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I am playing with only a bishop (black squares) and king vs a lone king. Is it possible for me to win?

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No. Even if you got your opponent's king to a corner and checked it with your bishop, your king can control only two of the three flight squares.

The most you could achieve is a stalemate, if your opponent helps. Therefore king and bishop vs king is considered a draw by insufficient material ("dead position" is the term used in the FIDE Laws of Chess; see article 5.2.2).

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