I think this is probably the nicest proof I've done yet; I'm quite proud of it. Of course talking to the other participants afterwards I realised I was unnecessarily wordy at parts, and maybe there wasn't quite a need for the amount of rigour I went into... Still, this was probably my worst test so far and I wasn't making progress with the other questions, so spending time on making this one pretty helped give me something to do.
Of course, the first draft isn't quite as nice...
But then the memo is literally two sentences... And shows that I didn't need to go into the whole wordy explanation of how moves are duplicated, and I just could've numbered the cells instead...
Oh well, I at least quite like my final write-up.