, 2025

Fix two positive integers $a$ and $b$. Let $n$ be a positive integer, and let $T$ be a black–white coloring of the cells of an $an\times bn$ grid, containing at least two white cells. Csigusz the snail starts on a white cell, visits all white cells exactly once, and then returns to the starting cell, always moving between side-adjacent white cells. He then notices that he was able to do this in exactly one way (that is, once he made his first move, there was a unique way to complete the cycle). Let $\mathcal{T}_n$ denote the set of all colorings $T$ satisfying this property, and let $\phi(T)$ denote the number of white lattice points in $T$. Show that