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Yes, a piece can check even if it can't move because of a pin. This is explicitly written in the FIDE Laws of Chess (emphasis added): 3.9.1 The king is said to be 'in check' if it is attacked by one or more of the opponent's pieces, even if such pieces are constrained from moving to the square occupied by the king because they would then leave or place ...


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You are in checkmate if you are in check and have no legal move by which to escape it. In this case you cannot escape check by moving your king. B4 or D4 would leave you in check from the knight, C4 would leave you in check from the bishop and anything on rows 2 or 3 would leave you in check from a rook. However moving your king is not the only way to escape ...


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