A beautiful draw
Please refer to the chess position here. http://home.comcast.net/~joyner.david/wdj/chess/Elkies05_files/Elkies05D02.gif
White to play and draw. Here the difficulty for white is that he cannot penetrate into black's fortess whereas black can get through via 2 ways into white's territory. Inspite of this, White moves marvelously to gain draw. How?
Answer:
1 Kb2! Kg8 2 Ka1!!.
Begin by observing that the White king is permanently restrained by the rigid pawn structure, but the Black king has two potential routes for penetration, via g4-f3 or c6. Thus, whenever Black plays Kg4 White must respond Ke2 or Kf2, and actually Ke2 is forced since the White king must be able to reach a5 in the four moves it takes the Black king to reach c6 from g4 (there's no time for White to reach b5 first, and conversely if White ever responds Kb5 to Black's Kd7 he'll never get back to the kingside in time to stop Kf3). Next note that there's only one way for the Black king to get from g4 to c6 in four moves, and likewise for the White king from e2 to a5; so we have five pairs of corresponding squares: we've already seen that Black's g4 corresponds to White's e2, and c6 to a5, and now we also have f5, e6, d7 corresponding to d2, c3, b4 respectively (for instance on Black's Kf5 White must respond Kd2 so as to be within one move of f2 and within three of a5). What, then, if Black plays Kf6? This threatens both Kf5 and Ke6, to which White must respond Kd2 and Kc3 respectively; so on Kf6 White must play Kc2. Again, Kg5 threatens Kg4, Kf5 and Kf6 and so forces Kd1 to control the corresponding e2, d2, c2. Proceeding in this manner we find the corresponding pairs e7:b3, g6:c1, f7:b2, e8:a3, d8:a4, h5:d2, h6:c2, g7:b1, f8:a2, and finally h7:b2 and g8:a1 which explains White's mysterious retreat to the corner. Adding to this the pair h8:a2 we further find that all these corresponding squares give mutual zugzwangs: White to move must abandon the corresponding square and lose, but with Black to move the White king can always stay on the corresponding square and rebuff Black's incursions via g4 or c6 indefinitely, assuring the draw.
White to play and draw. Here the difficulty for white is that he cannot penetrate into black's fortess whereas black can get through via 2 ways into white's territory. Inspite of this, White moves marvelously to gain draw. How?
Answer:
1 Kb2! Kg8 2 Ka1!!.
Begin by observing that the White king is permanently restrained by the rigid pawn structure, but the Black king has two potential routes for penetration, via g4-f3 or c6. Thus, whenever Black plays Kg4 White must respond Ke2 or Kf2, and actually Ke2 is forced since the White king must be able to reach a5 in the four moves it takes the Black king to reach c6 from g4 (there's no time for White to reach b5 first, and conversely if White ever responds Kb5 to Black's Kd7 he'll never get back to the kingside in time to stop Kf3). Next note that there's only one way for the Black king to get from g4 to c6 in four moves, and likewise for the White king from e2 to a5; so we have five pairs of corresponding squares: we've already seen that Black's g4 corresponds to White's e2, and c6 to a5, and now we also have f5, e6, d7 corresponding to d2, c3, b4 respectively (for instance on Black's Kf5 White must respond Kd2 so as to be within one move of f2 and within three of a5). What, then, if Black plays Kf6? This threatens both Kf5 and Ke6, to which White must respond Kd2 and Kc3 respectively; so on Kf6 White must play Kc2. Again, Kg5 threatens Kg4, Kf5 and Kf6 and so forces Kd1 to control the corresponding e2, d2, c2. Proceeding in this manner we find the corresponding pairs e7:b3, g6:c1, f7:b2, e8:a3, d8:a4, h5:d2, h6:c2, g7:b1, f8:a2, and finally h7:b2 and g8:a1 which explains White's mysterious retreat to the corner. Adding to this the pair h8:a2 we further find that all these corresponding squares give mutual zugzwangs: White to move must abandon the corresponding square and lose, but with Black to move the White king can always stay on the corresponding square and rebuff Black's incursions via g4 or c6 indefinitely, assuring the draw.
posted by padmaram @ 6:25 PM
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