Hm... solvability of the problem for a board of size n is equivalent to labeling the nodes of the n-cube with n colors in such a way that any node's neighbors t

Hm... solvability of the problem for a board of size n is equivalent to labeling the nodes of the n-cube with n colors in such a way that any node's neighbors take on all n colors (thus, each node will have one neighbor of each color). This means the number of pairs (A, B) of adjacent nodes such that B has any particular color must be the same as the number of ways to just choose A, which is 2^n. This is also the number of ways to choose B of a given color, times the n many ways to choose a neighbor for B. This means the number of nodes of any particular color is 2^n/n. As this must be an integer, we must have that n divides 2^n, and thus, n must itself be a power of 2.