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## [My Computational Complexity Web Log] The Beauty of the Magic Number

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• Baseball has a large number of mathematical nuggets but since my childhood I have always liked the simplicity of the magic number. In a division race, the
Message 1 of 1 , Sep 21, 2004
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Baseball has a large number of mathematical nuggets but since my childhood I have always liked the simplicity of the magic number. In a division race, the magic number is the minimum number of wins by the first place team and number of losses of the second place team to guarantee the first place team wins the division.

Let's do an example. As I write this The New York Yankees have 94 wins, the Boston Red Sox have 60 losses. The easiest way to compute the magic number comes from working backwards from the definition. There are 162 games in a season so the Yankees magic number is 162+1-(94+60) = 9. Any combination of nine Yankees wins and Red Sox losses and the Yankees wins the American League East. The "+1" comes from the fact that in a tie the Yankees would still need to win a one-game playoff to win the division.

What can the magic number teach us about complexity? Consider the RIOT Baseball Project at Berkeley. Not satisfied with the magic number, the project computes the First Place Clinch Number as the "Number of additional games, if won, guarantees a first-place finish." To compute this number one has to look not only at the current standings but the schedule of remaining games between the teams.

My main issue of the clinch number relates to complexity. Not only is it more complicated to compute; to update the clinch number after a game sometimes requires recomputing the number from scratch. The magic number has a simple update function counting down like a rocket launch. Yankees win the magic number drops by one. Red Sox lose the magic number drops by one. If the Yankees beat the Red Sox, both events happen so the magic number drops by two. And once the magic number hits zero you pop the champagne. That's the beauty of the magic number.

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Posted by Lance to My Computational Complexity Web Log at 9/21/2004 07:39:53 AM

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