Question 153
Anja has a number of $1 \times 1$ square tiles $\Box$, while Bojan has the L-shaped tiles (each consisting of three unit squares, as in the figure below). They take turns putting their tiles onto a rectangular board, one tile at a time. Anja wins if Bojan cannot fit another one of his tiles onto the board when his turn comes despite there being squares left uncovered. Otherwise, Bojan wins. Prove:
a) given a $6 \times 9$ board, Bojan cannot win regardless of who has the first move;
b) given an $8 \times 8$ board, Bojan can put the tiles onto the board in a way that ensures his winning, regardless of how Anja plays and who had the first move.
(Note: All tiles have to lie entirely on the board and they must not cover any of the other tiles).
a) given a $6 \times 9$ board, Bojan cannot win regardless of who has the first move;
b) given an $8 \times 8$ board, Bojan can put the tiles onto the board in a way that ensures his winning, regardless of how Anja plays and who had the first move.
(Note: All tiles have to lie entirely on the board and they must not cover any of the other tiles).