Problem
COM-B2-M05-P010 Corners of a Rectangle
On a \(6\times6\) board, all cells are white. In one move, choose a rectangle with sides along grid lines and change the colours of its four corner cells. Prove that a position with exactly one black cell cannot be obtained.
Each move changes the parity in each affected row and each affected column twice.
Consider the parity of the number of black cells in each row. One move affects two cells in one chosen row and two cells in another chosen row. Hence the parity in every row is preserved. Similarly, the parity in every column is preserved.
Initially all row and column parities are \(0\). If there is exactly one black cell, its row and its column have odd parity. This is impossible.
An important template: there is not one invariant, but a whole family of row and column invariants.