^ How the painters doing?
OK, so there are 2^17 - 1 possible initial "starting colors" for the houses. Each acceptable coloring arrangement can be represented by a binary number. Each painter, when painting according to the rules, is essentially a function from one acceptable binary number to another. More precisely, each painter can be thought of as a permutation of the set of the 2^17 - 1 acceptable binary numbers.
My
hunch is that these 17 "painter permutations" are elements of a group that is isomorphic to the integers mod 2^17 - 1 under addition. And more speciffically, each painter permutation corresponds to 2^17 - 1 - i, where i = 1, 2, 4...2^16. And hence, the composition of all painter permutations corresponds to (2^17 - 1 - 1) + (2^17 - 1 - 2) + ... + (2^17 - 1 - 2^16) = 16 * (2^17 -1) = 0 mod 2^17 -1.
And hence, the composition of all painter permutations is the identity function. So, after each has painted, the house colorings return to their initial state.
And the reason for my hunch is that I tabulated it for n = 2 and n = 3 (so, by golly, it should hold for n = 17).
Am I on the right track???