Warning: spoilers for day 24 part 2 below.

I read some reddit posts to help me solve this one. Many of them suggested visualizing the graph of wires

to better grasp what's happening, so I output a [graphviz](https://graphviz.org/) dot file and stared at it for a

while.

It gave me the idea to build a list of dependencies of the incorrect z wires, and filter out any wires that contributed

to correct z wires: this would reduce the swap candidates to just a handful of wires that I could viably test by trying

every possible combination of 4 swaps. This was misguided, however, since a "correct" z wire is dependent on the inputs,

and the solution has to work with ANY values in the x and y wires. Even using randomized x and y values to figure out

likely dependents eliminated too many candidates.

So I gave up solving it on my own, and dove more deeply into the reddit discussions. Most of them focus on

understanding the ["full adder"](https://en.wikipedia.org/wiki/Adder_(electronics)#Full_adder): the puzzle input

basically constructs one of these (except for the 4 swapped gates, of course). Interestingly, many people solved this

puzzle without code: they did it using pen and paper, and/or by inspecting a visual graph by eye and noticing how it

"obviously" deviated from the composition of gates for a full adder. But this wasn't obvious to me. I could see the

general shape of the graph and the places where it seemed off, but beyond that, it gave me a headache to try to

identify what the configuration of the full adder should look like for each z wire.

I ended up implementing [a solution by someone](https://www.reddit.com/r/adventofcode/comments/1hneuf0/comment/m41ms44/)

who used a brute force approach I thought was pretty clever: set up a bunch of test devices with randomized x and y

values, find a swap that results in the most improvement for all of them, and repeat until you find the set of 4 swaps

that works for every test device. After adding caching to the evaluation of wire values, my solution ran in only 35s

(the author reported it taking 2.5 mins on their machine). More details are in the code.

I suppose slogging through it was worth the insight into how mathematical addition works at the level of logical gates

(you need a carry bit!), even if I didn't use that knowledge for the actual solution.

Anyway, day 17 part 2 is the last remaining puzzle for me. Like day 24, it involves implementing a computer. Ugh,

I dislike these types of problems so much.