$\left[\begin{array}{l}{\left[\begin{array}{l}\text { Games and jokes }\end{array}\right]} \\ {[\underline{\text { Evenness and oddness }}]}\end{array}\right.$

On the board, there are 10 ones and 10 twos. In one move, it is allowed to erase any two digits, and if they were the same, write a two, and if they were different, write a one. If the last remaining digit on the board is a one, the first player wins; if it is a two, the second player wins.
The parity of the number of ones on the board after each move does not change. Since there was an even number of ones at the beginning, after the last move, there cannot be one (an odd number!) one left on the board.

## Answer

Second.

![](https://cdn.mathpix.com/cropped/2024_05_06_c33e2e8ade684f1ef192g-05.jpg?height=95&width=1215&top_left_y=14&top_left_x=11)

Two grandmasters take turns placing rooks on a chessboard (one rook per move) such that they do not attack each other. The player who cannot place a rook loses. Who will win with correct play - the first or the second grandmaster?

## Solution

A rook can be placed on a certain square if and only if there is no rook on the row or column containing that square. Therefore, the described game is equivalent to the following: the grandmasters take turns crossing out one letter and one number from the set of letters a, b, ..., h and numbers $1,2, \ldots, 8$. Clearly, no matter how the players move, after the eighth move, all letters and numbers will be crossed out. The eighth move belongs to the second player, so the first player will not be able to make the next move.

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