There is a certain endgame concept that most of us learn the hard way. You have an extra piece and a pawn against the lone king and you are waiting for your opponent to resign. But due to some mysterious reason, you can’t make any headway. The whole vibe just doesn’t feel right. Your opponent is seemingly trying hard not to smirk while the onlooking experts nod knowingly (or sympathetically?). What’s going on, you wonder. We are of course talking about the remarkable case Hans Kmoch termed the “Impotent pair” - a pawn on the edge of the board and a bishop that does not control the pawn’s promotion square. [caption id="attachment_136274" align="aligncenter" width="300"] “The impotent pair” - White can’t win[/caption] The black king can simply shuffle between h8 and g8 and there is nothing white can do to prevent it. Moving the bishop to the a2-g8 diagonal would only lead to a stalemate. If we go all technical, The endgame “king, bishop and pawn vs king” is always a draw when three conditions are met. Firstly, the pawn must be on a rook file (either a or h files). Secondly, the bishop should not be able to control the promotion square of the pawn. Finally, the defending king should be able to reach the promotion square. Undo one of the three, and we got ourselves a winning position. Troitsky and Vančura show how [caption id="attachment_136275" align="aligncenter" width="300"] White to play and win - Alexei Troitsky[/caption] The position appears dead drawn. The only condition yet to be fulfilled is the third one, but it feels like it’s just a matter of time before the black king reaches h8. That’s exactly what the master composer Troitsky wants you to think. 1.Be6! (planning to create a ‘blockade’ with 2.h6) Ke7 2.h6! Kf6 3.Bf5! Kf7 4.Bh7! Kf6 and the white king arrives in the nick of time with 5.Kf4 and proceeds to slowly push his counterpart away with Kf5-f6-g7. Here is another famous study with a similar idea: [caption id="attachment_136276" align="aligncenter" width="300"] White to play and win - Josef Vančura (1922)[/caption] Black is threatening to play 1...Kg1 so white must move his bishop, but to where? It's highly recommended to analyze each of the bishop moves and discover why only one works while the rest fail. First, here are the moves that do not work and why. The defensive maneuvers of the black king is also noteworthy. 1.Bg4 Ke3 2.h4 Kf4 3.h5 Kg5 = ; 1.Be6 Ke3 2.h4 Ke4 3.h5 Ke5 4.h6 Kf6 5.Bf5 Kf7 6.Bh7 Kf6 = ; 1.Bf1 Ke3 2.Bd3 Kf4 3.Kd4 Kg4 = ; 1.Bc8 Ke3 2.h4 Ke4 3.h5 Ke5 4.h6 Kf6 5.Bf5 Kf7 6.Bh7 Kf6 = ; 1.Bf5 Ke3 2.h4 Kf4 = The key to solve the problem is visualizing this position. [caption id="attachment_136277" align="aligncenter" width="300"] The final position with a beautiful blockade[/caption] It can be reached by 1.Bd7! Ke3 2.h4 Ke4 3.h5! Ke5 4.h6 Kf6 5.Be8! When more is less Put some supporting actors on the board, and the positions can get extra tricky. The following interesting position appeared on the board in a game between two International Masters during the 1996 Olympiad.