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Playing regular rules for both white and black try to eliminate all the square formations by black and/or white. From the above starting setup all rooks form a square. Also white's queen rook pawn, king knight with black's king rook pawn and queen knight(a2,g1,h7,b8) form the second largest square. The smallest are the 4 pieces on each corner of the chessboard. If captures are allowed what is the largest number of pieces that remain on the board without any square formed by 4 pieces?

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Here is my answer, with 32 pieces. Can add another couple of objects on a5 & b5 without causing a square to appear.

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