We're given a mathematical sequence of numbers of the form Xn+1 = Xn - k % j. We
have to find a number somewhat deep in the sequence.
Luckily, this was doable with simple iterating. I did not have to find a closed form solution. For a tougher number theory problem, see 2019 Day 22.
2015 Day 25 on AdventOfCode.com
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?
The machine springs to life, then falls silent again. It beeps. "Insufficient fuel", the console reads. "Fifty stars are required before proceeding. One star is available."
..."one star is available"? You check the fuel tank; sure enough, a lone star sits at the bottom, awaiting its friends. Looks like you need to provide 49 yourself.