Problem 3. (5 points) On the first row of a chessboard, there are 8 indistinguishable black queens, and on the last row, there are eight indistinguishable white queens. What is the minimum number of moves required for the white and black queens to swap places? Moves of the white and black queens alternate, with exactly one queen being moved each turn. (Queens can move any number of squares horizontally, vertically, or diagonally, but they are not allowed to jump over other pieces in their path.)
Solution: We will prove that the white and black queens can swap places with a minimum of 23 moves. The queen from the four corner positions that first leaves its place must make at least two moves, i.e., the corner queens together must make at least $1 \cdot 2 + 3 \cdot 1 = 5$ moves (see Fig. 1).

![](https://cdn.mathpix.com/cropped/2024_06_03_5d371ccc65ea9d468529g-2.jpg?height=383&width=371&top_left_y=891&top_left_x=454)

Figure 1:
![](https://cdn.mathpix.com/cropped/2024_06_03_5d371ccc65ea9d468529g-2.jpg?height=386&width=764&top_left_y=888&top_left_x=862)

Figure 2:

The remaining queens are divided into pairs of opposite (standing in the same column). The queen from two opposite queens that first leaves its place must make at least two moves to reach its new position; thus, for two opposite queens, a minimum of $1 \cdot 2 + 1 = 3$ moves are required (see Fig. 2). For 6 such pairs, a total of $6 \cdot 3 = 18$ moves are required. Thus, the total number of moves is 23.

The problem was solved by Nikolay Belukhov from Stara Zagora, Slavina Kitova, Viktor Konstantinov, and Vladimir Sotirov from AK Sofia.