Problem
COM-B1-M08-P010 Dominoes on a \(6\times6\) Board
#10
★★☆☆☆ Level 2 of 5
Players alternately place a domino on two adjacent empty cells of a \(6\times6\) board. A player who cannot move loses. Prove that the second player wins.
Respond with the centrally symmetric domino.
After the first player’s move, the second places a domino on the cells centrally symmetric to those just occupied. On a \(6\times6\) board, no domino coincides with its central reflection, and the reflection of adjacent cells is again adjacent. If the reflected cells had already been occupied, then by symmetry the cells of the first player’s move would also have been occupied, impossible. Hence the response is always legal, and the second player makes the last move.
Good check of the correctness of a symmetry strategy.