"Interesting" being a relative term

The deadline for A1 has passed, and I feel pretty confident about the answers I submitted. I did notice a few interesting discrepancies between the sample solutions and mine though:

1) The sample solutions consisted of fewer than half as many lines as mine, which reaffirms my fear that my proofs are too long. I think this was partially - in question 3 specifically - from my proof taking a circuitous path. In general though, I may have tried to be more precise than was necessary. I think switching to a more, um, prosey (my proofs are anything but prosaic) style cut down on their size a good bit, but I'm still working on cutting out more redundant or obvious stuff.

2) My solution to question 2 was completely different from the sample solution, which I thought was pretty cool. The technique in the sample solution is a bit easier to follow, but I like mine (which consisted of splitting the nth cycle into 2^n-1 pairs, and inserting a pair of new values in the middle of each one) just fine. I'd be really curious to know how many people used my solution, and how many used the other, and if there are even more techniques that work that people submitted.

3) The solution to question 3b made me realized that I had made the careless mistake of conflating "cannot be expressed as a ratio of natural numbers" and "irrational". This didn't actually meaningfully affect my proof, which would apply if I went back in time and replaced the latter term with the former, so I hope I won't be punished too harshly for my slip-up.

Also, no matter how many times I look at the solution to question 3b, I can't make sense of it. I can't decide whether it's more likely that this is the result of a mistake or transcription error on the part of the author, or if I'm just missing something stunningly obvious.

Speaking of missing the stunningly obvious, I almost fell for the pseudo-proof of the six-sidedness of hexagons during yesterday's lecture. It wasn't until after a minute or two that I thought to myself "it sure is neat that he proved that without using any properties of hexagons", and then the inevitable "wait a minute...". I'm embarrassed to admit that I briefly fell for a proof that could easily be adapted to "prove" that all horses are the same colour.