Finally cracked this puzzle over lunch today. Very satisfying, I think it's the first one of these Big Hard Logic Puzzles I've solved start to finish on my own.

This also means I'm no longer messing up Matt's viewer retention stats from having paused it a week ago pretty near the beginning. I'd be interested to see the drop-off in the graph at the point when they say "pause the video now and have a go at it".

youtube.com/watch?v=as7Gkm7Y7h

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Spoilery discussion of my solution 

I got hung up for ages on parity — because every possible arrangement you can reach by flipping one coin has the same parity, the 8x8 chessboard can be modelled as colouring a 63-dimensional cube, rather than a 64D one. Somehow I thought the loss of that one dimension would be more useful than the nice factorisability of 64. No idea why. And you can get to it that way but it's inelegant.

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Spoilery discussion of my solution 

Then I drew out some coloured tesseracts for low-N cases and that worked well. I found you could step up to the next solution by pulling out a copy of the solution into some new dimension, giving it N new colours, and then pulling out as many permuted versions as you need to fill in the missing flips — but it didn't give me an algorithm to devise the permutations.

Spoilery discussion of my solution 

Then I wrote the N=8 version in binary and it clicked and I felt simultaneously like a genius and an idiot.

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