Question 3
Let $n$ be an odd positive integer. A maximal grid-aligned configuration $C$ consists of $(n^2-1)/2$ identical dominoes on an $n\times n$ board of unit squares. Each covers two neighboring squares, with no overlap, so one square is uncovered. A move slides, without rotation, a domino to cover the uncovered square and leave another square uncovered. Let $k(C)$ be the number of distinct configurations obtainable by any number of such moves, including zero. Find the maximum of $k(C)$ in terms of $n$.
Proposed by Holden Mui. Source: Evan Chen’s third-party compilation; https://aops.com/community/p27349423. One editorial point; manual grading.
Proposed by Holden Mui. Source: Evan Chen’s third-party compilation; https://aops.com/community/p27349423. One editorial point; manual grading.