MathDB

A domino is a

1 \times 2

2 \times 1

tile. Let

n \ge 3

be an integer. Dominoes are placed on an

n \times n

board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some

k \ge 1

such that each row and each column has a value of

k

. Prove that a balanced configuration exists for every

n \ge 3

, and find the minimum number of dominoes needed in such a configuration.

Problem Statement