Question 161
a) On a $5 \times 5$ board, $21$ tokens are placed white side up so that each token lies on a separate $1 \times 1$ square (each token is two-coloured: it has one white and one black side). In each move, Marta takes one "white" token from the board, flips it and puts it back on some free $1 \times 1$ square. Her goal is to flip all $21$ tokens, so that at no moment are a "white" and a "black" token placed on neighbouring (sharing a side) $1 \times 1$ squares. Show that, regardless of the initial arrangement of the tokens, Marta cannot achieve her goal.
b) If instead of $21$ tokens there are $20$ tokens placed white side up on the board, is there an initial arrangement for which Marta can achieve her goal?
b) If instead of $21$ tokens there are $20$ tokens placed white side up on the board, is there an initial arrangement for which Marta can achieve her goal?