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Checkmate problems

Every time I get checkmate I can easily move out of check by taking that piece if possible but it won't! :-(
you cannot make illegal moves. the king cannot take a piece that is guarded by another piece. an example from your games : http://en.lichess.org/afpvds2j . You are in checkmate because the pawn attacks your king and any free square is covered. The pawn attacking the king is defended by two pawns and a knight, and the bishop is guarded by the rook.
another example: http://en.lichess.org/v5kfe4h2/black the queen is attacking the king as well as any escape squares your king had. The queen is defended by both knight and bishop so you are unable to take the queen.

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