Problem 4, 2008

Anja owns tiles shaped like a single unit square, Bojan tiles shaped like an L-tromino: three unit squares forming an L, as drawn below. The two players alternately place one tile of their own onto a rectangular board ruled into unit cells. If at some moment it is Bojan's turn and he cannot place a tile although at least one cell of the board is still uncovered, then Anja wins; otherwise Bojan wins. Prove that

(a) on a board of size \(6 \times 9\), Bojan cannot win, no matter who moves first;

(b) on a board of size \(8 \times 8\), Bojan can place his tiles so that he wins, no matter how Anja plays and no matter who moves first.

(Each tile must lie entirely on the board and cover whole cells of it, and no two tiles may overlap. A tile may be placed in any of its rotated positions.)