[Chess boards and chess pieces] $[\quad \underline{\text { Graph traversal }}$

## Authors: Buchin A.A, Ivaniuk A.V.

Two players take turns placing one checker on a $65 \times 65$ board. At no point should any row (horizontal or vertical) contain more than two checkers. The player who cannot make a move loses. Who wins with correct play?

#
Let the first player place the first checker on the board. Note that the permutation of the board's rows does not change anything. The same applies to the permutation of columns. Therefore, we will assume that the second player is additionally allowed to swap any rows and columns.

After the first player's first move, the second player will rearrange the rows and columns so that the first checker is in the middle column, but not in the center of the board.

Then he makes moves symmetrically to the first player's moves relative to the center of the board. In particular, the second checker will also end up in the middle column, and the first player will not be able to occupy the central cell. It is easy to verify that the second player will always be able to make a symmetrical move (the case of a move to the middle row should be considered separately).

## Answer

Second.