I solved this puzzle properly, by hand! It only took like four hours.
You know there's a simple algorithm for this puzzle: you just have to construct and solve a 54x54 linear system in ℤ₂! Each cell influences some subset of the other cells (and sets them to 1), so you make a column out of each cell to form a matrix 𝐀, as well as a vector for the solution 𝐛, and then solve 𝐀𝐱 + 𝐛 = 0 in ℤ₂.
More in-depth details below the cut, I'll explain all of that at least briefly for folks who don't know what those terms mean. And you won't see the solution unless you want to, if you want to try this method for yourself.
this is literally my favorite thing about this website. the fact that both of you exist. the creator who builds insane puzzle. the genius that solves insane puzzle mathematically and shows their work.
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