Problem
COM-B2-M05-P015 Rectangle Corners: A Complete Invariant
On a \(5\times7\) board, all cells are white. In one move, choose a rectangle and change the colours of its four corner cells. Prove that in every reachable colouring, every row and every column contains an even number of black cells.
One move affects two cells in each of two rows and two columns.
In the initial colouring, every row and every column contains \(0\) black cells, an even number.
One move chooses two rows and two columns. In each chosen row, exactly two cells change colour, so the parity of the number of black cells in that row is preserved. Other rows do not change. The same argument applies to columns.
Therefore after any number of moves, all rows and columns preserve parity \(0\).
This problem prepares for a stronger reachability criterion using rectangle parities.