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Tennis Balls (Posted on 2008-01-29) Difficulty: 3 of 5
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?

  Submitted by Charlie    
Rating: 5.0000 (1 votes)
Solution: (Hide)
During the first set, anywhere from 6 to 10, or 12 or 13, games would have been played, corresponding to scores of 6-0 through 6-4, and 7-5 or 7-6. If it were 6, then the standard procedure would have been to call for new balls after 1 game in the second set and again after game 10. If 7 through 10, new balls would be called for after 9, 8, 7 or 6 games of the second set; if 12 or 13, that would be after games 4 or 3 of the new set and then again after game 13 or 12.

So possible calls for new balls could come after 1, 3, 4, 6, 7, 8, 9, 10, 12 or 13 games. So the number of games after which it would be impossible, under the ordinary rules, to call for new balls would be 2, 5 or 11. The possible scores after such numbers of games have been played in the second set are:

 2   1-1, 2-0
 5   5-0, 4-1, 3-2
11   6-5

Dick's "impossible" call for new balls must have been after games 2, 5 and 11 of the second set, obviously in that order. The only possible shared score between end of game 5 and end of game 11 is 5, shared by 5-0 and 6-5, so at the second miscall, Tom was leading 5-0, but then, at the last miscall, Harry was leading 6-5.

Since Harry was scoreless after 5 games in the second set, the score couldn't have been 1-1 after 2 games, and so Tom must have led Harry 2-0 at that time.

Based on Enigma No. 1472 by Richard England, New Scientist, 8 December, 2007.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Puzzle Thoughts K Sengupta2023-06-19 03:19:22
Solutionre(2): SolutionDej Mar2008-01-30 22:51:30
Hints/Tipsre: SolutionCharlie2008-01-30 15:53:30
SolutionDej Mar2008-01-30 04:05:22
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