3. Simona and Lenka are playing a game. For a given integer $k$ such that $0 \leqq k \leqq 64$, Simona selects $k$ squares on an $8 \times 8$ chessboard and marks each of them with a cross. Lenka then fills the chessboard with thirty-two domino tiles in some way. If the number of tiles covering two crosses is odd, Lenka wins; otherwise, Simona wins. Depending on $k$, determine which of the girls has a winning strategy.
SOLUTION. Let's divide the solution according to the value of the number $k$.

If $k=0$, the number of dominoes covering two crosses is zero, so Simona wins.

If $0 < k \leq 32$, $k$ is even, Simona places 32 crosses on white squares and the remaining crosses anywhere. Then, under an even number of dominoes, there are two crosses (there are exactly $k-32$ such dominoes, because each domino covers one white and one black square of the chessboard), so Simona wins.

If $32 < k \leq 61$, $k$ is odd, Simona does not write crosses into three squares in one of the "white corners," i.e., into the corner white square and two adjacent black squares, but writes them into all other 31 white squares and the rest into any black squares (except the two mentioned). On white squares, there is an odd number of crosses, and on black squares, an even number of crosses. Around each black square with a cross, all white squares are also with crosses, so each domino that covers a black square with a cross covers two crosses. Other dominoes do not cover two crosses. Therefore, Simona wins again.

If $k=63$, there are not two crosses under just one domino, so in this case, Lenka wins automatically, without the need for any strategy.

Summary: For every $0 \leq k \leq 64, k \neq 63$, Simona has a winning strategy, and at $k=63$, Lenka wins automatically.

## GUIDING AND SUPPLEMENTARY PROBLEMS: