--- Day 9: Encoding Error ---
With your neighbor happily enjoying their video game, you turn your attention to an open data port on the little screen in the seat in front of you.
Though the port is non-standard, you manage to connect it to your computer through the clever use of several paperclips. Upon connection, the port outputs a series of numbers (your puzzle input).
The data appears to be encrypted with the eXchange-Masking Addition System (XMAS) which, conveniently for you, is an old cypher with an important weakness.
XMAS starts by transmitting a preamble of 25 numbers. After that, each number you receive should be the sum of any two of the 25 immediately previous numbers. The two numbers will have different values, and there might be more than one such pair.
For example, suppose your preamble consists of the numbers
25 in a random order. To be valid, the next number must be the sum of two of those numbers:
26would be a valid next number, as it could be
25(or many other pairs, like
49would be a valid next number, as it is the sum of
100would not be valid; no two of the previous 25 numbers sum to
50would also not be valid; although
25appears in the previous 25 numbers, the two numbers in the pair must be different.
Suppose the 26th number is
45, and the first number (no longer an option, as it is more than 25 numbers ago) was
20. Now, for the next number to be valid, there needs to be some pair of numbers among
45 that add up to it:
26would still be a valid next number, as
25are still within the previous 25 numbers.
65would not be valid, as no two of the available numbers sum to it.
66would both be valid, as they are the result of
Here is a larger example which only considers the previous 5 numbers (and has a preamble of length 5):
35 20 15 25 47 40 62 55 65 95 102 117 150 182 127 219 299 277 309 576
In this example, after the 5-number preamble, almost every number is the sum of two of the previous 5 numbers; the only number that does not follow this rule is
The first step of attacking the weakness in the XMAS data is to find the first number in the list (after the preamble) which is not the sum of two of the 25 numbers before it. What is the first number that does not have this property?