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Puzzled: Wins in a Row


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chess pieces

Credit: Paolo Gallo / PerseoMedusa

Each of these puzzles involves game-playing strategy. If you are sufficiently cleverand sufficiently unmotivated to work hard at being cleveryou can solve them all without resorting to algebra. Here is the premise: You have applied to join a chess club and been told that to qualify you must play three games against Ioana (the last new member), winning two games in a row. "Who gets the white pieces?" you ask and are told you and Ioana alternate and you get to decide whether to start with white or with black.

  1. Knowing that the probability of beating Ioana is better with the white pieces (first-move advantage), should you choose white or black for the first game?
  2. Suppose now that prior to the match, you discover Ioana is a former City Champion of Bucharest. Complaining, you persuade the club to give you the following concession: You must still beat Ioana two games in a row, but now you get to play her as many as 17 times, alternating sides as before. Should you choose white or black for the first game?
  3. You managed to win two games in a row from Ioana and are now a member of the club. However, to become an officer of the club, you must beat Ioana 10 times in a row, with 49 games to do it. Yipes! This may be more than you can handle, but to maximize your chances, should you start with white or with black?

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Author

Peter Winkler (puzzled@cacm.acm.org) is William Morrill Professor of Mathematics and Computer Science at Dartmouth College, Hanover, NH.

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Footnotes

Readers are encouraged to submit prospective puzzles for future columns to puzzled@cacm.acm.org.


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