Problem
COM-B1-M08-P017 Rooks on a Rectangle
#17
★★★☆☆ Level 3 of 5
Players alternately place rooks on an \(8\times10\) board so that no two rooks share a row or column. A player who cannot move loses. Who wins?
The maximum number of rooks equals the smaller side of the rectangle.
After \(k\) moves, \(k\) rows and \(k\) columns are occupied. As long as \(k<8\), there is a free row and a free column, so another rook can be placed. After \(8\) moves no free row remains, and no move is possible. The game lasts exactly \(8\) moves, so the second player wins.
Generalizes the square-board problem.