Problem
COM-B1-M08-P023 Dominoes on a Board with a Hole
The central cell is removed from a \(7\times7\) board. Players alternately place dominoes on two adjacent empty cells. A player who cannot move loses. Prove that the second player wins.
After removing the center, the remaining region has central symmetry with no fixed cells.
The second player responds to each first-player domino by placing the centrally symmetric domino. Since the center cell is removed, no remaining cell maps to itself. The reflection of two adjacent cells is again two adjacent cells. If the reflected domino were already occupied, then by the symmetry strategy the domino just placed by the first player would already have been occupied too, impossible. Thus the second player’s response is always legal, and the second player makes the last move.
Stronger than ordinary symmetry: the board is not rectangular, but symmetry remains.