KöMaL 2025 A922

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 $L=\lim_{n\to \infty}\frac{\max_{T\in \mathcal{T}_n} \phi(T)}{abn^2}$ exists for every choice of positive integers $a$ and $b$, and determine the value of $L$.