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Dominoes placed on a rectangular board

Source: RMM 2016 Day 1 Problem 2

February 27, 2016
combinatoricsRMM

Problem Statement

Given positive integers mm and n≥mn \ge m, determine the largest number of dominoes (1×21\times2 or 2×12 \times 1 rectangles) that can be placed on a rectangular board with mm rows and 2n2n columns consisting of cells (1×11 \times 1 squares) so that: (i) each domino covers exactly two adjacent cells of the board; (ii) no two dominoes overlap; (iii) no two form a 2×22 \times 2 square; and (iv) the bottom row of the board is completely covered by nn dominoes.
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