Advent of Code is a website that releases programming puzzles every December from the 1st to the 25th. The puzzles have two parts, and the second part isn't revealed until you solve the first part.

This year, there have been many questions regarding part 2 of day 22, as it involves modular arithmetic in a way that hasn't been seen in previous puzzles there. I have been asked for a tutorial for it on Reddit, but I'm posting it here due to better support for mathematical notation.

First off, if you are unfamiliar with modular arithmetic, I encourage you to read my other blog post, Modular Arithmetic for Beginners. The most important thing to understand from there is how the four basic operations (addition, subtraction, multiplication, division) can be redefined to work in modular ($$$\mathbb Z / p \mathbb Z$$$ field) arithmetic. I also recommend you try some of the puzzles linked. The terminology and notations I'll use will also be similar.

It is possible to solve this without explicitly invoking modular arithmetic (see this post for an example) but they basically amount to rediscovering certain properties of modular arithmetic anyway, as such that's the "language" I'll use for this tutorial.