Tokarev S.I.

On the first horizontal line of the chessboard, there are 8 black queens, and on the last - 8 white queens. In what minimum number of moves can the white queens exchange places with the black ones? The white and black queens take turns, moving one queen per move.
The queen on the b1 or b8 square that moved first could not have moved to the opposite rank with that move, meaning these two queens together made at least three moves. Similarly, the pairs of queens on the c, ..., g files each made at least three moves. The first of the four corner queens to move did not land on a corner square with that move, meaning they made at least five moves in total. Therefore, a total of at least \(6 \cdot 3 + 5 = 23\) moves are required.

An example of 23 moves is shown in the figure (the black queen moves first):
![](https://cdn.mathpix.com/cropped/2024_05_06_c07b2c4be0d3dcbcc4e5g-25.jpg?height=846&width=1292&top_left_y=362&top_left_x=386)

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