# Tennis

Alice, Bob, and Charlie play tennis matches consisting of one set only: two of them play a set and the winner stays on the court for the next set, with the loser replaced by the player who was idle. At the end of the day Alice played 15 sets, Bob played 14 sets, and Charlie played 9 sets. Who played in the 13th set? (This problem came from the Macalester Math Problem of the Week, which gave its origin as Problem 33 in Dick Hess’s book, All-Star Mathlete Puzzles, Sterling, 2009.) The best answer submitted to dhousman@goshen.edu will win a small prize.

Ruthi Wien’s answer: The game consists of 19 sets which is simply all the sets played added together (14+15+9)= 38 and divided by the number of players who play in a single set which is 2. So 38/2 = 19 total sets which were played. Because the winner always must play the player sitting idle, no two players can play each other twice in a row. Therefore, every player must play at least once in the course of two games. Since we know that Charlie must play at least once every two games, if he is one of the first players in the first round playing against either Bob or Alice and only plays once every two rounds (meaning, he always loses) he will have played 10 games (every other game for the 1st 18 rounds (9 times) and then because he started he is also in line to play the last round, resulting in 10 total games) However we are told that he only plays 9 times. So he must have not played in the first round, and therefore would only have played 9 times in the remaining 18 rounds by losing consistently and playing only every other game. If we number every round in ascending order – the first round #1, the last round #19, we can say that Charlie will only play games that are even numbered because he does not play in the first round and only plays every other round from then on. Therefore we can look at round number 13, and without having to know how Bob or Alice played, we know that Charlie is NOT playing in round #13 because it is an odd numbered game, and therefore the only two players who could be playing in it are Bob and Alice.

Hannah Geiser’s answer: The 13th set must be played by Alice and Bob. After each set a new player must enter the game, and since there are only three people, the least a person can play is once every other game, and then that person loses. This position falls to Charlie. Alice and Bob switch off between winning against each other, beating Charlie, and then losing to the other. Alice and Bob start off playing each other and then they play together every other games, so on the odd games they play each other. There are 19 total games because that is the sum of the games each person played divided by two because each game requires two people.
1.  AB
2.  AC
3.  AB
4.  CB
5.  AB
6.  AC
7.  AB
8.  CB
9.  AB
10. AC
11. AB
12. CB
13. AB
14. AC
15. AB
16. CB
17. AB
18. AC
19. AB