Bogdanov I.I.

On a chessboard, 32 white and 32 black pawns are placed in all squares. A pawn can capture pawns of the opposite color by moving diagonally one square and taking the place of the captured pawn (white pawns can only capture to the right-up and left-up, while black pawns can only capture to the left-down and right-down). Pawns cannot move in any other way. What is the minimum number of pawns that can remain on the board?
Note that a pawn that stood on a black square will always move only on black squares. Then after each move (on black squares) there will always be at least one pawn on black squares - the one that made the move. Similarly, at least one pawn will remain on white squares, and there will be no less than two pawns in total. The diagrams show how to arrange the pawns on white squares and how to move them so that only one remains. The arrangement and actions on black squares are analogous.
![](https://cdn.mathpix.com/cropped/2024_05_06_ed1a9e9dd82795d4db2eg-32.jpg?height=1126&width=1382&top_left_y=19&top_left_x=336)

## Answer

Two pawns.