherwise, register C is set to 0.
eqri (equal register/immediate) sets register C to 1 if register A is equal to value B. Otherwise, register C is set to 0.
eqrr (equal register/register) sets register C to 1 if register A is equal to register B. Otherwise, register C is set to 0.
Unfortunately, while the manual gives the name of each opcode, it doesn't seem to indicate the number. However, you can monitor the CPU to see the contents of the registers before and after instructions are executed to try to work them out. Each opcode has a number from 0 through 15, but the manual doesn't say which is which. For example, suppose you capture the following sample:
Before: [3, 2, 1, 1]
9 2 1 2
After: [3, 2, 2, 1]
This sample shows the effect of the instruction 9 2 1 2 on the registers. Before the instruction is executed, register 0 has value 3, register 1 has value 2, and registers 2 and 3 have value 1. After the instruction is executed, register 2's value becomes 2.
The instruction itself, 9 2 1 2, means that opcode 9 was executed with A=2, B=1, and C=2. Opcode 9 could be any of the 16 opcodes listed above, but only three of them behave in a way that would cause the result shown in the sample:
Opcode 9 could be mulr: register 2 (which has a value of 1) times register 1 (which has a value of 2) produces 2, which matches the value stored in the output register, register 2.
Opcode 9 could be addi: register 2 (which has a value of 1) plus value 1 produces 2, which matches the value stored in the output register, register 2.
Opcode 9 could be seti: value 2 matches the value stored in the output register, register 2; the number given for B is irrelevant.
None of the other opcodes produce the result captured in the sample. Because of this, the sample above behaves like three opcodes.
You collect many of these samples (the first section of your puzzle input). The manual also includes a small test program (the second section of your puzzle input) - you can ignore it for now.
Ignoring the opcode numbers, how many samples in your puzzle input behave like three or more opcodes?
Your puzzle answer was 642.
--- Part Two ---
Using the samples you collected, work out the number of each opcode and execute the test program (the second section of your puzzle input).
What value is contained in register 0 after executing the test program?
| 453
|
--- Day 19: A Series of Tubes ---
Somehow, a network packet got lost and ended up here. It's trying to follow a routing diagram (your puzzle input), but it's confused about where to go.
Its starting point is just off the top of the diagram. Lines (drawn with |, -, and +) show the path it needs to take, starting by going down onto the only line connected to the top of the diagram. It needs to follow this path until it reaches the end (located somewhere within the diagram) and stop there.
Sometimes, the lines cross over each other; in these cases, it needs to continue going the same direction, and only turn left or right when there's no other option. In addition, someone has left letters on the line; these also don't change its direction, but it can use them to keep track of where it's been. For example:
|
| +--+
A | C
F---|----E|--+
| | | D
+B-+ +--+
Given this diagram, the packet needs to take the following path:
Starting at the only line touching the top of the diagram, it must go down, pass through A, and continue onward to the first +.
Travel right, up, and right, passing through B in the process.
Continue down (collecting C), right, and up (collecting D).
Finally, go all the way left through E and stopping at F.
Following the path to the end, the letters it sees on its path are ABCDEF.
The little packet looks up at you, hoping you can help it find the way. What letters will it see (in the order it would see them) if it follows the path? (The routing diagram is very wide; make sure you view it without line wrapping.)
| 454
|
--- Day 14: Regolith Reservoir ---
The distress signal leads you to a giant waterfall! Actually, hang on - the signal seems like it's coming from the waterfall itself, and that doesn't make any sense. However, you do notice a little path that leads behind the waterfall.
Correction: the distress signal leads you behind a giant waterfall! There seems to be a large cave system here, and the signal definitely leads further inside.
As you begin to make your way deeper underground, you feel the ground rumble for a moment. Sand begins pouring into the cave! If you don't quickly figure out where the sand is going, you could quickly become trapped!
Fortunately, your familiarity with analyzing the path of falling material will come in handy here. You scan a two-dimensional vertical slice of the cave above you (your puzzle input) and discover that it is mostly air with structures made of rock.
Your scan traces the path of each solid rock structure and reports the x,y coordinates that form the shape of the path, where x represents distance to the right and y represents distance down. Each path appears as a single line of text in your scan. After the first point of each path, each point indicates the end of a straight horizontal or vertical line to be drawn from the previous point. For example:
498,4 -> 498,6 -> 496,6
503,4 -> 502,4 -> 502,9 -> 494,9
This scan means that there are two paths of rock; the first path consists of two straight lines, and the second path consists of three straight lines. (Specifically, the first path consists of a line of rock from 498,4 through 498,6 and another line of rock from 498,6 through 496,6.)
The sand is pouring into the cave from point 500,0.
Drawing rock as #, air as ., and the source of the sand as +, this becomes:
4 5 5
9 0 0
4 0 3
0 ......+...
1 ..........
2 ..........
3 ..........
4 ....#...##
5 ....#...#.
6 ..###...#.
7 ........#.
8 ........#.
9 #########.
Sand is produced one unit at a time, and the next unit of sand is not produced until the previous unit of sand comes to rest. A unit of sand is large enough to fill one tile of air in your scan.
A unit of sand always falls down one step if possible. If the tile immediately below is blocked (by rock or sand), the unit of sand attempts to instead move diagonally one step down and to the left. If that tile is blocked, the unit of sand attempts to instead move diagonally one step down and to the right. Sand keeps moving as long as it is able to do so, at each step trying to move down, then down-left, then down-right. If all three possible destinations are blocked, the unit of sand comes to rest and no longer moves, at which point the next unit of sand is created back at the source.
So, drawing sand that has come to rest as o, the first unit of sand simply falls straight down and then stops:
......+...
..........
..........
..........
....#...##
....#...#.
..###...#.
........#.
......o.#.
#########.
The second unit of sand then falls straight down, lands on the first one, and then comes to rest to its left:
......+...
..........
..........
..........
....#...##
....#...#.
..###...#.
........#.
.....oo.#.
#########.
After a total of five units of sand have come to rest, they form this pattern:
......+...
..........
..........
..........
....#...##
....#...#.
..###...#.
......o.#.
....oooo#.
#########.
After a total of 22 units of sand:
......+...
..........
......o...
.....ooo..
....#ooo##
....#ooo#.
..###ooo#.
....oooo#.
...ooooo#.
#########.
Finally, only two more units of sand can possibly come to rest:
......+...
..........
......o...
.....ooo..
....#ooo##
...o#ooo#.
..###ooo#.
....oooo#.
.o.ooooo#.
#########.
Once all 24 units of sand shown above have come to rest, all further sand flows out the bottom, falling into the endless void. Just for fun, the path any new sand takes before falling forever is shown here with ~:
.......+...
.......~...
......~o...
.....~ooo..
....~#ooo##
...~o#ooo#.
..~###ooo#.
..~..oooo#.
.~o.ooooo#.
~#########.
~..........
~..........
~..........
Using your scan, simulate the falling sand. How many units of sand come to rest before sand starts flowing into the abyss below?
Your puzzle answer was 696.
--- Part Two ---
You realize you misread the scan. There isn't an endless void at the bottom of the scan - there's floor, and you're standing on it!
You don't have time to scan the floor, so assume the floor is an infinite horizontal line with a y coordinate equal to two plus the highest y coordinate of any point in your scan.
In the example above, the highest y coordinate of any point is 9, and so the floor is at y=11. (This is as if your scan contained one extra rock path like -infinity,11 -> infinity,11.) With the added floor, the example above now looks like this:
...........+........
....................
....................
....................
.........#...##.....
.........#...#......
.......###...#......
.............#......
.............#......
.....#########......
....................
<-- etc #################### etc -->
To find somewhere safe to stand, you'll need to simulate falling sand until a unit of sand comes to rest at 500,0, blocking the source entirely and stopping the flow of sand into the cave. In the example above, the situation finally looks like this after 93 units of sand come to rest:
............o............
...........ooo...........
..........ooooo..........
.........ooooooo.........
........oo#ooo##o........
.......ooo#ooo#ooo.......
......oo###ooo#oooo......
.....oooo.oooo#ooooo.....
....oooooooooo#oooooo....
...ooo#########ooooooo...
..ooooo.......ooooooooo..
#########################
Using your scan, simulate the falling sand until the source of the sand becomes blocked. How many units of sand come to rest?
| 455
|
--- Day 11: Chronal Charge ---
You watch the Elves and their sleigh fade into the distance as they head toward the North Pole.
Actually, you're the one fading. The falling sensation returns.
The low fuel warning light is illuminated on your wrist-mounted device. Tapping it once causes it to project a hologram of the situation: a 300x300 grid of fuel cells and their current power levels, some negative. You're not sure what negative power means in the context of time travel, but it can't be good.
Each fuel cell has a coordinate ranging from 1 to 300 in both the X (horizontal) and Y (vertical) direction. In X,Y notation, the top-left cell is 1,1, and the top-right cell is 300,1.
The interface lets you select any 3x3 square of fuel cells. To increase your chances of getting to your destination, you decide to choose the 3x3 square with the largest total power.
The power level in a given fuel cell can be found through the following process:
Find the fuel cell's rack ID, which is its X coordinate plus 10.
Begin with a power level of the rack ID times the Y coordinate.
Increase the power level by the value of the grid serial number (your puzzle input).
Set the power level to itself multiplied by the rack ID.
Keep only the hundreds digit of the power level (so 12345 becomes 3; numbers with no hundreds digit become 0).
Subtract 5 from the power level.
For example, to find the power level of the fuel cell at 3,5 in a grid with serial number 8:
The rack ID is 3 + 10 = 13.
The power level starts at 13 * 5 = 65.
Adding the serial number produces 65 + 8 = 73.
Multiplying by the rack ID produces 73 * 13 = 949.
The hundreds digit of 949 is 9.
Subtracting 5 produces 9 - 5 = 4.
So, the power level of this fuel cell is 4.
Here are some more example power levels:
Fuel cell at 122,79, grid serial number 57: power level -5.
Fuel cell at 217,196, grid serial number 39: power level 0.
Fuel cell at 101,153, grid serial number 71: power level 4.
Your goal is to find the 3x3 square which has the largest total power. The square must be entirely within the 300x300 grid. Identify this square using the X,Y coordinate of its top-left fuel cell. For example:
For grid serial number 18, the largest total 3x3 square has a top-left corner of 33,45 (with a total power of 29); these fuel cells appear in the middle of this 5x5 region:
-2 -4 4 4 4
-4 4 4 4 -5
4 3 3 4 -4
1 1 2 4 -3
-1 0 2 -5 -2
For grid serial number 42, the largest 3x3 square's top-left is 21,61 (with a total power of 30); they are in the middle of this region:
-3 4 2 2 2
-4 4 3 3 4
-5 3 3 4 -4
4 3 3 4 -3
3 3 3 -5 -1
What is the X,Y coordinate of the top-left fuel cell of the 3x3 square with the largest total power?
| 456
|
--- Day 11: Radioisotope Thermoelectric Generators ---
You come upon a column of four floors that have been entirely sealed off from the rest of the building except for a small dedicated lobby. There are some radiation warnings and a big sign which reads "Radioisotope Testing Facility".
According to the project status board, this facility is currently being used to experiment with Radioisotope Thermoelectric Generators (RTGs, or simply "generators") that are designed to be paired with specially-constructed microchips. Basically, an RTG is a highly radioactive rock that generates electricity through heat.
The experimental RTGs have poor radiation containment, so they're dangerously radioactive. The chips are prototypes and don't have normal radiation shielding, but they do have the ability to generate an electromagnetic radiation shield when powered. Unfortunately, they can only be powered by their corresponding RTG. An RTG powering a microchip is still dangerous to other microchips.
In other words, if a chip is ever left in the same area as another RTG, and it's not connected to its own RTG, the chip will be fried. Therefore, it is assumed that you will follow procedure and keep chips connected to their corresponding RTG when they're in the same room, and away from other RTGs otherwise.
These microchips sound very interesting and useful to your current activities, and you'd like to try to retrieve them. The fourth floor of the facility has an assembling machine which can make a self-contained, shielded computer for you to take with you - that is, if you can bring it all of the RTGs and microchips.
Within the radiation-shielded part of the facility (in which it's safe to have these pre-assembly RTGs), there is an elevator that can move between the four floors. Its capacity rating means it can carry at most yourself and two RTGs or microchips in any combination. (They're rigged to some heavy diagnostic equipment - the assembling machine will detach it for you.) As a security measure, the elevator will only function if it contains at least one RTG or microchip. The elevator always stops on each floor to recharge, and this takes long enough that the items within it and the items on that floor can irradiate each other. (You can prevent this if a Microchip and its Generator end up on the same floor in this way, as they can be connected while the elevator is recharging.)
You make some notes of the locations of each component of interest (your puzzle input). Before you don a hazmat suit and start moving things around, you'd like to have an idea of what you need to do.
When you enter the containment area, you and the elevator will start on the first floor.
For example, suppose the isolated area has the following arrangement:
The first floor contains a hydrogen-compatible microchip and a lithium-compatible microchip.
The second floor contains a hydrogen generator.
The third floor contains a lithium generator.
The fourth floor contains nothing relevant.
As a diagram (F# for a Floor number, E for Elevator, H for Hydrogen, L for Lithium, M for Microchip, and G for Generator), the initial state looks like this:
F4 . . . . .
F3 . . . LG .
F2 . HG . . .
F1 E . HM . LM
Then, to get everything up to the assembling machine on the fourth floor, the following steps could be taken:
Bring the Hydrogen-compatible Microchip to the second floor, which is safe because it can get power from the Hydrogen Generator:
F4 . . . . .
F3 . . . LG .
F2 E HG HM . .
F1 . . . . LM
Bring both Hydrogen-related items to the third floor, which is safe because the Hydrogen-compatible microchip is getting power from its generator:
F4 . . . . .
F3 E HG HM LG .
F2 . . . . .
F1 . . . . LM
Leave the Hydrogen Generator on floor three, but bring the Hydrogen-compatible Microchip back down with you so you can still use the elevator:
F4 . . . . .
F3 . HG . LG .
F2 E . HM . .
F1 . . . . LM
At the first floor, grab the Lithium-compatible Microchip, which is safe because Microchips don't affect each other:
F4 . . . . .
F3 . HG . LG .
F2 . . . . .
F1 E . HM . LM
Bring both Microchips up one floor, where there is nothing to fry them:
F4 . . . . .
F3 . HG . LG .
F2 E . HM . LM
F1 . . . . .
Bring both Microchips up again to floor three, where they can be temporarily connected to their corresponding generators while the elevator recharges, preventing either of them from being fried:
F4 . . . . .
F3 E HG HM LG LM
F2 . . . . .
F1 . . . . .
Bring both Microchips to the fourth floor:
F4 E . HM . LM
F3 . HG . LG .
F2 . . . . .
F1 . . . . .
Leave the Lithium-compatible microchip on the fourth floor, but bring the Hydrogen-compatible one so you can still use the elevator; this is safe because although the Lithium Generator is on the destination floor, you can connect Hydrogen-compatible microchip to the Hydrogen Generator there:
F4 . . . . LM
F3 E HG HM LG .
F2 . . . . .
F1 . . . . .
Bring both Generators up to the fourth floor, which is safe because you can connect the Lithium-compatible Microchip to the Lithium Generator upon arrival:
F4 E HG . LG LM
F3 . . HM . .
F2 . . . . .
F1 . . . . .
Bring the Lithium Microchip with you to the third floor so you can use the elevator:
F4 . HG . LG .
F3 E . HM . LM
F2 . . . . .
F1 . . . . .
Bring both Microchips to the fourth floor:
F4 E HG HM LG LM
F3 . . . . .
F2 . . . . .
F1 . . . . .
In this arrangement, it takes 11 steps to collect all of the objects at the fourth floor for assembly. (Each elevator stop counts as one step, even if nothing is added to or removed from it.)
In your situation, what is the minimum number of steps required to bring all of the objects to the fourth floor?
Your puzzle answer was 31.
--- Part Two ---
You step into the cleanroom separating the lobby from the isolated area and put on the hazmat suit.
Upon entering the isolated containment area, however, you notice some extra parts on the first floor that weren't listed on the record outside:
An elerium generator.
An elerium-compatible microchip.
A dilithium generator.
A dilithium-compatible microchip.
These work just like the other generators and microchips. You'll have to get them up to assembly as well.
What is the minimum number of steps required to bring all of the objects, including these four new ones, to the fourth floor?
| 457
|
--- Day 22: Sand Slabs ---
Enough sand has fallen; it can finally filter water for Snow Island.
Well, almost.
The sand has been falling as large compacted bricks of sand, piling up to form an impressive stack here near the edge of Island Island. In order to make use of the sand to filter water, some of the bricks will need to be broken apart - nay, disintegrated - back into freely flowing sand.
The stack is tall enough that you'll have to be careful about choosing which bricks to disintegrate; if you disintegrate the wrong brick, large portions of the stack could topple, which sounds pretty dangerous.
The Elves responsible for water filtering operations took a snapshot of the bricks while they were still falling (your puzzle input) which should let you work out which bricks are safe to disintegrate. For example:
1,0,1~1,2,1
0,0,2~2,0,2
0,2,3~2,2,3
0,0,4~0,2,4
2,0,5~2,2,5
0,1,6~2,1,6
1,1,8~1,1,9
Each line of text in the snapshot represents the position of a single brick at the time the snapshot was taken. The position is given as two x,y,z coordinates - one for each end of the brick - separated by a tilde (~). Each brick is made up of a single straight line of cubes, and the Elves were even careful to choose a time for the snapshot that had all of the free-falling bricks at integer positions above the ground, so the whole snapshot is aligned to a three-dimensional cube grid.
A line like 2,2,2~2,2,2 means that both ends of the brick are at the same coordinate - in other words, that the brick is a single cube.
Lines like 0,0,10~1,0,10 or 0,0,10~0,1,10 both represent bricks that are two cubes in volume, both oriented horizontally. The first brick extends in the x direction, while the second brick extends in the y direction.
A line like 0,0,1~0,0,10 represents a ten-cube brick which is oriented vertically. One end of the brick is the cube located at 0,0,1, while the other end of the brick is located directly above it at 0,0,10.
The ground is at z=0 and is perfectly flat; the lowest z value a brick can have is therefore 1. So, 5,5,1~5,6,1 and 0,2,1~0,2,5 are both resting on the ground, but 3,3,2~3,3,3 was above the ground at the time of the snapshot.
Because the snapshot was taken while the bricks were still falling, some bricks will still be in the air; you'll need to start by figuring out where they will end up. Bricks are magically stabilized, so they never rotate, even in weird situations like where a long horizontal brick is only supported on one end. Two bricks cannot occupy the same position, so a falling brick will come to rest upon the first other brick it encounters.
Here is the same example again, this time with each brick given a letter so it can be marked in diagrams:
1,0,1~1,2,1 <- A
0,0,2~2,0,2 <- B
0,2,3~2,2,3 <- C
0,0,4~0,2,4 <- D
2,0,5~2,2,5 <- E
0,1,6~2,1,6 <- F
1,1,8~1,1,9 <- G
At the time of the snapshot, from the side so the x axis goes left to right, these bricks are arranged like this:
x
012
.G. 9
.G. 8
... 7
FFF 6
..E 5 z
D.. 4
CCC 3
BBB 2
.A. 1
--- 0
Rotating the perspective 90 degrees so the y axis now goes left to right, the same bricks are arranged like this:
y
012
.G. 9
.G. 8
... 7
.F. 6
EEE 5 z
DDD 4
..C 3
B.. 2
AAA 1
--- 0
Once all of the bricks fall downward as far as they can go, the stack looks like this, where ? means bricks are hidden behind other bricks at that location:
x
012
.G. 6
.G. 5
FFF 4
D.E 3 z
??? 2
.A. 1
--- 0
Again from the side:
y
012
.G. 6
.G. 5
.F. 4
??? 3 z
B.C 2
AAA 1
--- 0
Now that all of the bricks have settled, it becomes easier to tell which bricks are supporting which other bricks:
Brick A is the only brick supporting bricks B and C.
Brick B is one of two bricks supporting brick D and brick E.
Brick C is the other brick supporting brick D and brick E.
Brick D supports brick F.
Brick E also supports brick F.
Brick F supports brick G.
Brick G isn't supporting any bricks.
Your first task is to figure out which bricks are safe to disintegrate. A brick can be safely disintegrated if, after removing it, no other bricks would fall further directly downward. Don't actually disintegrate any bricks - just determine what would happen if, for each brick, only that brick were disintegrated. Bricks can be disintegrated even if they're completely surrounded by other bricks; you can squeeze between bricks if you need to.
In this example, the bricks can be disintegrated as follows:
Brick A cannot be disintegrated safely; if it were disintegrated, bricks B and C would both fall.
Brick B can be disintegrated; the bricks above it (D and E) would still be supported by brick C.
Brick C can be disintegrated; the bricks above it (D and E) would still be supported by brick B.
Brick D can be disintegrated; the brick above it (F) would still be supported by brick E.
Brick E can be disintegrated; the brick above it (F) would still be supported by brick D.
Brick F cannot be disintegrated; the brick above it (G) would fall.
Brick G can be disintegrated; it does not support any other bricks.
So, in this example, 5 bricks can be safely disintegrated.
Figure how the blocks will settle based on the snapshot. Once they've settled, consider disintegrating a single brick; how many bricks could be safely chosen as the one to get disintegrated?
| 458
|
--- Day 5: Supply Stacks ---
The expedition can depart as soon as the final supplies have been unloaded from the ships. Supplies are stored in stacks of marked crates, but because the needed supplies are buried under many other crates, the crates need to be rearranged.
The ship has a giant cargo crane capable of moving crates between stacks. To ensure none of the crates get crushed or fall over, the crane operator will rearrange them in a series of carefully-planned steps. After the crates are rearranged, the desired crates will be at the top of each stack.
The Elves don't want to interrupt the crane operator during this delicate procedure, but they forgot to ask her which crate will end up where, and they want to be ready to unload them as soon as possible so they can embark.
They do, however, have a drawing of the starting stacks of crates and the rearrangement procedure (your puzzle input). For example:
[D]
[N] [C]
[Z] [M] [P]
1 2 3
move 1 from 2 to 1
move 3 from 1 to 3
move 2 from 2 to 1
move 1 from 1 to 2
In this example, there are three stacks of crates. Stack 1 contains two crates: crate Z is on the bottom, and crate N is on top. Stack 2 contains three crates; from bottom to top, they are crates M, C, and D. Finally, stack 3 contains a single crate, P.
Then, the rearrangement procedure is given. In each step of the procedure, a quantity of crates is moved from one stack to a different stack. In the first step of the above rearrangement procedure, one crate is moved from stack 2 to stack 1, resulting in this configuration:
[D]
[N] [C]
[Z] [M] [P]
1 2 3
In the second step, three crates are moved from stack 1 to stack 3. Crates are moved one at a time, so the first crate to be moved (D) ends up below the second and third crates:
[Z]
[N]
[C] [D]
[M] [P]
1 2 3
Then, both crates are moved from stack 2 to stack 1. Again, because crates are moved one at a time, crate C ends up below crate M:
[Z]
[N]
[M] [D]
[C] [P]
1 2 3
Finally, one crate is moved from stack 1 to stack 2:
[Z]
[N]
[D]
[C] [M] [P]
1 2 3
The Elves just need to know which crate will end up on top of each stack; in this example, the top crates are C in stack 1, M in stack 2, and Z in stack 3, so you should combine these together and give the Elves the message CMZ.
After the rearrangement procedure completes, what crate ends up on top of each stack?
| 459
|
--- Day 4: Secure Container ---
You arrive at the Venus fuel depot only to discover it's protected by a password. The Elves had written the password on a sticky note, but someone threw it out.
However, they do remember a few key facts about the password:
It is a six-digit number.
The value is within the range given in your puzzle input.
Two adjacent digits are the same (like 22 in 122345).
Going from left to right, the digits never decrease; they only ever increase or stay the same (like 111123 or 135679).
Other than the range rule, the following are true:
111111 meets these criteria (double 11, never decreases).
223450 does not meet these criteria (decreasing pair of digits 50).
123789 does not meet these criteria (no double).
How many different passwords within the range given in your puzzle input meet these criteria?
Your puzzle answer was 1873.
--- Part Two ---
An Elf just remembered one more important detail: the two adjacent matching digits are not part of a larger group of matching digits.
Given this additional criterion, but still ignoring the range rule, the following are now true:
112233 meets these criteria because the digits never decrease and all repeated digits are exactly two digits long.
123444 no longer meets the criteria (the repeated 44 is part of a larger group of 444).
111122 meets the criteria (even though 1 is repeated more than twice, it still contains a double 22).
How many different passwords within the range given in your puzzle input meet all of the criteria?
| 460
|
--- Day 7: Bridge Repair ---
The Historians take you to a familiar rope bridge over a river in the middle of a jungle. The Chief isn't on this side of the bridge, though; maybe he's on the other side?
When you go to cross the bridge, you notice a group of engineers trying to repair it. (Apparently, it breaks pretty frequently.) You won't be able to cross until it's fixed.
You ask how long it'll take; the engineers tell you that it only needs final calibrations, but some young elephants were playing nearby and stole all the operators from their calibration equations! They could finish the calibrations if only someone could determine which test values could possibly be produced by placing any combination of operators into their calibration equations (your puzzle input).
For example:
190: 10 19
3267: 81 40 27
83: 17 5
156: 15 6
7290: 6 8 6 15
161011: 16 10 13
192: 17 8 14
21037: 9 7 18 13
292: 11 6 16 20
Each line represents a single equation. The test value appears before the colon on each line; it is your job to determine whether the remaining numbers can be combined with operators to produce the test value.
Operators are always evaluated left-to-right, not according to precedence rules. Furthermore, numbers in the equations cannot be rearranged. Glancing into the jungle, you can see elephants holding two different types of operators: add (+) and multiply (*).
Only three of the above equations can be made true by inserting operators:
190: 10 19 has only one position that accepts an operator: between 10 and 19. Choosing + would give 29, but choosing * would give the test value (10 * 19 = 190).
3267: 81 40 27 has two positions for operators. Of the four possible configurations of the operators, two cause the right side to match the test value: 81 + 40 * 27 and 81 * 40 + 27 both equal 3267 (when evaluated left-to-right)!
292: 11 6 16 20 can be solved in exactly one way: 11 + 6 * 16 + 20.
The engineers just need the total calibration result, which is the sum of the test values from just the equations that could possibly be true. In the above example, the sum of the test values for the three equations listed above is 3749.
Determine which equations could possibly be true. What is their total calibration result?
| 461
|
--- Day 5: How About a Nice Game of Chess? ---
You are faced with a security door designed by Easter Bunny engineers that seem to have acquired most of their security knowledge by watching hacking movies.
The eight-character password for the door is generated one character at a time by finding the MD5 hash of some Door ID (your puzzle input) and an increasing integer index (starting with 0).
A hash indicates the next character in the password if its hexadecimal representation starts with five zeroes. If it does, the sixth character in the hash is the next character of the password.
For example, if the Door ID is abc:
The first index which produces a hash that starts with five zeroes is 3231929, which we find by hashing abc3231929; the sixth character of the hash, and thus the first character of the password, is 1.
5017308 produces the next interesting hash, which starts with 000008f82..., so the second character of the password is 8.
The third time a hash starts with five zeroes is for abc5278568, discovering the character f.
In this example, after continuing this search a total of eight times, the password is 18f47a30.
Given the actual Door ID, what is the password?
| 462
|
--- Day 20: A Regular Map ---
While you were learning about instruction pointers, the Elves made considerable progress. When you look up, you discover that the North Pole base construction project has completely surrounded you.
The area you are in is made up entirely of rooms and doors. The rooms are arranged in a grid, and rooms only connect to adjacent rooms when a door is present between them.
For example, drawing rooms as ., walls as #, doors as | or -, your current position as X, and where north is up, the area you're in might look like this:
#####
#.|.#
#-###
#.|X#
#####
You get the attention of a passing construction Elf and ask for a map. "I don't have time to draw out a map of this place - it's huge. Instead, I can give you directions to every room in the facility!" He writes down some directions on a piece of parchment and runs off. In the example above, the instructions might have been ^WNE$, a regular expression or "regex" (your puzzle input).
The regex matches routes (like WNE for "west, north, east") that will take you from your current room through various doors in the facility. In aggregate, the routes will take you through every door in the facility at least once; mapping out all of these routes will let you build a proper map and find your way around.
^ and $ are at the beginning and end of your regex; these just mean that the regex doesn't match anything outside the routes it describes. (Specifically, ^ matches the start of the route, and $ matches the end of it.) These characters will not appear elsewhere in the regex.
The rest of the regex matches various sequences of the characters N (north), S (south), E (east), and W (west). In the example above, ^WNE$ matches only one route, WNE, which means you can move west, then north, then east from your current position. Sequences of letters like this always match that exact route in the same order.
Sometimes, the route can branch. A branch is given by a list of options separated by pipes (|) and wrapped in parentheses. So, ^N(E|W)N$ contains a branch: after going north, you must choose to go either east or west before finishing your route by going north again. By tracing out the possible routes after branching, you can determine where the doors are and, therefore, where the rooms are in the facility.
For example, consider this regex: ^ENWWW(NEEE|SSE(EE|N))$
This regex begins with ENWWW, which means that from your current position, all routes must begin by moving east, north, and then west three times, in that order. After this, there is a branch. Before you consider the branch, this is what you know about the map so far, with doors you aren't sure about marked with a ?:
#?#?#?#?#
?.|.|.|.?
#?#?#?#-#
?X|.?
#?#?#
After this point, there is (NEEE|SSE(EE|N)). This gives you exactly two options: NEEE and SSE(EE|N). By following NEEE, the map now looks like this:
#?#?#?#?#
?.|.|.|.?
#-#?#?#?#
?.|.|.|.?
#?#?#?#-#
?X|.?
#?#?#
Now, only SSE(EE|N) remains. Because it is in the same parenthesized group as NEEE, it starts from the same room NEEE started in. It states that starting from that point, there exist doors which will allow you to move south twice, then east; this ends up at another branch. After that, you can either move east twice or north once. This information fills in the rest of the doors:
#?#?#?#?#
?.|.|.|.?
#-#?#?#?#
?.|.|.|.?
#-#?#?#-#
?.?.?X|.?
#-#-#?#?#
?.|.|.|.?
#?#?#?#?#
Once you've followed all possible routes, you know the remaining unknown parts are all walls, producing a finished map of the facility:
#########
#.|.|.|.#
#-#######
#.|.|.|.#
#-#####-#
#.#.#X|.#
#-#-#####
#.|.|.|.#
#########
Sometimes, a list of options can have an empty option, like (NEWS|WNSE|). This means that routes at this point could effectively skip the options in parentheses and move on immediately. For example, consider this regex and the corresponding map:
^ENNWSWW(NEWS|)SSSEEN(WNSE|)EE(SWEN|)NNN$
###########
#.|.#.|.#.#
#-###-#-#-#
#.|.|.#.#.#
#-#####-#-#
#.#.#X|.#.#
#-#-#####-#
#.#.|.|.|.#
#-###-###-#
#.|.|.#.|.#
###########
This regex has one main route which, at three locations, can optionally include additional detours and be valid: (NEWS|), (WNSE|), and (SWEN|). Regardless of which option is taken, the route continues from the position it is left at after taking those steps. So, for example, this regex matches all of the following routes (and more that aren't listed here):
ENNWSWWSSSEENEENNN
ENNWSWWNEWSSSSEENEENNN
ENNWSWWNEWSSSSEENEESWENNNN
ENNWSWWSSSEENWNSEEENNN
By following the various routes the regex matches, a full map of all of the doors and rooms in the facility can be assembled.
To get a sense for the size of this facility, you'd like to determine which room is furthest from you: specifically, you would like to find the room for which the shortest path to that room would require passing through the most doors.
In the first example (^WNE$), this would be the north-east corner 3 doors away.
In the second example (^ENWWW(NEEE|SSE(EE|N))$), this would be the south-east corner 10 doors away.
In the third example (^ENNWSWW(NEWS|)SSSEEN(WNSE|)EE(SWEN|)NNN$), this would be the north-east corner 18 doors away.
Here are a few more examples:
Regex: ^ESSWWN(E|NNENN(EESS(WNSE|)SSS|WWWSSSSE(SW|NNNE)))$
Furthest room requires passing 23 doors
#############
#.|.|.|.|.|.#
#-#####-###-#
#.#.|.#.#.#.#
#-#-###-#-#-#
#.#.#.|.#.|.#
#-#-#-#####-#
#.#.#.#X|.#.#
#-#-#-###-#-#
#.|.#.|.#.#.#
###-#-###-#-#
#.|.#.|.|.#.#
#############
Regex: ^WSSEESWWWNW(S|NENNEEEENN(ESSSSW(NWSW|SSEN)|WSWWN(E|WWS(E|SS))))$
Furthest room requires passing 31 doors
###############
#.|.|.|.#.|.|.#
#-###-###-#-#-#
#.|.#.|.|.#.#.#
#-#########-#-#
#.#.|.|.|.|.#.#
#-#-#########-#
#.#.#.|X#.|.#.#
###-#-###-#-#-#
#.|.#.#.|.#.|.#
#-###-#####-###
#.|.#.|.|.#.#.#
#-#-#####-#-#-#
#.#.|.|.|.#.|.#
###############
What is the largest number of doors you would be required to pass through to reach a room? That is, find the room for which the shortest path from your starting location to that room would require passing through the most doors; what is the fewest doors you can pass through to reach it?
| 463
|
--- Day 17: Set and Forget ---
An early warning system detects an incoming solar flare and automatically activates the ship's electromagnetic shield. Unfortunately, this has cut off the Wi-Fi for many small robots that, unaware of the impending danger, are now trapped on exterior scaffolding on the unsafe side of the shield. To rescue them, you'll have to act quickly!
The only tools at your disposal are some wired cameras and a small vacuum robot currently asleep at its charging station. The video quality is poor, but the vacuum robot has a needlessly bright LED that makes it easy to spot no matter where it is.
An Intcode program, the Aft Scaffolding Control and Information Interface (ASCII, your puzzle input), provides access to the cameras and the vacuum robot. Currently, because the vacuum robot is asleep, you can only access the cameras.
Running the ASCII program on your Intcode computer will provide the current view of the scaffolds. This is output, purely coincidentally, as ASCII code: 35 means #, 46 means ., 10 starts a new line of output below the current one, and so on. (Within a line, characters are drawn left-to-right.)
In the camera output, # represents a scaffold and . represents open space. The vacuum robot is visible as ^, v, <, or > depending on whether it is facing up, down, left, or right respectively. When drawn like this, the vacuum robot is always on a scaffold; if the vacuum robot ever walks off of a scaffold and begins tumbling through space uncontrollably, it will instead be visible as X.
In general, the scaffold forms a path, but it sometimes loops back onto itself. For example, suppose you can see the following view from the cameras:
..#..........
..#..........
#######...###
#.#...#...#.#
#############
..#...#...#..
..#####...^..
Here, the vacuum robot, ^ is facing up and sitting at one end of the scaffold near the bottom-right of the image. The scaffold continues up, loops across itself several times, and ends at the top-left of the image.
The first step is to calibrate the cameras by getting the alignment parameters of some well-defined points. Locate all scaffold intersections; for each, its alignment parameter is the distance between its left edge and the left edge of the view multiplied by the distance between its top edge and the top edge of the view. Here, the intersections from the above image are marked O:
..#..........
..#..........
##O####...###
#.#...#...#.#
##O###O###O##
..#...#...#..
..#####...^..
For these intersections:
The top-left intersection is 2 units from the left of the image and 2 units from the top of the image, so its alignment parameter is 2 * 2 = 4.
The bottom-left intersection is 2 units from the left and 4 units from the top, so its alignment parameter is 2 * 4 = 8.
The bottom-middle intersection is 6 from the left and 4 from the top, so its alignment parameter is 24.
The bottom-right intersection's alignment parameter is 40.
To calibrate the cameras, you need the sum of the alignment parameters. In the above example, this is 76.
Run your ASCII program. What is the sum of the alignment parameters for the scaffold intersections?
| 464
|
--- Day 24: Planet of Discord ---
You land on Eris, your last stop before reaching Santa. As soon as you do, your sensors start picking up strange life forms moving around: Eris is infested with bugs! With an over 24-hour roundtrip for messages between you and Earth, you'll have to deal with this problem on your own.
Eris isn't a very large place; a scan of the entire area fits into a 5x5 grid (your puzzle input). The scan shows bugs (#) and empty spaces (.).
Each minute, The bugs live and die based on the number of bugs in the four adjacent tiles:
A bug dies (becoming an empty space) unless there is exactly one bug adjacent to it.
An empty space becomes infested with a bug if exactly one or two bugs are adjacent to it.
Otherwise, a bug or empty space remains the same. (Tiles on the edges of the grid have fewer than four adjacent tiles; the missing tiles count as empty space.) This process happens in every location simultaneously; that is, within the same minute, the number of adjacent bugs is counted for every tile first, and then the tiles are updated.
Here are the first few minutes of an example scenario:
Initial state:
....#
#..#.
#..##
..#..
#....
After 1 minute:
#..#.
####.
###.#
##.##
.##..
After 2 minutes:
#####
....#
....#
...#.
#.###
After 3 minutes:
#....
####.
...##
#.##.
.##.#
After 4 minutes:
####.
....#
##..#
.....
##...
To understand the nature of the bugs, watch for the first time a layout of bugs and empty spaces matches any previous layout. In the example above, the first layout to appear twice is:
.....
.....
.....
#....
.#...
To calculate the biodiversity rating for this layout, consider each tile left-to-right in the top row, then left-to-right in the second row, and so on. Each of these tiles is worth biodiversity points equal to increasing powers of two: 1, 2, 4, 8, 16, 32, and so on. Add up the biodiversity points for tiles with bugs; in this example, the 16th tile (32768 points) and 22nd tile (2097152 points) have bugs, a total biodiversity rating of 2129920.
What is the biodiversity rating for the first layout that appears twice?
Your puzzle answer was 18844281.
--- Part Two ---
After careful analysis, one thing is certain: you have no idea where all these bugs are coming from.
Then, you remember: Eris is an old Plutonian settlement! Clearly, the bugs are coming from recursively-folded space.
This 5x5 grid is only one level in an infinite number of recursion levels. The tile in the middle of the grid is actually another 5x5 grid, the grid in your scan is contained as the middle tile of a larger 5x5 grid, and so on. Two levels of grids look like this:
| | | |
| | | |
| | | |
-----+-----+---------+-----+-----
| | | |
| | | |
| | | |
-----+-----+---------+-----+-----
| | | | | | | |
| |-+-+-+-+-| |
| | | | | | | |
| |-+-+-+-+-| |
| | | |?| | | |
| |-+-+-+-+-| |
| | | | | | | |
| |-+-+-+-+-| |
| | | | | | | |
-----+-----+---------+-----+-----
| | | |
| | | |
| | | |
-----+-----+---------+-----+-----
| | | |
| | | |
| | | |
(To save space, some of the tiles are not drawn to scale.) Remember, this is only a small part of the infinitely recursive grid; there is a 5x5 grid that contains this diagram, and a 5x5 grid that contains that one, and so on. Also, the ? in the diagram contains another 5x5 grid, which itself contains another 5x5 grid, and so on.
The scan you took (your puzzle input) shows where the bugs are on a single level of this structure. The middle tile of your scan is empty to accommodate the recursive grids within it. Initially, no other levels contain bugs.
Tiles still count as adjacent if they are directly up, down, left, or right of a given tile. Some tiles have adjacent tiles at a recursion level above or below its own level. For example:
| | | |
1 | 2 | 3 | 4 | 5
| | | |
-----+-----+---------+-----+-----
| | | |
6 | 7 | 8 | 9 | 10
| | | |
-----+-----+---------+-----+-----
| |A|B|C|D|E| |
| |-+-+-+-+-| |
| |F|G|H|I|J| |
| |-+-+-+-+-| |
11 | 12 |K|L|?|N|O| 14 | 15
| |-+-+-+-+-| |
| |P|Q|R|S|T| |
| |-+-+-+-+-| |
| |U|V|W|X|Y| |
-----+-----+---------+-----+-----
| | | |
16 | 17 | 18 | 19 | 20
| | | |
-----+-----+---------+-----+-----
| | | |
21 | 22 | 23 | 24 | 25
| | | |
Tile 19 has four adjacent tiles: 14, 18, 20, and 24.
Tile G has four adjacent tiles: B, F, H, and L.
Tile D has four adjacent tiles: 8, C, E, and I.
Tile E has four adjacent tiles: 8, D, 14, and J.
Tile 14 has eight adjacent tiles: 9, E, J, O, T, Y, 15, and 19.
Tile N has eight adjacent tiles: I, O, S, and five tiles within the sub-grid marked ?.
The rules about bugs living and dying are the same as before.
For example, consider the same initial state as above:
....#
#..#.
#.?##
..#..
#....
The center tile is drawn as ? to indicate the next recursive grid. Call this level 0; the grid within this one is level 1, and the grid that contains this one is level -1. Then, after ten minutes, the grid at each level would look like this:
Depth -5:
..#..
.#.#.
..?.#
.#.#.
..#..
Depth -4:
...#.
...##
..?..
...##
...#.
Depth -3:
#.#..
.#...
..?..
.#...
#.#..
Depth -2:
.#.##
....#
..?.#
...##
.###.
Depth -1:
#..##
...##
..?..
...#.
.####
Depth 0:
.#...
.#.##
.#?..
.....
.....
Depth 1:
.##..
#..##
..?.#
##.##
#####
Depth 2:
###..
##.#.
#.?..
.#.##
#.#..
Depth 3:
..###
.....
#.?..
#....
#...#
Depth 4:
.###.
#..#.
#.?..
##.#.
.....
Depth 5:
####.
#..#.
#.?#.
####.
.....
In this example, after 10 minutes, a total of 99 bugs are present.
Starting with your scan, how many bugs are present after 200 minutes?
| 465
|
--- Day 13: Point of Incidence ---
With your help, the hot springs team locates an appropriate spring which launches you neatly and precisely up to the edge of Lava Island.
There's just one problem: you don't see any lava.
You do see a lot of ash and igneous rock; there are even what look like gray mountains scattered around. After a while, you make your way to a nearby cluster of mountains only to discover that the valley between them is completely full of large mirrors. Most of the mirrors seem to be aligned in a consistent way; perhaps you should head in that direction?
As you move through the valley of mirrors, you find that several of them have fallen from the large metal frames keeping them in place. The mirrors are extremely flat and shiny, and many of the fallen mirrors have lodged into the ash at strange angles. Because the terrain is all one color, it's hard to tell where it's safe to walk or where you're about to run into a mirror.
You note down the patterns of ash (.) and rocks (#) that you see as you walk (your puzzle input); perhaps by carefully analyzing these patterns, you can figure out where the mirrors are!
For example:
#.##..##.
..#.##.#.
##......#
##......#
..#.##.#.
..##..##.
#.#.##.#.
#...##..#
#....#..#
..##..###
#####.##.
#####.##.
..##..###
#....#..#
To find the reflection in each pattern, you need to find a perfect reflection across either a horizontal line between two rows or across a vertical line between two columns.
In the first pattern, the reflection is across a vertical line between two columns; arrows on each of the two columns point at the line between the columns:
123456789
><
#.##..##.
..#.##.#.
##......#
##......#
..#.##.#.
..##..##.
#.#.##.#.
><
123456789
In this pattern, the line of reflection is the vertical line between columns 5 and 6. Because the vertical line is not perfectly in the middle of the pattern, part of the pattern (column 1) has nowhere to reflect onto and can be ignored; every other column has a reflected column within the pattern and must match exactly: column 2 matches column 9, column 3 matches 8, 4 matches 7, and 5 matches 6.
The second pattern reflects across a horizontal line instead:
1 #...##..# 1
2 #....#..# 2
3 ..##..### 3
4v#####.##.v4
5^#####.##.^5
6 ..##..### 6
7 #....#..# 7
This pattern reflects across the horizontal line between rows 4 and 5. Row 1 would reflect with a hypothetical row 8, but since that's not in the pattern, row 1 doesn't need to match anything. The remaining rows match: row 2 matches row 7, row 3 matches row 6, and row 4 matches row 5.
To summarize your pattern notes, add up the number of columns to the left of each vertical line of reflection; to that, also add 100 multiplied by the number of rows above each horizontal line of reflection. In the above example, the first pattern's vertical line has 5 columns to its left and the second pattern's horizontal line has 4 rows above it, a total of 405.
Find the line of reflection in each of the patterns in your notes. What number do you get after summarizing all of your notes?
| 466
|
--- Day 8: Matchsticks ---
Space on the sleigh is limited this year, and so Santa will be bringing his list as a digital copy. He needs to know how much space it will take up when stored.
It is common in many programming languages to provide a way to escape special characters in strings. For example, C, JavaScript, Perl, Python, and even PHP handle special characters in very similar ways.
However, it is important to realize the difference between the number of characters in the code representation of the string literal and the number of characters in the in-memory string itself.
For example:
"" is 2 characters of code (the two double quotes), but the string contains zero characters.
"abc" is 5 characters of code, but 3 characters in the string data.
"aaa\"aaa" is 10 characters of code, but the string itself contains six "a" characters and a single, escaped quote character, for a total of 7 characters in the string data.
"\x27" is 6 characters of code, but the string itself contains just one - an apostrophe ('), escaped using hexadecimal notation.
Santa's list is a file that contains many double-quoted string literals, one on each line. The only escape sequences used are \\ (which represents a single backslash), \" (which represents a lone double-quote character), and \x plus two hexadecimal characters (which represents a single character with that ASCII code).
Disregarding the whitespace in the file, what is the number of characters of code for string literals minus the number of characters in memory for the values of the strings in total for the entire file?
For example, given the four strings above, the total number of characters of string code (2 + 5 + 10 + 6 = 23) minus the total number of characters in memory for string values (0 + 3 + 7 + 1 = 11) is 23 - 11 = 12.
| 467
|
--- Day 10: Syntax Scoring ---
You ask the submarine to determine the best route out of the deep-sea cave, but it only replies:
Syntax error in navigation subsystem on line: all of them
All of them?! The damage is worse than you thought. You bring up a copy of the navigation subsystem (your puzzle input).
The navigation subsystem syntax is made of several lines containing chunks. There are one or more chunks on each line, and chunks contain zero or more other chunks. Adjacent chunks are not separated by any delimiter; if one chunk stops, the next chunk (if any) can immediately start. Every chunk must open and close with one of four legal pairs of matching characters:
If a chunk opens with (, it must close with ).
If a chunk opens with [, it must close with ].
If a chunk opens with {, it must close with }.
If a chunk opens with <, it must close with >.
So, () is a legal chunk that contains no other chunks, as is []. More complex but valid chunks include ([]), {()()()}, <([{}])>, [<>({}){}[([])<>]], and even (((((((((()))))))))).
Some lines are incomplete, but others are corrupted. Find and discard the corrupted lines first.
A corrupted line is one where a chunk closes with the wrong character - that is, where the characters it opens and closes with do not form one of the four legal pairs listed above.
Examples of corrupted chunks include (], {()()()>, (((()))}, and <([]){()}[{}]). Such a chunk can appear anywhere within a line, and its presence causes the whole line to be considered corrupted.
For example, consider the following navigation subsystem:
[({(<(())[]>[[{[]{<()<>>
[(()[<>])]({[<{<<[]>>(
{([(<{}[<>[]}>{[]{[(<()>
(((({<>}<{<{<>}{[]{[]{}
[[<[([]))<([[{}[[()]]]
[{[{({}]{}}([{[{{{}}([]
{<[[]]>}<{[{[{[]{()[[[]
[<(<(<(<{}))><([]([]()
<{([([[(<>()){}]>(<<{{
<{([{{}}[<[[[<>{}]]]>[]]
Some of the lines aren't corrupted, just incomplete; you can ignore these lines for now. The remaining five lines are corrupted:
{([(<{}[<>[]}>{[]{[(<()> - Expected ], but found } instead.
[[<[([]))<([[{}[[()]]] - Expected ], but found ) instead.
[{[{({}]{}}([{[{{{}}([] - Expected ), but found ] instead.
[<(<(<(<{}))><([]([]() - Expected >, but found ) instead.
<{([([[(<>()){}]>(<<{{ - Expected ], but found > instead.
Stop at the first incorrect closing character on each corrupted line.
Did you know that syntax checkers actually have contests to see who can get the high score for syntax errors in a file? It's true! To calculate the syntax error score for a line, take the first illegal character on the line and look it up in the following table:
): 3 points.
]: 57 points.
}: 1197 points.
>: 25137 points.
In the above example, an illegal ) was found twice (2*3 = 6 points), an illegal ] was found once (57 points), an illegal } was found once (1197 points), and an illegal > was found once (25137 points). So, the total syntax error score for this file is 6+57+1197+25137 = 26397 points!
Find the first illegal character in each corrupted line of the navigation subsystem. What is the total syntax error score for those errors?
Your puzzle answer was 299793.
--- Part Two ---
Now, discard the corrupted lines. The remaining lines are incomplete.
Incomplete lines don't have any incorrect characters - instead, they're missing some closing characters at the end of the line. To repair the navigation subsystem, you just need to figure out the sequence of closing characters that complete all open chunks in the line.
You can only use closing characters (), ], }, or >), and you must add them in the correct order so that only legal pairs are formed and all chunks end up closed.
In the example above, there are five incomplete lines:
[({(<(())[]>[[{[]{<()<>> - Complete by adding }}]])})].
[(()[<>])]({[<{<<[]>>( - Complete by adding )}>]}).
(((({<>}<{<{<>}{[]{[]{} - Complete by adding }}>}>)))).
{<[[]]>}<{[{[{[]{()[[[] - Complete by adding ]]}}]}]}>.
<{([{{}}[<[[[<>{}]]]>[]] - Complete by adding ])}>.
Did you know that autocomplete tools also have contests? It's true! The score is determined by considering the completion string character-by-character. Start with a total score of 0. Then, for each character, multiply the total score by 5 and then increase the total score by the point value given for the character in the following table:
): 1 point.
]: 2 points.
}: 3 points.
>: 4 points.
So, the last completion string above - ])}> - would be scored as follows:
Start with a total score of 0.
Multiply the total score by 5 to get 0, then add the value of ] (2) to get a new total score of 2.
Multiply the total score by 5 to get 10, then add the value of ) (1) to get a new total score of 11.
Multiply the total score by 5 to get 55, then add the value of } (3) to get a new total score of 58.
Multiply the total score by 5 to get 290, then add the value of > (4) to get a new total score of 294.
The five lines' completion strings have total scores as follows:
}}]])})] - 288957 total points.
)}>]}) - 5566 total points.
}}>}>)))) - 1480781 total points.
]]}}]}]}> - 995444 total points.
])}> - 294 total points.
Autocomplete tools are an odd bunch: the winner is found by sorting all of the scores and then taking the middle score. (There will always be an odd number of scores to consider.) In this example, the middle score is 288957 because there are the same number of scores smaller and larger than it.
Find the completion string for each incomplete line, score the completion strings, and sort the scores. What is the middle score?
| 468
|
--- Day 2: Corruption Checksum ---
As you walk through the door, a glowing humanoid shape yells in your direction. "You there! Your state appears to be idle. Come help us repair the corruption in this spreadsheet - if we take another millisecond, we'll have to display an hourglass cursor!"
The spreadsheet consists of rows of apparently-random numbers. To make sure the recovery process is on the right track, they need you to calculate the spreadsheet's checksum. For each row, determine the difference between the largest value and the smallest value; the checksum is the sum of all of these differences.
For example, given the following spreadsheet:
5 1 9 5
7 5 3
2 4 6 8
The first row's largest and smallest values are 9 and 1, and their difference is 8.
The second row's largest and smallest values are 7 and 3, and their difference is 4.
The third row's difference is 6.
In this example, the spreadsheet's checksum would be 8 + 4 + 6 = 18.
What is the checksum for the spreadsheet in your puzzle input?
Your puzzle answer was 58975.
--- Part Two ---
"Great work; looks like we're on the right track after all. Here's a star for your effort." However, the program seems a little worried. Can programs be worried?
"Based on what we're seeing, it looks like all the User wanted is some information about the evenly divisible values in the spreadsheet. Unfortunately, none of us are equipped for that kind of calculation - most of us specialize in bitwise operations."
It sounds like the goal is to find the only two numbers in each row where one evenly divides the other - that is, where the result of the division operation is a whole number. They would like you to find those numbers on each line, divide them, and add up each line's result.
For example, given the following spreadsheet:
5 9 2 8
9 4 7 3
3 8 6 5
In the first row, the only two numbers that evenly divide are 8 and 2; the result of this division is 4.
In the second row, the two numbers are 9 and 3; the result is 3.
In the third row, the result is 2.
In this example, the sum of the results would be 4 + 3 + 2 = 9.
What is the sum of each row's result in your puzzle input?
| 469
|
--- Day 4: Ceres Search ---
"Looks like the Chief's not here. Next!" One of The Historians pulls out a device and pushes the only button on it. After a brief flash, you recognize the interior of the Ceres monitoring station!
As the search for the Chief continues, a small Elf who lives on the station tugs on your shirt; she'd like to know if you could help her with her word search (your puzzle input). She only has to find one word: XMAS.
This word search allows words to be horizontal, vertical, diagonal, written backwards, or even overlapping other words. It's a little unusual, though, as you don't merely need to find one instance of XMAS - you need to find all of them. Here are a few ways XMAS might appear, where irrelevant characters have been replaced with .:
..X...
.SAMX.
.A..A.
XMAS.S
.X....
The actual word search will be full of letters instead. For example:
MMMSXXMASM
MSAMXMSMSA
AMXSXMAAMM
MSAMASMSMX
XMASAMXAMM
XXAMMXXAMA
SMSMSASXSS
SAXAMASAAA
MAMMMXMMMM
MXMXAXMASX
In this word search, XMAS occurs a total of 18 times; here's the same word search again, but where letters not involved in any XMAS have been replaced with .:
....XXMAS.
.SAMXMS...
...S..A...
..A.A.MS.X
XMASAMX.MM
X.....XA.A
S.S.S.S.SS
.A.A.A.A.A
..M.M.M.MM
.X.X.XMASX
Take a look at the little Elf's word search. How many times does XMAS appear?
Your puzzle answer was 2447.
The first half of this puzzle is complete! It provides one gold star: *
--- Part Two ---
The Elf looks quizzically at you. Did you misunderstand the assignment?
Looking for the instructions, you flip over the word search to find that this isn't actually an XMAS puzzle; it's an X-MAS puzzle in which you're supposed to find two MAS in the shape of an X. One way to achieve that is like this:
M.S
.A.
M.S
Irrelevant characters have again been replaced with . in the above diagram. Within the X, each MAS can be written forwards or backwards.
Here's the same example from before, but this time all of the X-MASes have been kept instead:
.M.S......
..A..MSMS.
.M.S.MAA..
..A.ASMSM.
.M.S.M....
..........
S.S.S.S.S.
.A.A.A.A..
M.M.M.M.M.
..........
In this example, an X-MAS appears 9 times.
Flip the word search from the instructions back over to the word search side and try again. How many times does an X-MAS appear?
| 470
|
--- Day 24: Never Tell Me The Odds ---
It seems like something is going wrong with the snow-making process. Instead of forming snow, the water that's been absorbed into the air seems to be forming hail!
Maybe there's something you can do to break up the hailstones?
Due to strong, probably-magical winds, the hailstones are all flying through the air in perfectly linear trajectories. You make a note of each hailstone's position and velocity (your puzzle input). For example:
19, 13, 30 @ -2, 1, -2
18, 19, 22 @ -1, -1, -2
20, 25, 34 @ -2, -2, -4
12, 31, 28 @ -1, -2, -1
20, 19, 15 @ 1, -5, -3
Each line of text corresponds to the position and velocity of a single hailstone. The positions indicate where the hailstones are right now (at time 0). The velocities are constant and indicate exactly how far each hailstone will move in one nanosecond.
Each line of text uses the format px py pz @ vx vy vz. For instance, the hailstone specified by 20, 19, 15 @ 1, -5, -3 has initial X position 20, Y position 19, Z position 15, X velocity 1, Y velocity -5, and Z velocity -3. After one nanosecond, the hailstone would be at 21, 14, 12.
Perhaps you won't have to do anything. How likely are the hailstones to collide with each other and smash into tiny ice crystals?
To estimate this, consider only the X and Y axes; ignore the Z axis. Looking forward in time, how many of the hailstones' paths will intersect within a test area? (The hailstones themselves don't have to collide, just test for intersections between the paths they will trace.)
In this example, look for intersections that happen with an X and Y position each at least 7 and at most 27; in your actual data, you'll need to check a much larger test area. Comparing all pairs of hailstones' future paths produces the following results:
Hailstone A: 19, 13, 30 @ -2, 1, -2
Hailstone B: 18, 19, 22 @ -1, -1, -2
Hailstones' paths will cross inside the test area (at x=14.333, y=15.333).
Hailstone A: 19, 13, 30 @ -2, 1, -2
Hailstone B: 20, 25, 34 @ -2, -2, -4
Hailstones' paths will cross inside the test area (at x=11.667, y=16.667).
Hailstone A: 19, 13, 30 @ -2, 1, -2
Hailstone B: 12, 31, 28 @ -1, -2, -1
Hailstones' paths will cross outside the test area (at x=6.2, y=19.4).
Hailstone A: 19, 13, 30 @ -2, 1, -2
Hailstone B: 20, 19, 15 @ 1, -5, -3
Hailstones' paths crossed in the past for hailstone A.
Hailstone A: 18, 19, 22 @ -1, -1, -2
Hailstone B: 20, 25, 34 @ -2, -2, -4
Hailstones' paths are parallel; they never intersect.
Hailstone A: 18, 19, 22 @ -1, -1, -2
Hailstone B: 12, 31, 28 @ -1, -2, -1
Hailstones' paths will cross outside the test area (at x=-6, y=-5).
Hailstone A: 18, 19, 22 @ -1, -1, -2
Hailstone B: 20, 19, 15 @ 1, -5, -3
Hailstones' paths crossed in the past for both hailstones.
Hailstone A: 20, 25, 34 @ -2, -2, -4
Hailstone B: 12, 31, 28 @ -1, -2, -1
Hailstones' paths will cross outside the test area (at x=-2, y=3).
Hailstone A: 20, 25, 34 @ -2, -2, -4
Hailstone B: 20, 19, 15 @ 1, -5, -3
Hailstones' paths crossed in the past for hailstone B.
Hailstone A: 12, 31, 28 @ -1, -2, -1
Hailstone B: 20, 19, 15 @ 1, -5, -3
Hailstones' paths crossed in the past for both hailstones.
So, in this example, 2 hailstones' future paths cross inside the boundaries of the test area.
However, you'll need to search a much larger test area if you want to see if any hailstones might collide. Look for intersections that happen with an X and Y position each at least 200000000000000 and at most 400000000000000. Disregard the Z axis entirely.
Considering only the X and Y axes, check all pairs of hailstones' future paths for intersections. How many of these intersections occur within the test area?
| 471
|
--- Day 11: Dumbo Octopus ---
You enter a large cavern full of rare bioluminescent dumbo octopuses! They seem to not like the Christmas lights on your submarine, so you turn them off for now.
There are 100 octopuses arranged neatly in a 10 by 10 grid. Each octopus slowly gains energy over time and flashes brightly for a moment when its energy is full. Although your lights are off, maybe you could navigate through the cave without disturbing the octopuses if you could predict when the flashes of light will happen.
Each octopus has an energy level - your submarine can remotely measure the energy level of each octopus (your puzzle input). For example:
5483143223
2745854711
5264556173
6141336146
6357385478
4167524645
2176841721
6882881134
4846848554
5283751526
The energy level of each octopus is a value between 0 and 9. Here, the top-left octopus has an energy level of 5, the bottom-right one has an energy level of 6, and so on.
You can model the energy levels and flashes of light in steps. During a single step, the following occurs:
First, the energy level of each octopus increases by 1.
Then, any octopus with an energy level greater than 9 flashes. This increases the energy level of all adjacent octopuses by 1, including octopuses that are diagonally adjacent. If this causes an octopus to have an energy level greater than 9, it also flashes. This process continues as long as new octopuses keep having their energy level increased beyond 9. (An octopus can only flash at most once per step.)
Finally, any octopus that flashed during this step has its energy level set to 0, as it used all of its energy to flash.
Adjacent flashes can cause an octopus to flash on a step even if it begins that step with very little energy. Consider the middle octopus with 1 energy in this situation:
Before any steps:
11111
19991
19191
19991
11111
After step 1:
34543
40004
50005
40004
34543
After step 2:
45654
51115
61116
51115
45654
An octopus is highlighted when it flashed during the given step.
Here is how the larger example above progresses:
Before any steps:
5483143223
2745854711
5264556173
6141336146
6357385478
4167524645
2176841721
6882881134
4846848554
5283751526
After step 1:
6594254334
3856965822
6375667284
7252447257
7468496589
5278635756
3287952832
7993992245
5957959665
6394862637
After step 2:
8807476555
5089087054
8597889608
8485769600
8700908800
6600088989
6800005943
0000007456
9000000876
8700006848
After step 3:
0050900866
8500800575
9900000039
9700000041
9935080063
7712300000
7911250009
2211130000
0421125000
0021119000
After step 4:
2263031977
0923031697
0032221150
0041111163
0076191174
0053411122
0042361120
5532241122
1532247211
1132230211
After step 5:
4484144000
2044144000
2253333493
1152333274
1187303285
1164633233
1153472231
6643352233
2643358322
2243341322
After step 6:
5595255111
3155255222
3364444605
2263444496
2298414396
2275744344
2264583342
7754463344
3754469433
3354452433
After step 7:
6707366222
4377366333
4475555827
3496655709
3500625609
3509955566
3486694453
8865585555
4865580644
4465574644
After step 8:
7818477333
5488477444
5697666949
4608766830
4734946730
4740097688
6900007564
0000009666
8000004755
6800007755
After step 9:
9060000644
7800000976
6900000080
5840000082
5858000093
6962400000
8021250009
2221130009
9111128097
7911119976
After step 10:
0481112976
0031112009
0041112504
0081111406
0099111306
0093511233
0442361130
5532252350
0532250600
0032240000
After step 10, there have been a total of 204 flashes. Fast forwarding, here is the same configuration every 10 steps:
After step 20:
3936556452
5686556806
4496555690
4448655580
4456865570
5680086577
7000009896
0000000344
6000000364
4600009543
After step 30:
0643334118
4253334611
3374333458
2225333337
2229333338
2276733333
2754574565
5544458511
9444447111
7944446119
After step 40:
6211111981
0421111119
0042111115
0003111115
0003111116
0065611111
0532351111
3322234597
2222222976
2222222762
After step 50:
9655556447
4865556805
4486555690
4458655580
4574865570
5700086566
6000009887
8000000533
6800000633
5680000538
After step 60:
2533334200
2743334640
2264333458
2225333337
2225333338
2287833333
3854573455
1854458611
1175447111
1115446111
After step 70:
8211111164
0421111166
0042111114
0004211115
0000211116
0065611111
0532351111
7322235117
5722223475
4572222754
After step 80:
1755555697
5965555609
4486555680
4458655580
4570865570
5700086566
7000008666
0000000990
0000000800
0000000000
After step 90:
7433333522
2643333522
2264333458
2226433337
2222433338
2287833333
2854573333
4854458333
3387779333
3333333333
After step 100:
0397666866
0749766918
0053976933
0004297822
0004229892
0053222877
0532222966
9322228966
7922286866
6789998766
After 100 steps, there have been a total of 1656 flashes.
Given the starting energy levels of the dumbo octopuses in your cavern, simulate 100 steps. How many total flashes are there after 100 steps?
| 472
|
--- Day 6: Chronal Coordinates ---
The device on your wrist beeps several times, and once again you feel like you're falling.
"Situation critical," the device announces. "Destination indeterminate. Chronal interference detected. Please specify new target coordinates."
The device then produces a list of coordinates (your puzzle input). Are they places it thinks are safe or dangerous? It recommends you check manual page 729. The Elves did not give you a manual.
If they're dangerous, maybe you can minimize the danger by finding the coordinate that gives the largest distance from the other points.
Using only the Manhattan distance, determine the area around each coordinate by counting the number of integer X,Y locations that are closest to that coordinate (and aren't tied in distance to any other coordinate).
Your goal is to find the size of the largest area that isn't infinite. For example, consider the following list of coordinates:
1, 1
1, 6
8, 3
3, 4
5, 5
8, 9
If we name these coordinates A through F, we can draw them on a grid, putting 0,0 at the top left:
..........
.A........
..........
........C.
...D......
.....E....
.B........
..........
..........
........F.
This view is partial - the actual grid extends infinitely in all directions. Using the Manhattan distance, each location's closest coordinate can be determined, shown here in lowercase:
aaaaa.cccc
aAaaa.cccc
aaaddecccc
aadddeccCc
..dDdeeccc
bb.deEeecc
bBb.eeee..
bbb.eeefff
bbb.eeffff
bbb.ffffFf
Locations shown as . are equally far from two or more coordinates, and so they don't count as being closest to any.
In this example, the areas of coordinates A, B, C, and F are infinite - while not shown here, their areas extend forever outside the visible grid. However, the areas of coordinates D and E are finite: D is closest to 9 locations, and E is closest to 17 (both including the coordinate's location itself). Therefore, in this example, the size of the largest area is 17.
What is the size of the largest area that isn't infinite?
Your puzzle answer was 4016.
--- Part Two ---
On the other hand, if the coordinates are safe, maybe the best you can do is try to find a region near as many coordinates as possible.
For example, suppose you want the sum of the Manhattan distance to all of the coordinates to be less than 32. For each location, add up the distances to all of the given coordinates; if the total of those distances is less than 32, that location is within the desired region. Using the same coordinates as above, the resulting region looks like this:
..........
.A........
..........
...###..C.
..#D###...
..###E#...
.B.###....
..........
..........
........F.
In particular, consider the highlighted location 4,3 located at the top middle of the region. Its calculation is as follows, where abs() is the absolute value function:
Distance to coordinate A: abs(4-1) + abs(3-1) = 5
Distance to coordinate B: abs(4-1) + abs(3-6) = 6
Distance to coordinate C: abs(4-8) + abs(3-3) = 4
Distance to coordinate D: abs(4-3) + abs(3-4) = 2
Distance to coordinate E: abs(4-5) + abs(3-5) = 3
Distance to coordinate F: abs(4-8) + abs(3-9) = 10
Total distance: 5 + 6 + 4 + 2 + 3 + 10 = 30
Because the total distance to all coordinates (30) is less than 32, the location is within the region.
This region, which also includes coordinates D and E, has a total size of 16.
Your actual region will need to be much larger than this example, though, instead including all locations with a total distance of less than 10000.
What is the size of the region containing all locations which have a total distance to all given coordinates of less than 10000?
| 473
|
--- Day 22: Slam Shuffle ---
There isn't much to do while you wait for the droids to repair your ship. At least you're drifting in the right direction. You decide to practice a new card shuffle you've been working on.
Digging through the ship's storage, you find a deck of space cards! Just like any deck of space cards, there are 10007 cards in the deck numbered 0 through 10006. The deck must be new - they're still in factory order, with 0 on the top, then 1, then 2, and so on, all the way through to 10006 on the bottom.
You've been practicing three different techniques that you use while shuffling. Suppose you have a deck of only 10 cards (numbered 0 through 9):
To deal into new stack, create a new stack of cards by dealing the top card of the deck onto the top of the new stack repeatedly until you run out of cards:
Top Bottom
0 1 2 3 4 5 6 7 8 9 Your deck
New stack
1 2 3 4 5 6 7 8 9 Your deck
0 New stack
2 3 4 5 6 7 8 9 Your deck
1 0 New stack
3 4 5 6 7 8 9 Your deck
2 1 0 New stack
Several steps later...
9 Your deck
8 7 6 5 4 3 2 1 0 New stack
Your deck
9 8 7 6 5 4 3 2 1 0 New stack
Finally, pick up the new stack you've just created and use it as the deck for the next technique.
To cut N cards, take the top N cards off the top of the deck and move them as a single unit to the bottom of the deck, retaining their order. For example, to cut 3:
Top Bottom
0 1 2 3 4 5 6 7 8 9 Your deck
3 4 5 6 7 8 9 Your deck
0 1 2 Cut cards
3 4 5 6 7 8 9 Your deck
0 1 2 Cut cards
3 4 5 6 7 8 9 0 1 2 Your deck
You've also been getting pretty good at a version of this technique where N is negative! In that case, cut (the absolute value of) N cards from the bottom of the deck onto the top. For example, to cut -4:
Top Bottom
0 1 2 3 4 5 6 7 8 9 Your deck
0 1 2 3 4 5 Your deck
6 7 8 9 Cut cards
0 1 2 3 4 5 Your deck
6 7 8 9 Cut cards
6 7 8 9 0 1 2 3 4 5 Your deck
To deal with increment N, start by clearing enough space on your table to lay out all of the cards individually in a long line. Deal the top card into the leftmost position. Then, move N positions to the right and deal the next card there. If you would move into a position past the end of the space on your table, wrap around and keep counting from the leftmost card again. Continue this process until you run out of cards.
For example, to deal with increment 3:
0 1 2 3 4 5 6 7 8 9 Your deck
. . . . . . . . . . Space on table
^ Current position
Deal the top card to the current position:
1 2 3 4 5 6 7 8 9 Your deck
0 . . . . . . . . . Space on table
^ Current position
Move the current position right 3:
1 2 3 4 5 6 7 8 9 Your deck
0 . . . . . . . . . Space on table
^ Current position
Deal the top card:
2 3 4 5 6 7 8 9 Your deck
0 . . 1 . . . . . . Space on table
^ Current position
Move right 3 and deal:
3 4 5 6 7 8 9 Your deck
0 . . 1 . . 2 . . . Space on table
^ Current position
Move right 3 and deal:
4 5 6 7 8 9 Your deck
0 . . 1 . . 2 . . 3 Space on table
^ Current position
Move right 3, wrapping around, and deal:
5 6 7 8 9 Your deck
0 . 4 1 . . 2 . . 3 Space on table
^ Current position
And so on:
0 7 4 1 8 5 2 9 6 3 Space on table
Positions on the table which already contain cards are still counted; they're not skipped. Of course, this technique is carefully designed so it will never put two cards in the same position or leave a position empty.
Finally, collect the cards on the table so that the leftmost card ends up at the top of your deck, the card to its right ends up just below the top card, and so on, until the rightmost card ends up at the bottom of the deck.
The complete shuffle process (your puzzle input) consists of applying many of these techniques. Here are some examples that combine techniques; they all start with a factory order deck of 10 cards:
deal with increment 7
deal into new stack
deal into new stack
Result: 0 3 6 9 2 5 8 1 4 7
cut 6
deal with increment 7
deal into new stack
Result: 3 0 7 4 1 8 5 2 9 6
deal with increment 7
deal with increment 9
cut -2
Result: 6 3 0 7 4 1 8 5 2 9
deal into new stack
cut -2
deal with increment 7
cut 8
cut -4
deal with increment 7
cut 3
deal with increment 9
deal with increment 3
cut -1
Result: 9 2 5 8 1 4 7 0 3 6
Positions within the deck count from 0 at the top, then 1 for the card immediately below the top card, and so on to the bottom. (That is, cards start in the position matching their number.)
After shuffling your factory order deck of 10007 cards, what is the position of card 2019?
| 474
|
--- Day 9: Disk Fragmenter ---
Another push of the button leaves you in the familiar hallways of some friendly amphipods! Good thing you each somehow got your own personal mini submarine. The Historians jet away in search of the Chief, mostly by driving directly into walls.
While The Historians quickly figure out how to pilot these things, you notice an amphipod in the corner struggling with his computer. He's trying to make more contiguous free space by compacting all of the files, but his program isn't working; you offer to help.
He shows you the disk map (your puzzle input) he's already generated. For example:
2333133121414131402
The disk map uses a dense format to represent the layout of files and free space on the disk. The digits alternate between indicating the length of a file and the length of free space.
So, a disk map like 12345 would represent a one-block file, two blocks of free space, a three-block file, four blocks of free space, and then a five-block file. A disk map like 90909 would represent three nine-block files in a row (with no free space between them).
Each file on disk also has an ID number based on the order of the files as they appear before they are rearranged, starting with ID 0. So, the disk map 12345 has three files: a one-block file with ID 0, a three-block file with ID 1, and a five-block file with ID 2. Using one character for each block where digits are the file ID and . is free space, the disk map 12345 represents these individual blocks:
0..111....22222
The first example above, 2333133121414131402, represents these individual blocks:
00...111...2...333.44.5555.6666.777.888899
The amphipod would like to move file blocks one at a time from the end of the disk to the leftmost free space block (until there are no gaps remaining between file blocks). For the disk map 12345, the process looks like this:
0..111....22222
02.111....2222.
022111....222..
0221112...22...
02211122..2....
022111222......
The first example requires a few more steps:
00...111...2...333.44.5555.6666.777.888899
009..111...2...333.44.5555.6666.777.88889.
0099.111...2...333.44.5555.6666.777.8888..
00998111...2...333.44.5555.6666.777.888...
009981118..2...333.44.5555.6666.777.88....
0099811188.2...333.44.5555.6666.777.8.....
009981118882...333.44.5555.6666.777.......
0099811188827..333.44.5555.6666.77........
00998111888277.333.44.5555.6666.7.........
009981118882777333.44.5555.6666...........
009981118882777333644.5555.666............
00998111888277733364465555.66.............
0099811188827773336446555566..............
The final step of this file-compacting process is to update the filesystem checksum. To calculate the checksum, add up the result of multiplying each of these blocks' position with the file ID number it contains. The leftmost block is in position 0. If a block contains free space, skip it instead.
Continuing the first example, the first few blocks' position multiplied by its file ID number are 0 * 0 = 0, 1 * 0 = 0, 2 * 9 = 18, 3 * 9 = 27, 4 * 8 = 32, and so on. In this example, the checksum is the sum of these, 1928.
Compact the amphipod's hard drive using the process he requested. What is the resulting filesystem checksum? (Be careful copy/pasting the input for this puzzle; it is a single, very long line.)
Your puzzle answer was 6154342787400.
The first half of this puzzle is complete! It provides one gold star: *
--- Part Two ---
Upon completion, two things immediately become clear. First, the disk definitely has a lot more contiguous free space, just like the amphipod hoped. Second, the computer is running much more slowly! Maybe introducing all of that file system fragmentation was a bad idea?
The eager amphipod already has a new plan: rather than move individual blocks, he'd like to try compacting the files on his disk by moving whole files instead.
This time, attempt to move whole files to the leftmost span of free space blocks that could fit the file. Attempt to move each file exactly once in order of decreasing file ID number starting with the file with the highest file ID number. If there is no span of free space to the left of a file that is large enough to fit the file, the file does not move.
The first example from above now proceeds differently:
00...111...2...333.44.5555.6666.777.888899
0099.111...2...333.44.5555.6666.777.8888..
0099.1117772...333.44.5555.6666.....8888..
0099.111777244.333....5555.6666.....8888..
00992111777.44.333....5555.6666.....8888..
The process of updating the filesystem checksum is the same; now, this example's checksum would be 2858.
Start over, now compacting the amphipod's hard drive using this new method instead. What is the resulting filesystem checksum?
| 475
|
--- Day 15: Oxygen System ---
Out here in deep space, many things can go wrong. Fortunately, many of those things have indicator lights. Unfortunately, one of those lights is lit: the oxygen system for part of the ship has failed!
According to the readouts, the oxygen system must have failed days ago after a rupture in oxygen tank two; that section of the ship was automatically sealed once oxygen levels went dangerously low. A single remotely-operated repair droid is your only option for fixing the oxygen system.
The Elves' care package included an Intcode program (your puzzle input) that you can use to remotely control the repair droid. By running that program, you can direct the repair droid to the oxygen system and fix the problem.
The remote control program executes the following steps in a loop forever:
Accept a movement command via an input instruction.
Send the movement command to the repair droid.
Wait for the repair droid to finish the movement operation.
Report on the status of the repair droid via an output instruction.
Only four movement commands are understood: north (1), south (2), west (3), and east (4). Any other command is invalid. The movements differ in direction, but not in distance: in a long enough east-west hallway, a series of commands like 4,4,4,4,3,3,3,3 would leave the repair droid back where it started.
The repair droid can reply with any of the following status codes:
0: The repair droid hit a wall. Its position has not changed.
1: The repair droid has moved one step in the requested direction.
2: The repair droid has moved one step in the requested direction; its new position is the location of the oxygen system.
You don't know anything about the area around the repair droid, but you can figure it out by watching the status codes.
For example, we can draw the area using D for the droid, # for walls, . for locations the droid can traverse, and empty space for unexplored locations. Then, the initial state looks like this:
D
To make the droid go north, send it 1. If it replies with 0, you know that location is a wall and that the droid didn't move:
#
D
To move east, send 4; a reply of 1 means the movement was successful:
#
.D
Then, perhaps attempts to move north (1), south (2), and east (4) are all met with replies of 0:
##
.D#
#
Now, you know the repair droid is in a dead end. Backtrack with 3 (which you already know will get a reply of 1 because you already know that location is open):
##
D.#
#
Then, perhaps west (3) gets a reply of 0, south (2) gets a reply of 1, south again (2) gets a reply of 0, and then west (3) gets a reply of 2:
##
#..#
D.#
#
Now, because of the reply of 2, you know you've found the oxygen system! In this example, it was only 2 moves away from the repair droid's starting position.
What is the fewest number of movement commands required to move the repair droid from its starting position to the location of the oxygen system?
Your puzzle answer was 238.
--- Part Two ---
You quickly repair the oxygen system; oxygen gradually fills the area.
Oxygen starts in the location containing the repaired oxygen system. It takes one minute for oxygen to spread to all open locations that are adjacent to a location that already contains oxygen. Diagonal locations are not adjacent.
In the example above, suppose you've used the droid to explore the area fully and have the following map (where locations that currently contain oxygen are marked O):
##
#..##
#.#..#
#.O.#
###
Initially, the only location which contains oxygen is the location of the repaired oxygen system. However, after one minute, the oxygen spreads to all open (.) locations that are adjacent to a location containing oxygen:
##
#..##
#.#..#
#OOO#
###
After a total of two minutes, the map looks like this:
##
#..##
#O#O.#
#OOO#
###
After a total of three minutes:
##
#O.##
#O#OO#
#OOO#
###
And finally, the whole region is full of oxygen after a total of four minutes:
##
#OO##
#O#OO#
#OOO#
###
So, in this example, all locations contain oxygen after 4 minutes.
Use the repair droid to get a complete map of the area. How many minutes will it take to fill with oxygen?
| 476
|
--- Day 4: Camp Cleanup ---
Space needs to be cleared before the last supplies can be unloaded from the ships, and so several Elves have been assigned the job of cleaning up sections of the camp. Every section has a unique ID number, and each Elf is assigned a range of section IDs.
However, as some of the Elves compare their section assignments with each other, they've noticed that many of the assignments overlap. To try to quickly find overlaps and reduce duplicated effort, the Elves pair up and make a big list of the section assignments for each pair (your puzzle input).
For example, consider the following list of section assignment pairs:
2-4,6-8
2-3,4-5
5-7,7-9
2-8,3-7
6-6,4-6
2-6,4-8
For the first few pairs, this list means:
Within the first pair of Elves, the first Elf was assigned sections 2-4 (sections 2, 3, and 4), while the second Elf was assigned sections 6-8 (sections 6, 7, 8).
The Elves in the second pair were each assigned two sections.
The Elves in the third pair were each assigned three sections: one got sections 5, 6, and 7, while the other also got 7, plus 8 and 9.
This example list uses single-digit section IDs to make it easier to draw; your actual list might contain larger numbers. Visually, these pairs of section assignments look like this:
.234..... 2-4
.....678. 6-8
.23...... 2-3
...45.... 4-5
....567.. 5-7
......789 7-9
.2345678. 2-8
..34567.. 3-7
.....6... 6-6
...456... 4-6
.23456... 2-6
...45678. 4-8
Some of the pairs have noticed that one of their assignments fully contains the other. For example, 2-8 fully contains 3-7, and 6-6 is fully contained by 4-6. In pairs where one assignment fully contains the other, one Elf in the pair would be exclusively cleaning sections their partner will already be cleaning, so these seem like the most in need of reconsideration. In this example, there are 2 such pairs.
In how many assignment pairs does one range fully contain the other?
| 477
|
--- Day 25: Let It Snow ---
Merry Christmas! Santa is booting up his weather machine; looks like you might get a white Christmas after all.
The weather machine beeps! On the console of the machine is a copy protection message asking you to enter a code from the instruction manual. Apparently, it refuses to run unless you give it that code. No problem; you'll just look up the code in the--
"Ho ho ho", Santa ponders aloud. "I can't seem to find the manual."
You look up the support number for the manufacturer and give them a call. Good thing, too - that 49th star wasn't going to earn itself.
"Oh, that machine is quite old!", they tell you. "That model went out of support six minutes ago, and we just finished shredding all of the manuals. I bet we can find you the code generation algorithm, though."
After putting you on hold for twenty minutes (your call is very important to them, it reminded you repeatedly), they finally find an engineer that remembers how the code system works.
The codes are printed on an infinite sheet of paper, starting in the top-left corner. The codes are filled in by diagonals: starting with the first row with an empty first box, the codes are filled in diagonally up and to the right. This process repeats until the infinite paper is covered. So, the first few codes are filled in in this order:
| 1 2 3 4 5 6
---+---+---+---+---+---+---+
1 | 1 3 6 10 15 21
2 | 2 5 9 14 20
3 | 4 8 13 19
4 | 7 12 18
5 | 11 17
6 | 16
For example, the 12th code would be written to row 4, column 2; the 15th code would be written to row 1, column 5.
The voice on the other end of the phone continues with how the codes are actually generated. The first code is 20151125. After that, each code is generated by taking the previous one, multiplying it by 252533, and then keeping the remainder from dividing that value by 33554393.
So, to find the second code (which ends up in row 2, column 1), start with the previous value, 20151125. Multiply it by 252533 to get 5088824049625. Then, divide that by 33554393, which leaves a remainder of 31916031. That remainder is the second code.
"Oh!", says the voice. "It looks like we missed a scrap from one of the manuals. Let me read it to you." You write down his numbers:
| 1 2 3 4 5 6
---+---------+---------+---------+---------+---------+---------+
1 | 20151125 18749137 17289845 30943339 10071777 33511524
2 | 31916031 21629792 16929656 7726640 15514188 4041754
3 | 16080970 8057251 1601130 7981243 11661866 16474243
4 | 24592653 32451966 21345942 9380097 10600672 31527494
5 | 77061 17552253 28094349 6899651 9250759 31663883
6 | 33071741 6796745 25397450 24659492 1534922 27995004
"Now remember", the voice continues, "that's not even all of the first few numbers; for example, you're missing the one at 7,1 that would come before 6,2. But, it should be enough to let your-- oh, it's time for lunch! Bye!" The call disconnects.
Santa looks nervous. Your puzzle input contains the message on the machine's console. What code do you give the machine?
| 478
|
--- Day 12: The N-Body Problem ---
The space near Jupiter is not a very safe place; you need to be careful of a big distracting red spot, extreme radiation, and a whole lot of moons swirling around. You decide to start by tracking the four largest moons: Io, Europa, Ganymede, and Callisto.
After a brief scan, you calculate the position of each moon (your puzzle input). You just need to simulate their motion so you can avoid them.
Each moon has a 3-dimensional position (x, y, and z) and a 3-dimensional velocity. The position of each moon is given in your scan; the x, y, and z velocity of each moon starts at 0.
Simulate the motion of the moons in time steps. Within each time step, first update the velocity of every moon by applying gravity. Then, once all moons' velocities have been updated, update the position of every moon by applying velocity. Time progresses by one step once all of the positions are updated.
To apply gravity, consider every pair of moons. On each axis (x, y, and z), the velocity of each moon changes by exactly +1 or -1 to pull the moons together. For example, if Ganymede has an x position of 3, and Callisto has a x position of 5, then Ganymede's x velocity changes by +1 (because 5 > 3) and Callisto's x velocity changes by -1 (because 3 < 5). However, if the positions on a given axis are the same, the velocity on that axis does not change for that pair of moons.
Once all gravity has been applied, apply velocity: simply add the velocity of each moon to its own position. For example, if Europa has a position of x=1, y=2, z=3 and a velocity of x=-2, y=0,z=3, then its new position would be x=-1, y=2, z=6. This process does not modify the velocity of any moon.
For example, suppose your scan reveals the following positions:
<x=-1, y=0, z=2>
<x=2, y=-10, z=-7>
<x=4, y=-8, z=8>
<x=3, y=5, z=-1>
Simulating the motion of these moons would produce the following:
After 0 steps:
pos=<x=-1, y= 0, z= 2>, vel=<x= 0, y= 0, z= 0>
pos=<x= 2, y=-10, z=-7>, vel=<x= 0, y= 0, z= 0>
pos=<x= 4, y= -8, z= 8>, vel=<x= 0, y= 0, z= 0>
pos=<x= 3, y= 5, z=-1>, vel=<x= 0, y= 0, z= 0>
After 1 step:
pos=<x= 2, y=-1, z= 1>, vel=<x= 3, y=-1, z=-1>
pos=<x= 3, y=-7, z=-4>, vel=<x= 1, y= 3, z= 3>
pos=<x= 1, y=-7, z= 5>, vel=<x=-3, y= 1, z=-3>
pos=<x= 2, y= 2, z= 0>, vel=<x=-1, y=-3, z= 1>
After 2 steps:
pos=<x= 5, y=-3, z=-1>, vel=<x= 3, y=-2, z=-2>
pos=<x= 1, y=-2, z= 2>, vel=<x=-2, y= 5, z= 6>
pos=<x= 1, y=-4, z=-1>, vel=<x= 0, y= 3, z=-6>
pos=<x= 1, y=-4, z= 2>, vel=<x=-1, y=-6, z= 2>
After 3 steps:
pos=<x= 5, y=-6, z=-1>, vel=<x= 0, y=-3, z= 0>
pos=<x= 0, y= 0, z= 6>, vel=<x=-1, y= 2, z= 4>
pos=<x= 2, y= 1, z=-5>, vel=<x= 1, y= 5, z=-4>
pos=<x= 1, y=-8, z= 2>, vel=<x= 0, y=-4, z= 0>
After 4 steps:
pos=<x= 2, y=-8, z= 0>, vel=<x=-3, y=-2, z= 1>
pos=<x= 2, y= 1, z= 7>, vel=<x= 2, y= 1, z= 1>
pos=<x= 2, y= 3, z=-6>, vel=<x= 0, y= 2, z=-1>
pos=<x= 2, y=-9, z= 1>, vel=<x= 1, y=-1, z=-1>
After 5 steps:
pos=<x=-1, y=-9, z= 2>, vel=<x=-3, y=-1, z= 2>
pos=<x= 4, y= 1, z= 5>, vel=<x= 2, y= 0, z=-2>
pos=<x= 2, y= 2, z=-4>, vel=<x= 0, y=-1, z= 2>
pos=<x= 3, y=-7, z=-1>, vel=<x= 1, y= 2, z=-2>
After 6 steps:
pos=<x=-1, y=-7, z= 3>, vel=<x= 0, y= 2, z= 1>
pos=<x= 3, y= 0, z= 0>, vel=<x=-1, y=-1, z=-5>
pos=<x= 3, y=-2, z= 1>, vel=<x= 1, y=-4, z= 5>
pos=<x= 3, y=-4, z=-2>, vel=<x= 0, y= 3, z=-1>
After 7 steps:
pos=<x= 2, y=-2, z= 1>, vel=<x= 3, y= 5, z=-2>
pos=<x= 1, y=-4, z=-4>, vel=<x=-2, y=-4, z=-4>
pos=<x= 3, y=-7, z= 5>, vel=<x= 0, y=-5, z= 4>
pos=<x= 2, y= 0, z= 0>, vel=<x=-1, y= 4, z= 2>
After 8 steps:
pos=<x= 5, y= 2, z=-2>, vel=<x= 3, y= 4, z=-3>
pos=<x= 2, y=-7, z=-5>, vel=<x= 1, y=-3, z=-1>
pos=<x= 0, y=-9, z= 6>, vel=<x=-3, y=-2, z= 1>
pos=<x= 1, y= 1, z= 3>, vel=<x=-1, y= 1, z= 3>
After 9 steps:
pos=<x= 5, y= 3, z=-4>, vel=<x= 0, y= 1, z=-2>
pos=<x= 2, y=-9, z=-3>, vel=<x= 0, y=-2, z= 2>
pos=<x= 0, y=-8, z= 4>, vel=<x= 0, y= 1, z=-2>
pos=<x= 1, y= 1, z= 5>, vel=<x= 0, y= 0, z= 2>
After 10 steps:
pos=<x= 2, y= 1, z=-3>, vel=<x=-3, y=-2, z= 1>
pos=<x= 1, y=-8, z= 0>, vel=<x=-1, y= 1, z= 3>
pos=<x= 3, y=-6, z= 1>, vel=<x= 3, y= 2, z=-3>
pos=<x= 2, y= 0, z= 4>, vel=<x= 1, y=-1, z=-1>
Then, it might help to calculate the total energy in the system. The total energy for a single moon is its potential energy multiplied by its kinetic energy. A moon's potential energy is the sum of the absolute values of its x, y, and z position coordinates. A moon's kinetic energy is the sum of the absolute values of its velocity coordinates. Below, each line shows the calculations for a moon's potential energy (pot), kinetic energy (kin), and total energy:
Energy after 10 steps:
pot: 2 + 1 + 3 = 6; kin: 3 + 2 + 1 = 6; total: 6 * 6 = 36
pot: 1 + 8 + 0 = 9; kin: 1 + 1 + 3 = 5; total: 9 * 5 = 45
pot: 3 + 6 + 1 = 10; kin: 3 + 2 + 3 = 8; total: 10 * 8 = 80
pot: 2 + 0 + 4 = 6; kin: 1 + 1 + 1 = 3; total: 6 * 3 = 18
Sum of total energy: 36 + 45 + 80 + 18 = 179
In the above example, adding together the total energy for all moons after 10 steps produces the total energy in the system, 179.
Here's a second example:
<x=-8, y=-10, z=0>
<x=5, y=5, z=10>
<x=2, y=-7, z=3>
<x=9, y=-8, z=-3>
Every ten steps of simulation for 100 steps produces:
After 0 steps:
pos=<x= -8, y=-10, z= 0>, vel=<x= 0, y= 0, z= 0>
pos=<x= 5, y= 5, z= 10>, vel=<x= 0, y= 0, z= 0>
pos=<x= 2, y= -7, z= 3>, vel=<x= 0, y= 0, z= 0>
pos=<x= 9, y= -8, z= -3>, vel=<x= 0, y= 0, z= 0>
After 10 steps:
pos=<x= -9, y=-10, z= 1>, vel=<x= -2, y= -2, z= -1>
pos=<x= 4, y= 10, z= 9>, vel=<x= -3, y= 7, z= -2>
pos=<x= 8, y=-10, z= -3>, vel=<x= 5, y= -1, z= -2>
pos=<x= 5, y=-10, z= 3>, vel=<x= 0, y= -4, z= 5>
After 20 steps:
pos=<x=-10, y= 3, z= -4>, vel=<x= -5, y= 2, z= 0>
pos=<x= 5, y=-25, z= 6>, vel=<x= 1, y= 1, z= -4>
pos=<x= 13, y= 1, z= 1>, vel=<x= 5, y= -2, z= 2>
pos=<x= 0, y= 1, z= 7>, vel=<x= -1, y= -1, z= 2>
After 30 steps:
pos=<x= 15, y= -6, z= -9>, vel=<x= -5, y= 4, z= 0>
pos=<x= -4, y=-11, z= 3>, vel=<x= -3, y=-10, z= 0>
pos=<x= 0, y= -1, z= 11>, vel=<x= 7, y= 4, z= 3>
pos=<x= -3, y= -2, z= 5>, vel=<x= 1, y= 2, z= -3>
After 40 steps:
pos=<x= 14, y=-12, z= -4>, vel=<x= 11, y= 3, z= 0>
pos=<x= -1, y= 18, z= 8>, vel=<x= -5, y= 2, z= 3>
pos=<x= -5, y=-14, z= 8>, vel=<x= 1, y= -2, z= 0>
pos=<x= 0, y=-12, z= -2>, vel=<x= -7, y= -3, z= -3>
After 50 steps:
pos=<x=-23, y= 4, z= 1>, vel=<x= -7, y= -1, z= 2>
pos=<x= 20, y=-31, z= 13>, vel=<x= 5, y= 3, z= 4>
pos=<x= -4, y= 6, z= 1>, vel=<x= -1, y= 1, z= -3>
pos=<x= 15, y= 1, z= -5>, vel=<x= 3, y= -3, z= -3>
After 60 steps:
pos=<x= 36, y=-10, z= 6>, vel=<x= 5, y= 0, z= 3>
pos=<x=-18, y= 10, z= 9>, vel=<x= -3, y= -7, z= 5>
pos=<x= 8, y=-12, z= -3>, vel=<x= -2, y= 1, z= -7>
pos=<x=-18, y= -8, z= -2>, vel=<x= 0, y= 6, z= -1>
After 70 steps:
pos=<x=-33, y= -6, z= 5>, vel=<x= -5, y= -4, z= 7>
pos=<x= 13, y= -9, z= 2>, vel=<x= -2, y= 11, z= 3>
pos=<x= 11, y= -8, z= 2>, vel=<x= 8, y= -6, z= -7>
pos=<x= 17, y= 3, z= 1>, vel=<x= -1, y= -1, z= -3>
After 80 steps:
pos=<x= 30, y= -8, z= 3>, vel=<x= 3, y= 3, z= 0>
pos=<x= -2, y= -4, z= 0>, vel=<x= 4, y=-13, z= 2>
pos=<x=-18, y= -7, z= 15>, vel=<x= -8, y= 2, z= -2>
pos=<x= -2, y= -1, z= -8>, vel=<x= 1, y= 8, z= 0>
After 90 steps:
pos=<x=-25, y= -1, z= 4>, vel=<x= 1, y= -3, z= 4>
pos=<x= 2, y= -9, z= 0>, vel=<x= -3, y= 13, z= -1>
pos=<x= 32, y= -8, z= 14>, vel=<x= 5, y= -4, z= 6>
pos=<x= -1, y= -2, z= -8>, vel=<x= -3, y= -6, z= -9>
After 100 steps:
pos=<x= 8, y=-12, z= -9>, vel=<x= -7, y= 3, z= 0>
pos=<x= 13, y= 16, z= -3>, vel=<x= 3, y=-11, z= -5>
pos=<x=-29, y=-11, z= -1>, vel=<x= -3, y= 7, z= 4>
pos=<x= 16, y=-13, z= 23>, vel=<x= 7, y= 1, z= 1>
Energy after 100 steps:
pot: 8 + 12 + 9 = 29; kin: 7 + 3 + 0 = 10; total: 29 * 10 = 290
pot: 13 + 16 + 3 = 32; kin: 3 + 11 + 5 = 19; total: 32 * 19 = 608
pot: 29 + 11 + 1 = 41; kin: 3 + 7 + 4 = 14; total: 41 * 14 = 574
pot: 16 + 13 + 23 = 52; kin: 7 + 1 + 1 = 9; total: 52 * 9 = 468
Sum of total energy: 290 + 608 + 574 + 468 = 1940
What is the total energy in the system after simulating the moons given in your scan for 1000 steps?
Your puzzle answer was 7636.
--- Part Two ---
All this drifting around in space makes you wonder about the nature of the universe. Does history really repeat itself? You're curious whether the moons will ever return to a previous state.
Determine the number of steps that must occur before all of the moons' positions and velocities exactly match a previous point in time.
For example, the first example above takes 2772 steps before they exactly match a previous point in time; it eventually returns to the initial state:
After 0 steps:
pos=<x= -1, y= 0, z= 2>, vel=<x= 0, y= 0, z= 0>
pos=<x= 2, y=-10, z= -7>, vel=<x= 0, y= 0, z= 0>
pos=<x= 4, y= -8, z= 8>, vel=<x= 0, y= 0, z= 0>
pos=<x= 3, y= 5, z= -1>, vel=<x= 0, y= 0, z= 0>
After 2770 steps:
pos=<x= 2, y= -1, z= 1>, vel=<x= -3, y= 2, z= 2>
pos=<x= 3, y= -7, z= -4>, vel=<x= 2, y= -5, z= -6>
pos=<x= 1, y= -7, z= 5>, vel=<x= 0, y= -3, z= 6>
pos=<x= 2, y= 2, z= 0>, vel=<x= 1, y= 6, z= -2>
After 2771 steps:
pos=<x= -1, y= 0, z= 2>, vel=<x= -3, y= 1, z= 1>
pos=<x= 2, y=-10, z= -7>, vel=<x= -1, y= -3, z= -3>
pos=<x= 4, y= -8, z= 8>, vel=<x= 3, y= -1, z= 3>
pos=<x= 3, y= 5, z= -1>, vel=<x= 1, y= 3, z= -1>
After 2772 steps:
pos=<x= -1, y= 0, z= 2>, vel=<x= 0, y= 0, z= 0>
pos=<x= 2, y=-10, z= -7>, vel=<x= 0, y= 0, z= 0>
pos=<x= 4, y= -8, z= 8>, vel=<x= 0, y= 0, z= 0>
pos=<x= 3, y= 5, z= -1>, vel=<x= 0, y= 0, z= 0>
Of course, the universe might last for a very long time before repeating. Here's a copy of the second example from above:
<x=-8, y=-10, z=0>
<x=5, y=5, z=10>
<x=2, y=-7, z=3>
<x=9, y=-8, z=-3>
This set of initial positions takes 4686774924 steps before it repeats a previous state! Clearly, you might need to find a more efficient way to simulate the universe.
How many steps does it take to reach the first state that exactly matches a previous state?
| 479
|
--- Day 6: Signals and Noise ---
Something is jamming your communications with Santa. Fortunately, your signal is only partially jammed, and protocol in situations like this is to switch to a simple repetition code to get the message through.
In this model, the same message is sent repeatedly. You've recorded the repeating message signal (your puzzle input), but the data seems quite corrupted - almost too badly to recover. Almost.
All you need to do is figure out which character is most frequent for each position. For example, suppose you had recorded the following messages:
eedadn
drvtee
eandsr
raavrd
atevrs
tsrnev
sdttsa
rasrtv
nssdts
ntnada
svetve
tesnvt
vntsnd
vrdear
dvrsen
enarar
The most common character in the first column is e; in the second, a; in the third, s, and so on. Combining these characters returns the error-corrected message, easter.
Given the recording in your puzzle input, what is the error-corrected version of the message being sent?
| 480
|
--- Day 8: Space Image Format ---
The Elves' spirits are lifted when they realize you have an opportunity to reboot one of their Mars rovers, and so they are curious if you would spend a brief sojourn on Mars. You land your ship near the rover.
When you reach the rover, you discover that it's already in the process of rebooting! It's just waiting for someone to enter a BIOS password. The Elf responsible for the rover takes a picture of the password (your puzzle input) and sends it to you via the Digital Sending Network.
Unfortunately, images sent via the Digital Sending Network aren't encoded with any normal encoding; instead, they're encoded in a special Space Image Format. None of the Elves seem to remember why this is the case. They send you the instructions to decode it.
Images are sent as a series of digits that each represent the color of a single pixel. The digits fill each row of the image left-to-right, then move downward to the next row, filling rows top-to-bottom until every pixel of the image is filled.
Each image actually consists of a series of identically-sized layers that are filled in this way. So, the first digit corresponds to the top-left pixel of the first layer, the second digit corresponds to the pixel to the right of that on the same layer, and so on until the last digit, which corresponds to the bottom-right pixel of the last layer.
For example, given an image 3 pixels wide and 2 pixels tall, the image data 123456789012 corresponds to the following image layers:
Layer 1: 123
456
Layer 2: 789
012
The image you received is 25 pixels wide and 6 pixels tall.
To make sure the image wasn't corrupted during transmission, the Elves would like you to find the layer that contains the fewest 0 digits. On that layer, what is the number of 1 digits multiplied by the number of 2 digits?
Your puzzle answer was 1224.
--- Part Two ---
Now you're ready to decode the image. The image is rendered by stacking the layers and aligning the pixels with the same positions in each layer. The digits indicate the color of the corresponding pixel: 0 is black, 1 is white, and 2 is transparent.
The layers are rendered with the first layer in front and the last layer in back. So, if a given position has a transparent pixel in the first and second layers, a black pixel in the third layer, and a white pixel in the fourth layer, the final image would have a black pixel at that position.
For example, given an image 2 pixels wide and 2 pixels tall, the image data 0222112222120000 corresponds to the following image layers:
Layer 1: 02
22
Layer 2: 11
22
Layer 3: 22
12
Layer 4: 00
00
Then, the full image can be found by determining the top visible pixel in each position:
The top-left pixel is black because the top layer is 0.
The top-right pixel is white because the top layer is 2 (transparent), but the second layer is 1.
The bottom-left pixel is white because the top two layers are 2, but the third layer is 1.
The bottom-right pixel is black because the only visible pixel in that position is 0 (from layer 4).
So, the final image looks like this:
01
10
What message is produced after decoding your image?
| 481
|
--- Day 12: Passage Pathing ---
With your submarine's subterranean subsystems subsisting suboptimally, the only way you're getting out of this cave anytime soon is by finding a path yourself. Not just a path - the only way to know if you've found the best path is to find all of them.
Fortunately, the sensors are still mostly working, and so you build a rough map of the remaining caves (your puzzle input). For example:
start-A
start-b
A-c
A-b
b-d
A-end
b-end
This is a list of how all of the caves are connected. You start in the cave named start, and your destination is the cave named end. An entry like b-d means that cave b is connected to cave d - that is, you can move between them.
So, the above cave system looks roughly like this:
start
/ c--A-----b--d
/
end
Your goal is to find the number of distinct paths that start at start, end at end, and don't visit small caves more than once. There are two types of caves: big caves (written in uppercase, like A) and small caves (written in lowercase, like b). It would be a waste of time to visit any small cave more than once, but big caves are large enough that it might be worth visiting them multiple times. So, all paths you find should visit small caves at most once, and can visit big caves any number of times.
Given these rules, there are 10 paths through this example cave system:
start,A,b,A,c,A,end
start,A,b,A,end
start,A,b,end
start,A,c,A,b,A,end
start,A,c,A,b,end
start,A,c,A,end
start,A,end
start,b,A,c,A,end
start,b,A,end
start,b,end
(Each line in the above list corresponds to a single path; the caves visited by that path are listed in the order they are visited and separated by commas.)
Note that in this cave system, cave d is never visited by any path: to do so, cave b would need to be visited twice (once on the way to cave d and a second time when returning from cave d), and since cave b is small, this is not allowed.
Here is a slightly larger example:
dc-end
HN-start
start-kj
dc-start
dc-HN
LN-dc
HN-end
kj-sa
kj-HN
kj-dc
The 19 paths through it are as follows:
start,HN,dc,HN,end
start,HN,dc,HN,kj,HN,end
start,HN,dc,end
start,HN,dc,kj,HN,end
start,HN,end
start,HN,kj,HN,dc,HN,end
start,HN,kj,HN,dc,end
start,HN,kj,HN,end
start,HN,kj,dc,HN,end
start,HN,kj,dc,end
start,dc,HN,end
start,dc,HN,kj,HN,end
start,dc,end
start,dc,kj,HN,end
start,kj,HN,dc,HN,end
start,kj,HN,dc,end
start,kj,HN,end
start,kj,dc,HN,end
start,kj,dc,end
Finally, this even larger example has 226 paths through it:
fs-end
he-DX
fs-he
start-DX
pj-DX
end-zg
zg-sl
zg-pj
pj-he
RW-he
fs-DX
pj-RW
zg-RW
start-pj
he-WI
zg-he
pj-fs
start-RW
How many paths through this cave system are there that visit small caves at most once?
| 482
|
--- Day 18: Like a Rogue ---
As you enter this room, you hear a loud click! Some of the tiles in the floor here seem to be pressure plates for traps, and the trap you just triggered has run out of... whatever it tried to do to you. You doubt you'll be so lucky next time.
Upon closer examination, the traps and safe tiles in this room seem to follow a pattern. The tiles are arranged into rows that are all the same width; you take note of the safe tiles (.) and traps (^) in the first row (your puzzle input).
The type of tile (trapped or safe) in each row is based on the types of the tiles in the same position, and to either side of that position, in the previous row. (If either side is off either end of the row, it counts as "safe" because there isn't a trap embedded in the wall.)
For example, suppose you know the first row (with tiles marked by letters) and want to determine the next row (with tiles marked by numbers):
ABCDE
12345
The type of tile 2 is based on the types of tiles A, B, and C; the type of tile 5 is based on tiles D, E, and an imaginary "safe" tile. Let's call these three tiles from the previous row the left, center, and right tiles, respectively. Then, a new tile is a trap only in one of the following situations:
Its left and center tiles are traps, but its right tile is not.
Its center and right tiles are traps, but its left tile is not.
Only its left tile is a trap.
Only its right tile is a trap.
In any other situation, the new tile is safe.
Then, starting with the row ..^^., you can determine the next row by applying those rules to each new tile:
The leftmost character on the next row considers the left (nonexistent, so we assume "safe"), center (the first ., which means "safe"), and right (the second ., also "safe") tiles on the previous row. Because all of the trap rules require a trap in at least one of the previous three tiles, the first tile on this new row is also safe, ..
The second character on the next row considers its left (.), center (.), and right (^) tiles from the previous row. This matches the fourth rule: only the right tile is a trap. Therefore, the next tile in this new row is a trap, ^.
The third character considers .^^, which matches the second trap rule: its center and right tiles are traps, but its left tile is not. Therefore, this tile is also a trap, ^.
The last two characters in this new row match the first and third rules, respectively, and so they are both also traps, ^.
After these steps, we now know the next row of tiles in the room: .^^^^. Then, we continue on to the next row, using the same rules, and get ^^..^. After determining two new rows, our map looks like this:
..^^.
.^^^^
^^..^
Here's a larger example with ten tiles per row and ten rows:
.^^.^.^^^^
^^^...^..^
^.^^.^.^^.
..^^...^^^
.^^^^.^^.^
^^..^.^^..
^^^^..^^^.
^..^^^^.^^
.^^^..^.^^
^^.^^^..^^
In ten rows, this larger example has 38 safe tiles.
Starting with the map in your puzzle input, in a total of 40 rows (including the starting row), how many safe tiles are there?
Your puzzle answer was 1951.
--- Part Two ---
How many safe tiles are there in a total of 400000 rows?
| 483
|
--- Day 5: Binary Boarding ---
You board your plane only to discover a new problem: you dropped your boarding pass! You aren't sure which seat is yours, and all of the flight attendants are busy with the flood of people that suddenly made it through passport control.
You write a quick program to use your phone's camera to scan all of the nearby boarding passes (your puzzle input); perhaps you can find your seat through process of elimination.
Instead of zones or groups, this airline uses binary space partitioning to seat people. A seat might be specified like FBFBBFFRLR, where F means "front", B means "back", L means "left", and R means "right".
The first 7 characters will either be F or B; these specify exactly one of the 128 rows on the plane (numbered 0 through 127). Each letter tells you which half of a region the given seat is in. Start with the whole list of rows; the first letter indicates whether the seat is in the front (0 through 63) or the back (64 through 127). The next letter indicates which half of that region the seat is in, and so on until you're left with exactly one row.
For example, consider just the first seven characters of FBFBBFFRLR:
Start by considering the whole range, rows 0 through 127.
F means to take the lower half, keeping rows 0 through 63.
B means to take the upper half, keeping rows 32 through 63.
F means to take the lower half, keeping rows 32 through 47.
B means to take the upper half, keeping rows 40 through 47.
B keeps rows 44 through 47.
F keeps rows 44 through 45.
The final F keeps the lower of the two, row 44.
The last three characters will be either L or R; these specify exactly one of the 8 columns of seats on the plane (numbered 0 through 7). The same process as above proceeds again, this time with only three steps. L means to keep the lower half, while R means to keep the upper half.
For example, consider just the last 3 characters of FBFBBFFRLR:
Start by considering the whole range, columns 0 through 7.
R means to take the upper half, keeping columns 4 through 7.
L means to take the lower half, keeping columns 4 through 5.
The final R keeps the upper of the two, column 5.
So, decoding FBFBBFFRLR reveals that it is the seat at row 44, column 5.
Every seat also has a unique seat ID: multiply the row by 8, then add the column. In this example, the seat has ID 44 * 8 + 5 = 357.
Here are some other boarding passes:
BFFFBBFRRR: row 70, column 7, seat ID 567.
FFFBBBFRRR: row 14, column 7, seat ID 119.
BBFFBBFRLL: row 102, column 4, seat ID 820.
As a sanity check, look through your list of boarding passes. What is the highest seat ID on a boarding pass?
Your puzzle answer was 828.
--- Part Two ---
Ding! The "fasten seat belt" signs have turned on. Time to find your seat.
It's a completely full flight, so your seat should be the only missing boarding pass in your list. However, there's a catch: some of the seats at the very front and back of the plane don't exist on this aircraft, so they'll be missing from your list as well.
Your seat wasn't at the very front or back, though; the seats with IDs +1 and -1 from yours will be in your list.
What is the ID of your seat?
| 484
|
--- Day 13: Shuttle Search ---
Your ferry can make it safely to a nearby port, but it won't get much further. When you call to book another ship, you discover that no ships embark from that port to your vacation island. You'll need to get from the port to the nearest airport.
Fortunately, a shuttle bus service is available to bring you from the sea port to the airport! Each bus has an ID number that also indicates how often the bus leaves for the airport.
Bus schedules are defined based on a timestamp that measures the number of minutes since some fixed reference point in the past. At timestamp 0, every bus simultaneously departed from the sea port. After that, each bus travels to the airport, then various other locations, and finally returns to the sea port to repeat its journey forever.
The time this loop takes a particular bus is also its ID number: the bus with ID 5 departs from the sea port at timestamps 0, 5, 10, 15, and so on. The bus with ID 11 departs at 0, 11, 22, 33, and so on. If you are there when the bus departs, you can ride that bus to the airport!
Your notes (your puzzle input) consist of two lines. The first line is your estimate of the earliest timestamp you could depart on a bus. The second line lists the bus IDs that are in service according to the shuttle company; entries that show x must be out of service, so you decide to ignore them.
To save time once you arrive, your goal is to figure out the earliest bus you can take to the airport. (There will be exactly one such bus.)
For example, suppose you have the following notes:
939
7,13,x,x,59,x,31,19
Here, the earliest timestamp you could depart is 939, and the bus IDs in service are 7, 13, 59, 31, and 19. Near timestamp 939, these bus IDs depart at the times marked D:
time bus 7 bus 13 bus 59 bus 31 bus 19
929 . . . . .
930 . . . D .
931 D . . . D
932 . . . . .
933 . . . . .
934 . . . . .
935 . . . . .
936 . D . . .
937 . . . . .
938 D . . . .
939 . . . . .
940 . . . . .
941 . . . . .
942 . . . . .
943 . . . . .
944 . . D . .
945 D . . . .
946 . . . . .
947 . . . . .
948 . . . . .
949 . D . . .
The earliest bus you could take is bus ID 59. It doesn't depart until timestamp 944, so you would need to wait 944 - 939 = 5 minutes before it departs. Multiplying the bus ID by the number of minutes you'd need to wait gives 295.
What is the ID of the earliest bus you can take to the airport multiplied by the number of minutes you'll need to wait for that bus?
| 485
|
--- Day 14: Restroom Redoubt ---
One of The Historians needs to use the bathroom; fortunately, you know there's a bathroom near an unvisited location on their list, and so you're all quickly teleported directly to the lobby of Easter Bunny Headquarters.
Unfortunately, EBHQ seems to have "improved" bathroom security again after your last visit. The area outside the bathroom is swarming with robots!
To get The Historian safely to the bathroom, you'll need a way to predict where the robots will be in the future. Fortunately, they all seem to be moving on the tile floor in predictable straight lines.
You make a list (your puzzle input) of all of the robots' current positions (p) and velocities (v), one robot per line. For example:
p=0,4 v=3,-3
p=6,3 v=-1,-3
p=10,3 v=-1,2
p=2,0 v=2,-1
p=0,0 v=1,3
p=3,0 v=-2,-2
p=7,6 v=-1,-3
p=3,0 v=-1,-2
p=9,3 v=2,3
p=7,3 v=-1,2
p=2,4 v=2,-3
p=9,5 v=-3,-3
Each robot's position is given as p=x,y where x represents the number of tiles the robot is from the left wall and y represents the number of tiles from the top wall (when viewed from above). So, a position of p=0,0 means the robot is all the way in the top-left corner.
Each robot's velocity is given as v=x,y where x and y are given in tiles per second. Positive x means the robot is moving to the right, and positive y means the robot is moving down. So, a velocity of v=1,-2 means that each second, the robot moves 1 tile to the right and 2 tiles up.
The robots outside the actual bathroom are in a space which is 101 tiles wide and 103 tiles tall (when viewed from above). However, in this example, the robots are in a space which is only 11 tiles wide and 7 tiles tall.
The robots are good at navigating over/under each other (due to a combination of springs, extendable legs, and quadcopters), so they can share the same tile and don't interact with each other. Visually, the number of robots on each tile in this example looks like this:
1.12.......
...........
...........
......11.11
1.1........
.........1.
.......1...
These robots have a unique feature for maximum bathroom security: they can teleport. When a robot would run into an edge of the space they're in, they instead teleport to the other side, effectively wrapping around the edges. Here is what robot p=2,4 v=2,-3 does for the first few seconds:
Initial state:
...........
...........
...........
...........
..1........
...........
...........
After 1 second:
...........
....1......
...........
...........
...........
...........
...........
After 2 seconds:
...........
...........
...........
...........
...........
......1....
...........
After 3 seconds:
...........
...........
........1..
...........
...........
...........
...........
After 4 seconds:
...........
...........
...........
...........
...........
...........
..........1
After 5 seconds:
...........
...........
...........
.1.........
...........
...........
...........
The Historian can't wait much longer, so you don't have to simulate the robots for very long. Where will the robots be after 100 seconds?
In the above example, the number of robots on each tile after 100 seconds has elapsed looks like this:
......2..1.
...........
1..........
.11........
.....1.....
...12......
.1....1....
To determine the safest area, count the number of robots in each quadrant after 100 seconds. Robots that are exactly in the middle (horizontally or vertically) don't count as being in any quadrant, so the only relevant robots are:
..... 2..1.
..... .....
1.... .....
..... .....
...12 .....
.1... 1....
In this example, the quadrants contain 1, 3, 4, and 1 robot. Multiplying these together gives a total safety factor of 12.
Predict the motion of the robots in your list within a space which is 101 tiles wide and 103 tiles tall. What will the safety factor be after exactly 100 seconds have elapsed?
| 486
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--- Day 25: Clock Signal ---
You open the door and find yourself on the roof. The city sprawls away from you for miles and miles.
There's not much time now - it's already Christmas, but you're nowhere near the North Pole, much too far to deliver these stars to the sleigh in time.
However, maybe the huge antenna up here can offer a solution. After all, the sleigh doesn't need the stars, exactly; it needs the timing data they provide, and you happen to have a massive signal generator right here.
You connect the stars you have to your prototype computer, connect that to the antenna, and begin the transmission.
Nothing happens.
You call the service number printed on the side of the antenna and quickly explain the situation. "I'm not sure what kind of equipment you have connected over there," he says, "but you need a clock signal." You try to explain that this is a signal for a clock.
"No, no, a clock signal - timing information so the antenna computer knows how to read the data you're sending it. An endless, alternating pattern of 0, 1, 0, 1, 0, 1, 0, 1, 0, 1...." He trails off.
You ask if the antenna can handle a clock signal at the frequency you would need to use for the data from the stars. "There's no way it can! The only antenna we've installed capable of that is on top of a top-secret Easter Bunny installation, and you're definitely not-" You hang up the phone.
You've extracted the antenna's clock signal generation assembunny code (your puzzle input); it looks mostly compatible with code you worked on just recently.
This antenna code, being a signal generator, uses one extra instruction:
out x transmits x (either an integer or the value of a register) as the next value for the clock signal.
The code takes a value (via register a) that describes the signal to generate, but you're not sure how it's used. You'll have to find the input to produce the right signal through experimentation.
What is the lowest positive integer that can be used to initialize register a and cause the code to output a clock signal of 0, 1, 0, 1... repeating forever?
| 487
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--- Day 14: Reindeer Olympics ---
This year is the Reindeer Olympics! Reindeer can fly at high speeds, but must rest occasionally to recover their energy. Santa would like to know which of his reindeer is fastest, and so he has them race.
Reindeer can only either be flying (always at their top speed) or resting (not moving at all), and always spend whole seconds in either state.
For example, suppose you have the following Reindeer:
Comet can fly 14 km/s for 10 seconds, but then must rest for 127 seconds.
Dancer can fly 16 km/s for 11 seconds, but then must rest for 162 seconds.
After one second, Comet has gone 14 km, while Dancer has gone 16 km. After ten seconds, Comet has gone 140 km, while Dancer has gone 160 km. On the eleventh second, Comet begins resting (staying at 140 km), and Dancer continues on for a total distance of 176 km. On the 12th second, both reindeer are resting. They continue to rest until the 138th second, when Comet flies for another ten seconds. On the 174th second, Dancer flies for another 11 seconds.
In this example, after the 1000th second, both reindeer are resting, and Comet is in the lead at 1120 km (poor Dancer has only gotten 1056 km by that point). So, in this situation, Comet would win (if the race ended at 1000 seconds).
Given the descriptions of each reindeer (in your puzzle input), after exactly 2503 seconds, what distance has the winning reindeer traveled?
Your puzzle answer was 2696.
--- Part Two ---
Seeing how reindeer move in bursts, Santa decides he's not pleased with the old scoring system.
Instead, at the end of each second, he awards one point to the reindeer currently in the lead. (If there are multiple reindeer tied for the lead, they each get one point.) He keeps the traditional 2503 second time limit, of course, as doing otherwise would be entirely ridiculous.
Given the example reindeer from above, after the first second, Dancer is in the lead and gets one point. He stays in the lead until several seconds into Comet's second burst: after the 140th second, Comet pulls into the lead and gets his first point. Of course, since Dancer had been in the lead for the 139 seconds before that, he has accumulated 139 points by the 140th second.
After the 1000th second, Dancer has accumulated 689 points, while poor Comet, our old champion, only has 312. So, with the new scoring system, Dancer would win (if the race ended at 1000 seconds).
Again given the descriptions of each reindeer (in your puzzle input), after exactly 2503 seconds, how many points does the winning reindeer have?
| 488
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--- Day 5: A Maze of Twisty Trampolines, All Alike ---
An urgent interrupt arrives from the CPU: it's trapped in a maze of jump instructions, and it would like assistance from any programs with spare cycles to help find the exit.
The message includes a list of the offsets for each jump. Jumps are relative: -1 moves to the previous instruction, and 2 skips the next one. Start at the first instruction in the list. The goal is to follow the jumps until one leads outside the list.
In addition, these instructions are a little strange; after each jump, the offset of that instruction increases by 1. So, if you come across an offset of 3, you would move three instructions forward, but change it to a 4 for the next time it is encountered.
For example, consider the following list of jump offsets:
0
3
0
1
-3
Positive jumps ("forward") move downward; negative jumps move upward. For legibility in this example, these offset values will be written all on one line, with the current instruction marked in parentheses. The following steps would be taken before an exit is found:
(0) 3 0 1 -3 - before we have taken any steps.
(1) 3 0 1 -3 - jump with offset 0 (that is, don't jump at all). Fortunately, the instruction is then incremented to 1.
2 (3) 0 1 -3 - step forward because of the instruction we just modified. The first instruction is incremented again, now to 2.
2 4 0 1 (-3) - jump all the way to the end; leave a 4 behind.
2 (4) 0 1 -2 - go back to where we just were; increment -3 to -2.
2 5 0 1 -2 - jump 4 steps forward, escaping the maze.
In this example, the exit is reached in 5 steps.
How many steps does it take to reach the exit?
| 489
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Title: Shockers
Time Limit: None seconds
Memory Limit: None megabytes
Problem Description:
Valentin participates in a show called "Shockers". The rules are quite easy: jury selects one letter which Valentin doesn't know. He should make a small speech, but every time he pronounces a word that contains the selected letter, he receives an electric shock. He can make guesses which letter is selected, but for each incorrect guess he receives an electric shock too. The show ends when Valentin guesses the selected letter correctly.
Valentin can't keep in mind everything, so he could guess the selected letter much later than it can be uniquely determined and get excessive electric shocks. Excessive electric shocks are those which Valentin got after the moment the selected letter can be uniquely determined. You should find out the number of excessive electric shocks.
Input Specification:
The first line contains a single integer *n* (1<=≤<=*n*<=≤<=105) — the number of actions Valentin did.
The next *n* lines contain descriptions of his actions, each line contains description of one action. Each action can be of one of three types:
1. Valentin pronounced some word and didn't get an electric shock. This action is described by the string ". w" (without quotes), in which "." is a dot (ASCII-code 46), and *w* is the word that Valentin said. 1. Valentin pronounced some word and got an electric shock. This action is described by the string "! w" (without quotes), in which "!" is an exclamation mark (ASCII-code 33), and *w* is the word that Valentin said. 1. Valentin made a guess about the selected letter. This action is described by the string "? s" (without quotes), in which "?" is a question mark (ASCII-code 63), and *s* is the guess — a lowercase English letter.
All words consist only of lowercase English letters. The total length of all words does not exceed 105.
It is guaranteed that last action is a guess about the selected letter. Also, it is guaranteed that Valentin didn't make correct guesses about the selected letter before the last action. Moreover, it's guaranteed that if Valentin got an electric shock after pronouncing some word, then it contains the selected letter; and also if Valentin didn't get an electric shock after pronouncing some word, then it does not contain the selected letter.
Output Specification:
Output a single integer — the number of electric shocks that Valentin could have avoided if he had told the selected letter just after it became uniquely determined.
Demo Input:
['5\n! abc\n. ad\n. b\n! cd\n? c\n', '8\n! hello\n! codeforces\n? c\n. o\n? d\n? h\n. l\n? e\n', '7\n! ababahalamaha\n? a\n? b\n? a\n? b\n? a\n? h\n']
Demo Output:
['1\n', '2\n', '0\n']
Note:
In the first test case after the first action it becomes clear that the selected letter is one of the following: *a*, *b*, *c*. After the second action we can note that the selected letter is not *a*. Valentin tells word "b" and doesn't get a shock. After that it is clear that the selected letter is *c*, but Valentin pronounces the word *cd* and gets an excessive electric shock.
In the second test case after the first two electric shocks we understand that the selected letter is *e* or *o*. Valentin tries some words consisting of these letters and after the second word it's clear that the selected letter is *e*, but Valentin makes 3 more actions before he makes a correct hypothesis.
In the third example the selected letter can be uniquely determined only when Valentin guesses it, so he didn't get excessive electric shocks.
| 490
|
Title: Flags
Time Limit: 2 seconds
Memory Limit: 256 megabytes
Problem Description:
When Igor K. was a freshman, his professor strictly urged him, as well as all other freshmen, to solve programming Olympiads. One day a problem called "Flags" from a website called Timmy's Online Judge caught his attention. In the problem one had to find the number of three-colored flags that would satisfy the condition... actually, it doesn't matter. Igor K. quickly found the formula and got the so passionately desired Accepted.
However, the professor wasn't very much impressed. He decided that the problem represented on Timmy's Online Judge was very dull and simple: it only had three possible colors of flag stripes and only two limitations. He suggested a complicated task to Igor K. and the fellow failed to solve it. Of course, we won't tell anybody that the professor couldn't solve it as well.
And how about you? Can you solve the problem?
The flags consist of one or several parallel stripes of similar width. The stripes can be one of the following colors: white, black, red or yellow. You should find the number of different flags with the number of stripes from *L* to *R*, if:
- a flag cannot have adjacent stripes of one color; - a flag cannot have adjacent white and yellow stripes; - a flag cannot have adjacent red and black stripes; - a flag cannot have the combination of black, white and red stripes following one after another in this or reverse order; - symmetrical flags (as, for example, a WB and a BW flag, where W and B stand for the white and black colors) are considered the same.
Input Specification:
The only line contains two integers *L* and *R* (1<=≤<=*L*<=≤<=*R*<=≤<=109). They are the lower and upper borders of the number of stripes on the flag.
Output Specification:
Print a single number — the number of different flags that would satisfy the condition of the problem and would have from *L* to *R* stripes, modulo 1000000007.
Demo Input:
['3 4\n', '5 6\n']
Demo Output:
['23', '64']
Note:
In the first test the following flags exist (they are listed in the lexicographical order, the letters B, R, W, Y stand for Black, Red, White and Yellow correspondingly):
3 stripes: BWB, BYB, BYR, RWR, RYR, WBW, WBY, WRW, WRY, YBY, YRY (overall 11 flags).
4 stripes: BWBW, BWBY, BYBW, BYBY, BYRW, BYRY, RWRW, RWRY, RYBW, RYBY, RYRW, RYRY (12 flags).
That's why the answer to test 1 is equal to 11 + 12 = 23.
| 491
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Title: Door Frames
Time Limit: None seconds
Memory Limit: None megabytes
Problem Description:
Petya has equal wooden bars of length *n*. He wants to make a frame for two equal doors. Each frame has two vertical (left and right) sides of length *a* and one top side of length *b*. A solid (i.e. continuous without breaks) piece of bar is needed for each side.
Determine a minimal number of wooden bars which are needed to make the frames for two doors. Petya can cut the wooden bars into any parts, but each side of each door should be a solid piece of a wooden bar (or a whole wooden bar).
Input Specification:
The first line contains a single integer *n* (1<=≤<=*n*<=≤<=1<=000) — the length of each wooden bar.
The second line contains a single integer *a* (1<=≤<=*a*<=≤<=*n*) — the length of the vertical (left and right) sides of a door frame.
The third line contains a single integer *b* (1<=≤<=*b*<=≤<=*n*) — the length of the upper side of a door frame.
Output Specification:
Print the minimal number of wooden bars with length *n* which are needed to make the frames for two doors.
Demo Input:
['8\n1\n2\n', '5\n3\n4\n', '6\n4\n2\n', '20\n5\n6\n']
Demo Output:
['1\n', '6\n', '4\n', '2\n']
Note:
In the first example one wooden bar is enough, since the total length of all six sides of the frames for two doors is 8.
In the second example 6 wooden bars is enough, because for each side of the frames the new wooden bar is needed.
| 492
|
Title: none
Time Limit: None seconds
Memory Limit: None megabytes
Problem Description:
Little Chris is very keen on his toy blocks. His teacher, however, wants Chris to solve more problems, so he decided to play a trick on Chris.
There are exactly *s* blocks in Chris's set, each block has a unique number from 1 to *s*. Chris's teacher picks a subset of blocks *X* and keeps it to himself. He will give them back only if Chris can pick such a non-empty subset *Y* from the remaining blocks, that the equality holds:
For example, consider a case where *s*<==<=8 and Chris's teacher took the blocks with numbers 1, 4 and 5. One way for Chris to choose a set is to pick the blocks with numbers 3 and 6, see figure. Then the required sums would be equal: (1<=-<=1)<=+<=(4<=-<=1)<=+<=(5<=-<=1)<==<=(8<=-<=3)<=+<=(8<=-<=6)<==<=7.
However, now Chris has exactly *s*<==<=106 blocks. Given the set *X* of blocks his teacher chooses, help Chris to find the required set *Y*!
Input Specification:
The first line of input contains a single integer *n* (1<=≤<=*n*<=≤<=5·105), the number of blocks in the set *X*. The next line contains *n* distinct space-separated integers *x*1, *x*2, ..., *x**n* (1<=≤<=*x**i*<=≤<=106), the numbers of the blocks in *X*.
Note: since the size of the input and output could be very large, don't use slow output techniques in your language. For example, do not use input and output streams (cin, cout) in C++.
Output Specification:
In the first line of output print a single integer *m* (1<=≤<=*m*<=≤<=106<=-<=*n*), the number of blocks in the set *Y*. In the next line output *m* distinct space-separated integers *y*1, *y*2, ..., *y**m* (1<=≤<=*y**i*<=≤<=106), such that the required equality holds. The sets *X* and *Y* should not intersect, i.e. *x**i*<=≠<=*y**j* for all *i*, *j* (1<=≤<=*i*<=≤<=*n*; 1<=≤<=*j*<=≤<=*m*). It is guaranteed that at least one solution always exists. If there are multiple solutions, output any of them.
Demo Input:
['3\n1 4 5\n', '1\n1\n']
Demo Output:
['2\n999993 1000000', '1\n1000000 \n']
Note:
none
| 493
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Title: Fibonacci Sums
Time Limit: None seconds
Memory Limit: None megabytes
Problem Description:
Fibonacci numbers have the following form:
Let's consider some non-empty set *S*<==<={*s*1,<=*s*2,<=...,<=*s**k*}, consisting of different Fibonacci numbers. Let's find the sum of values of this set's elements:
Let's call the set *S* a number *n*'s decomposition into Fibonacci sum.
It's easy to see that several numbers have several decompositions into Fibonacci sum. For example, for 13 we have 13,<=5<=+<=8,<=2<=+<=3<=+<=8 — three decompositions, and for 16: 3<=+<=13,<=1<=+<=2<=+<=13,<=3<=+<=5<=+<=8,<=1<=+<=2<=+<=5<=+<=8 — four decompositions.
By the given number *n* determine the number of its possible different decompositions into Fibonacci sum.
Input Specification:
The first line contains an integer *t* — the number of tests (1<=≤<=*t*<=≤<=105). Each of the following *t* lines contains one test.
Each test is an integer *n* (1<=≤<=*n*<=≤<=1018).
Please do not use the %lld specificator to read or write 64-bit integers in C++. It is preferred to use the cin, cout streams or the %I64d specificator.
Output Specification:
For each input data test print a single number on a single line — the answer to the problem.
Demo Input:
['2\n13\n16\n']
Demo Output:
['3\n4\n']
Note:
Two decompositions are different if there exists a number that is contained in the first decomposition, but is not contained in the second one. Decompositions that differ only in the order of summands are considered equal.
| 494
|
Title: Mishka and trip
Time Limit: None seconds
Memory Limit: None megabytes
Problem Description:
Little Mishka is a great traveller and she visited many countries. After thinking about where to travel this time, she chose XXX — beautiful, but little-known northern country.
Here are some interesting facts about XXX:
1. XXX consists of *n* cities, *k* of whose (just imagine!) are capital cities. 1. All of cities in the country are beautiful, but each is beautiful in its own way. Beauty value of *i*-th city equals to *c**i*. 1. All the cities are consecutively connected by the roads, including 1-st and *n*-th city, forming a cyclic route 1<=—<=2<=—<=...<=—<=*n*<=—<=1. Formally, for every 1<=≤<=*i*<=<<=*n* there is a road between *i*-th and *i*<=+<=1-th city, and another one between 1-st and *n*-th city. 1. Each capital city is connected with each other city directly by the roads. Formally, if city *x* is a capital city, then for every 1<=≤<=*i*<=≤<=*n*,<=<=*i*<=≠<=*x*, there is a road between cities *x* and *i*. 1. There is at most one road between any two cities. 1. Price of passing a road directly depends on beauty values of cities it connects. Thus if there is a road between cities *i* and *j*, price of passing it equals *c**i*·*c**j*.
Mishka started to gather her things for a trip, but didn't still decide which route to follow and thus she asked you to help her determine summary price of passing each of the roads in XXX. Formally, for every pair of cities *a* and *b* (*a*<=<<=*b*), such that there is a road between *a* and *b* you are to find sum of products *c**a*·*c**b*. Will you help her?
Input Specification:
The first line of the input contains two integers *n* and *k* (3<=≤<=*n*<=≤<=100<=000,<=1<=≤<=*k*<=≤<=*n*) — the number of cities in XXX and the number of capital cities among them.
The second line of the input contains *n* integers *c*1,<=*c*2,<=...,<=*c**n* (1<=≤<=*c**i*<=≤<=10<=000) — beauty values of the cities.
The third line of the input contains *k* distinct integers *id*1,<=*id*2,<=...,<=*id**k* (1<=≤<=*id**i*<=≤<=*n*) — indices of capital cities. Indices are given in ascending order.
Output Specification:
Print the only integer — summary price of passing each of the roads in XXX.
Demo Input:
['4 1\n2 3 1 2\n3\n', '5 2\n3 5 2 2 4\n1 4\n']
Demo Output:
['17', '71']
Note:
This image describes first sample case:
<img class="tex-graphics" src="https://espresso.codeforces.com/9dcb0e62694349977e064303a74efd880af5eb03.png" style="max-width: 100.0%;max-height: 100.0%;"/>
It is easy to see that summary price is equal to 17.
This image describes second sample case:
<img class="tex-graphics" src="https://espresso.codeforces.com/c30146637f3647c2fc77818bbbf3c080f94f116e.png" style="max-width: 100.0%;max-height: 100.0%;"/>
It is easy to see that summary price is equal to 71.
| 495
|
Title: Superhero's Job
Time Limit: None seconds
Memory Limit: None megabytes
Problem Description:
It's tough to be a superhero. And it's twice as tough to resist the supervillain who is cool at math. Suppose that you're an ordinary Batman in an ordinary city of Gotham. Your enemy Joker mined the building of the city administration and you only have several minutes to neutralize the charge. To do that you should enter the cancel code on the bomb control panel.
However, that mad man decided to give you a hint. This morning the mayor found a playing card under his pillow. There was a line written on the card:
The bomb has a note saying "*J*(*x*)<==<=*A*", where *A* is some positive integer. You suspect that the cancel code is some integer *x* that meets the equation *J*(*x*)<==<=*A*. Now in order to decide whether you should neutralize the bomb or run for your life, you've got to count how many distinct positive integers *x* meet this equation.
Input Specification:
The single line of the input contains a single integer *A* (1<=≤<=*A*<=≤<=1012).
Output Specification:
Print the number of solutions of the equation *J*(*x*)<==<=*A*.
Demo Input:
['3\n', '24\n']
Demo Output:
['1\n', '3\n']
Note:
Record *x*|*n* means that number *n* divides number *x*.
<img align="middle" class="tex-formula" src="https://espresso.codeforces.com/f651455cf24acab8e8b76da4abe0ec8f028ec17b.png" style="max-width: 100.0%;max-height: 100.0%;"/> is defined as the largest positive integer that divides both *a* and *b*.
In the first sample test the only suitable value of *x* is 2. Then *J*(2) = 1 + 2.
In the second sample test the following values of *x* match:
- *x* = 14, *J*(14) = 1 + 2 + 7 + 14 = 24 - *x* = 15, *J*(15) = 1 + 3 + 5 + 15 = 24 - *x* = 23, *J*(23) = 1 + 23 = 24
| 496
|
Title: Linear Kingdom Races
Time Limit: None seconds
Memory Limit: None megabytes
Problem Description:
You are a car race organizer and would like to arrange some races in Linear Kingdom.
Linear Kingdom has *n* consecutive roads spanning from left to right. The roads are numbered from 1 to *n* from left to right, thus the roads follow in the order of their numbers' increasing. There will be several races that may be held on these roads. Each race will use a consecutive subset of these roads. Also, each race will pay some amount of money to you if this race is held. No races overlap in time, so some roads can be used in several races.
Unfortunately, some of the roads are in a bad condition and they need repair. Each road has repair costs associated with it, you are required to pay this cost to repair the road. A race can only take place if all the roads used in the race are renovated. Your task is to repair such roads (possibly all or none) that will maximize your profit. Your profit is defined as the total money you get from the races that are held minus the total money you spent to repair the roads. Note that you may decide not to repair any road and gain zero profit.
Print the maximum profit you can gain.
Input Specification:
The first line contains two single-space separated integers, *n* and *m* (1<=≤<=*n*,<=*m*<=≤<=2·105), denoting the number of roads and the number of races, respectively.
Then *n* lines follow, each line will contain a single non-negative integer not exceeding 109 denoting the cost to repair a road. The costs are given in order from road 1 to road *n*.
Finally, *m* lines follow. Each line is single-space-separated triplets of integers. Each triplet will be given as *lb*, *ub*, and *p* (1<=≤<=*lb*<=≤<=*ub*<=≤<=*n*,<=1<=≤<=*p*<=≤<=109), which means that the race these three integers describe will use all the roads from *lb* to *ub*, inclusive, and if it's held you get *p*.
Output Specification:
Print a single integer denoting the maximum possible profit you can gain.
Please, do not use the %lld specificator to read or write 64-bit integers in C++. It is recommended to use cin, cout stream (also you may use %I64d specificator).
Demo Input:
['7 4\n3\n2\n3\n2\n1\n2\n3\n1 2 5\n2 3 5\n3 5 3\n7 7 5\n', '2 1\n0\n3\n1 2 5\n', '3 1\n10\n10\n10\n1 3 10\n']
Demo Output:
['4\n', '2\n', '0\n']
Note:
In the first sample the optimal solution is to repair roads 1, 2, 3, and 7. Three races will take place which nets you 15. The road repair costs 11, hence your profit is 4.
| 497
|
Title: Treasure Hunt
Time Limit: None seconds
Memory Limit: None megabytes
Problem Description:
Captain Bill the Hummingbird and his crew recieved an interesting challenge offer. Some stranger gave them a map, potion of teleportation and said that only this potion might help them to reach the treasure.
Bottle with potion has two values *x* and *y* written on it. These values define four moves which can be performed using the potion:
- - - -
Map shows that the position of Captain Bill the Hummingbird is (*x*1,<=*y*1) and the position of the treasure is (*x*2,<=*y*2).
You task is to tell Captain Bill the Hummingbird whether he should accept this challenge or decline. If it is possible for Captain to reach the treasure using the potion then output "YES", otherwise "NO" (without quotes).
The potion can be used infinite amount of times.
Input Specification:
The first line contains four integer numbers *x*1,<=*y*1,<=*x*2,<=*y*2 (<=-<=105<=≤<=*x*1,<=*y*1,<=*x*2,<=*y*2<=≤<=105) — positions of Captain Bill the Hummingbird and treasure respectively.
The second line contains two integer numbers *x*,<=*y* (1<=≤<=*x*,<=*y*<=≤<=105) — values on the potion bottle.
Output Specification:
Print "YES" if it is possible for Captain to reach the treasure using the potion, otherwise print "NO" (without quotes).
Demo Input:
['0 0 0 6\n2 3\n', '1 1 3 6\n1 5\n']
Demo Output:
['YES\n', 'NO\n']
Note:
In the first example there exists such sequence of moves:
1. <img align="middle" class="tex-formula" src="https://espresso.codeforces.com/7c939890fb4ed35688177327dac981bfa9216c00.png" style="max-width: 100.0%;max-height: 100.0%;"/> — the first type of move 1. <img align="middle" class="tex-formula" src="https://espresso.codeforces.com/afbfa42fbac4e0641e7466e3aac74cbbb08ed597.png" style="max-width: 100.0%;max-height: 100.0%;"/> — the third type of move
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Title: Bear and Colors
Time Limit: None seconds
Memory Limit: None megabytes
Problem Description:
Bear Limak has *n* colored balls, arranged in one long row. Balls are numbered 1 through *n*, from left to right. There are *n* possible colors, also numbered 1 through *n*. The *i*-th ball has color *t**i*.
For a fixed interval (set of consecutive elements) of balls we can define a dominant color. It's a color occurring the biggest number of times in the interval. In case of a tie between some colors, the one with the smallest number (index) is chosen as dominant.
There are non-empty intervals in total. For each color, your task is to count the number of intervals in which this color is dominant.
Input Specification:
The first line of the input contains a single integer *n* (1<=≤<=*n*<=≤<=5000) — the number of balls.
The second line contains *n* integers *t*1,<=*t*2,<=...,<=*t**n* (1<=≤<=*t**i*<=≤<=*n*) where *t**i* is the color of the *i*-th ball.
Output Specification:
Print *n* integers. The *i*-th of them should be equal to the number of intervals where *i* is a dominant color.
Demo Input:
['4\n1 2 1 2\n', '3\n1 1 1\n']
Demo Output:
['7 3 0 0 \n', '6 0 0 \n']
Note:
In the first sample, color 2 is dominant in three intervals:
- An interval [2, 2] contains one ball. This ball's color is 2 so it's clearly a dominant color. - An interval [4, 4] contains one ball, with color 2 again. - An interval [2, 4] contains two balls of color 2 and one ball of color 1.
There are 7 more intervals and color 1 is dominant in all of them.
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