2023 USAMO Problems/Problem 3
Problem
Consider an -by-
board of unit squares for some odd positive integer
. We say that a collection
of identical dominoes is a maximal grid-aligned configuration on the board if
consists of
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap:
then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let
be the number of distinct maximal grid-aligned configurations obtainable from
by repeatedly sliding dominoes. Find the maximum value of
as a function of
.
Solution
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