[math-fun] Non-existence from parity (was Re: Existence from parity)

For certain values of N, it's possible to arrange the first N positive integers such that the sum of each adjacent pair is a square. For instance: 8 1 15 10 6 3 13 12 4 5 11 14 2 7 9 I've been looking into whether the same is true of sequences other than squares. With cubes, I still don't know whether I didn't find any because there aren't any, or whether I just didn't try a large enough N. But with powers of 2, I soon came up with a parity-related proof that there can't be a (non-empty) solution no matter how large N is.