Ana plays a game on a 100×100 chessboard. Initially, there is a white pawn on each square of the bottom row and a black pawn on each square of the top row, and no other pawns anywhere else.\\
Each white pawn moves toward the top row and each black pawn moves toward the bottom row in one of the following ways:
[*] it moves to the square directly in front of it if there is no other pawn on it;
[*] it captures a pawn on one of the diagonally adjacent squares in the row immediately in front of it if there is a pawn of the opposite color on it.(We say a pawn P captures a pawn Q of the opposite color if we remove Q from the board and move P to the square that Q was previously on.)\\
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Ana can move any pawn (not necessarily alternating between black and white) according to those rules. What is the smallest number of pawns that can remain on the board after no more moves can be made?Proposed by José Alejandro Reyes González, Mexico combinatoricscombinatorics proposedgamegame strategyinduction