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Problem 4 (Tennis Anyone ?)

Anne, Brent, Cindy, and Dylan are playing tennis doubles (two versus two). First, Anne and Brent play against Cindy and Dylan; next Anne and Cindy play against Brent and Dylan; finally Anne and Dylan play against Brent and Cindy. Show that for every possible outcome of the three games, there is one player that is either on the winning team every game or on the losing team every game.

Solution
If we enumerate the different outcomes of the three games, we have a total of eight outcomes illustrated by the tree below. The labels on the edges of the tree indicate the winner of each game. For example, outcome 1 means ab (Anne and Brent) win game 1, ac (Anne and Cindy) win game 2, and ad (Anne and Dylan) win game three. Likewise, outcome 4 means ab win game 1, bd wins game 2, and bc win game 3.

If you look carefully, you will see that for each of the eight outcomes, there is exactly one player that will be either on the winning or losing team.

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Last modified on Tuesday, January 12, 1999