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Omni-MATH / Consider an n -by- n board of unit squares for some odd positive integer n . We say that a collection C of identical dominoes is a maximal grid-aligned configuration on the board i…

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problem

Consider an nn -by- nn board of unit squares for some odd positive integer nn . We say that a collection CC of identical dominoes is a maximal grid-aligned configuration on the board if CC consists of (n2−1)/2(n^2-1)/2 dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: CC 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 k(C)k(C) be the number of distinct maximal grid-aligned configurations obtainable from CC by repeatedly sliding dominoes. Find the maximum value of k(C)k(C) as a function of nn .
Plain-text mathematical notation (without MathML)
Consider an n -by- n board of unit squares for some odd positive integer n . We say that a collection C of identical dominoes is a maximal grid-aligned configuration on the board if C consists of (n²−1)/2 dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: C 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 k(C) be the number of distinct maximal grid-aligned configurations obtainable from C by repeatedly sliding dominoes. Find the maximum value of k(C) as a function of n .
Original LaTeX notation
Consider an $n$ -by- $n$ board of unit squares for some odd positive integer $n$ . We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ 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 $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$ .

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Source and history

task_id
d05cf1b6-4a91-52af-a8d9-825b9daa7700
task_key
test--d05cf1b6-4a91-52af-a8d9-825b9daa7700
task_revision_id
1
config / split
/ test

Official source

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