In tennis a set is won by the first player to win 6 games unless the score reaches 5-5, in which case the set is won 7-5 or 7-6.

At Wimbledon the first batch of tennis balls is used for the warmup and the first 7 games. Each subsequent batch of balls is used for 9 games. As a result of these rules, there are certain scores during the course of play of the second set, that, if this procedure is followed, a new batch of balls would never be called in upon being reached.

Dick was officiating at a match which he called for a new batch of balls in the second set at three points among these occasions when supposedly a new batch of balls could never be appropriate, as described in the preceding paragraph.

The two competitors were Tom and Harry, and the number of games that Tom had won in the second set was the same at the times of the second and third of these three "impossible" calls for new balls.

What was the score at each of the times that Dick called for new balls when in theory he should never call for them at such scores?

(In reply to re: Solution by Charlie)

The possible winning scores are 6-0, 6-1, 6-2, 6-3, 6-4, 7-5 and 7-6; thus the possible number of games in a set are 6, 7, 8, 9, 10, 12, and 13.
If the first set was won after 6 games, the call for a new balls would occur after the first game of the second set with the next "possible" call after the tenth game.
If the first set was won after 7 games, the call for new balls would have occurred after the first set and before the beginning of the second set with the next "possible" call, assuming the second set has not yet been won, after the the ninth game.
If the first set was won after 8 games, the call for new balls would have occured between game 7 and game 8 of the first set with the next "possible" call, assuming the second set has not yet been won, after the eighth game.
If the first set was won after 9 games, the call for new balls would have occured between game 7 and game 8 of the first set with the next "possible" call, assuming the second set has not yet been won, after the seventh game.
If the first set was won after 10 games, the call for new balls would have occured between game 7 and game 8 of the first set with the next "possible" call, assuming the second set has not yet been won, after the sixth game.
If the first set was won after 12 games, the call for new balls would have occured between game 7 and game 8 of the first set with the next "possible" call after the fourth game.
If the first set was won after 13 games, the call for new balls would have occured between game 7 and game 8 of the first set with the next "possible" call after the third game, and again, assuming the second set has not yet been won, after the twelfth game.

The "possible" call for new balls, therefore, do not occur after the second, fifth, and eleventh games of the second set.

As it is given that there were three "impossible" calls during the second set, the set must have been won 7-6 in the thirteen game or 7-5 in the twelfth game. As Tom's score remained the same in the fifth and twelfth games, the second set must have had one of the following scores: 
       
       1 2 3 4 5 6 7 8 9 10 11 12 13
Tom    1 2 3 4 5 5 5 5 5  5  5  6  6
Harry  0 0 0 0 0 1 2 3 4  5  6  6  7

       1 2 3 4 5 6 7 8 9 10 11 12 13
Tom    1 2 3 4 5 5 5 5 5  5  5  6  7
Harry  0 0 0 0 0 1 2 3 4  5  6  6  6

       1 2 3 4 5 6 7 8 9 10 11 12
Tom    1 2 3 4 5 5 5 5 5  5  5  5
Harry  0 0 0 0 0 1 2 3 4  5  6  7

The scores at the end of the second, fifth and eleventh games were 2-0, 5-0, and 5-6.

Posted by Dej Mar on 2008-01-30 22:51:30