Mistilings with dominoes (Q1343803)

Given a checker board and a type of a domino, the mistiling ratio is defined as being the minimum proportion of squares that are covered in a maximal packing of the board with dominoes of that type. It is proved that given an infinite checker board, the mistiling ratio of the hook is 6/11, the one of the \(m \times m\) square domino is \(m^2/(2m - 1)^2\), and the one of the \(1 \times m\) domino is at least \(2/(m + 1)\). It is conjectured that for that last case with \(m \geq 3\), the mistiling ratio is \(2m/(m^2 + 1)\).