[ Integer lattices (miscellaneous). ] $\qquad$
A rectangular grid of $m \times n$ cells is made of strings of unit length. Two players take turns. In one move, a player can cut (in the middle) an uncut unit string. If there are no closed string contours left, the player who made the last move is considered the loser. Which player will win with correct play, and how should they play to win?
At the beginning of the game, there were $m(n+1)+n(m+1)=2 m n+m+n$ strings of unit length. This number has the same parity as the number $m+n$. The last move in the game destroys the last closed loop.

However, the boundary of a closed loop contains an even number of unit-length strings (see the solution to problem 30310). The winning strategy for any player is to avoid destroying the last closed loop as long as possible.

## Answer

If $m+n$ is even, the second player wins; if $m+n$ is odd, the first player wins.

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