Newsgroups: rec.puzzles
From: Dan Hoey <haoyuep@aol.com>
Date: Thu, 21 Aug 2008 19:33:58 -0400
Subject: Re: Enigma 1501 - Five sets at Wimbledon

James Dow Allen wrote:
> Remark that in a fifth set going to extra points
> (score of 8-6, or 9-7, etc.) the server wins an odd
> number of games.

To rule out 4, 6, or 8 in the 6-4 set, you need the
slightly stronger result that if the point spread is
2, then the server wins an odd number of games.

Proof:  If the point spread is 2, the score is N+1
to N-1, and the number of games is 2N, with each
player serving N times.  If the winner wins K of
his serves, then he must win N+1-K of the loser's
serves.  The loser then wins N-(N+1-K) = K-1 of his
serves, and the total number of server wins is 2K-1,
QED.

This can be generalized to show that in any sport
consisting of games with alternating servers, the
server wins an odd number of games if the point
spread is 2 mod 4, and an even number of games if
the point spread is 0 mod 4.  The parity for odd
point spreads depends on whether the first server
wins.

Dan Hoey
haoyuep at aol.com