Given a square grid, initially all its unit squares are coloured white. A move consists of choosing a row, or a column, and changing the colours of all its unit squares, from white to black, and from black to white. Is it possible that after finitely many moves, the square grid contains exactly 574 black unit squares?
Solution
No, it is not possible.
Suppose it is possible that after finitely many moves, the square grid contains exactly 574 black unit squares. Then there are exactly white unit squares. No matter where the white unit squares are, we can always find a square that contains exactly 1 white unit square and 3 black unit squares. One checks that the parity of the number of white cells in this square is fixed no matter how we choose the rows or columns in a move. Therefore, it is
impossible that all the four cells are white initially. This is a contradiction, and
so it is impossible to have exactly 574 black unit squares.
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