10.3. Kolya and Dima are playing a game on an $8 \times 8$ board, taking turns, with Kolya starting. Kolya draws crosses in the cells, while Dima covers pairs of adjacent cells (dominoes) with $1 \times 2$ rectangles. On his turn, Kolya must place one cross in any empty cell (i.e., a cell that does not yet have a cross and is not covered by a domino). Dima, on his turn, must cover two cells (which are not yet covered by other dominoes) with a domino, where the total number of crosses in these two cells is even (0 or 2). The player who cannot make a move loses. Who has the winning strategy?
Answer. Dima.

First solution. We will outline a strategy for Dima. Before the game starts, we divide the board into 32 rectangles of $1 \times 2$. In the first 16 moves, the opponent can place their crosses in no more than 16 of these rectangles, so we can cover the remaining 16 rectangles with our dominoes, and all the opponent's crosses will be in some cells of the remaining 16 rectangles (we will call these rectangles new).

Now we will show that Dima can respond to Kolya's $(16+k)$-th move $(1 \leqslant k \leqslant 16)$ by covering some new rectangle. By this point, Kolya has placed $16+k$ crosses, and all these crosses are in 16 new rectangles. This means that Kolya has already filled at least $k$ new rectangles entirely with crosses, while Dima has covered no more than $k-1$ new rectangles by this point. Therefore, before the $(16+k)$-th move, there is at least one new rectangle with two crosses that has not yet been covered by a domino, and it can be covered by this move, as required.

After Dima's 32nd move, the entire board will be covered with dominoes, and Kolya will lose because he will have no place to move.

Second solution. We will outline a strategy for Dima. Before the game starts, we divide the board into $2 \times 2$ squares. Suppose Dima responds to a cross placed in a square for the first time with a domino covering two empty cells in the same square. When Kolya places a second cross in a square, Dima will cover this cross and the first cross placed in this square with a domino, resulting in the square being covered by two dominoes. Thus, Dima has a response to any move by Kolya.

Comment. A correct answer without presenting a winning strategy - 0 points.

A correct winning strategy for Dima is presented without justification that there is indeed a response to every move by Kolya - 4 points.

In the case of presenting a correct strategy, the solution can
be rated lower than 7 points if there are gaps in the proof that Dima indeed has a response to every move by Kolya.