### 3.7.89. Dominating queens

A constraint that can be used for modelling the dominating queens problem. Place a number of queens on an $n$ by $n$ chessboard in such a way that all squares are either attacked by a queen or are occupied by a queen. A queen can attack all squares located on the same column, on the same row or on the same diagonal. Values of the minimum number of queens for $n$ less than or equal to 120 are reported in [OstergardWeakley01]. Most of them are in fact either equal to $⌊\frac{n+1}{2}⌋$ or to $⌊\frac{n+1}{2}⌋+1$. Values $n=3$ and $n=11$ are the only two values below 120 for which the previous assertion is not true since we only need in these two cases $⌊\frac{n}{2}⌋$ queens.