Abe, Ben, and Catelyn were playing 1-on-1 tennis matches one afternoon. After each match, the winner stayed on the court and the loser was replaced by the person who sat out.

At the end of their session, Abe had played 8 matches, Ben had played 12 matches, and Catelyn had played 14 matches. Ben was particularly exhausted as he had played all of the last 7 matches.

Is it possible to figure out who played and who won in the 4th match?

#### Solution

Since the three played in 34 matches between them, that means there were 17 matches played.

The key insight is: since each player only sits out for one match, each player plays in at least every other match. This means no player can play less than 8 of the 17 matches, and this is only possible if that player loses every match.

Abe in fact only played in 8 matches, which means we know exactly which games he must have played (and lost): matches 2, 4, 6, 8, 10, 12, 14, and 16. This means Abe must have lost the 4th match.

So we just need to figure out who Abe played against in match 4.

Ben must have played against Catelyn in every match not involving Abe. And since Ben played the last 7 matches, we know he must have played games 1, 3, 5, 7, 9, 11, 12, 13, 14, 15, 16, and 17.

This captures all 12 matches that Ben played, so we know that Catelyn played in all of the remaining matches. Therefore, Catelyn beat Abe in the 4th match.