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Table Tennis Serve (Posted on 2013-07-17) Difficulty: 3 of 5
In a game of table tennis, 24 of the 37 points played were won by the player serving, and Smith beat Jones 21-16. Remembering in table tennis that service alternates every five points, who served first?

No Solution Yet Submitted by K Sengupta    
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Solution impossible outcome (spoiler) | Comment 4 of 6 |
Disagree.  The winner does not necessarily serve the winning point, as Ady assumed.

If the loser serves first, then the loser serves 20 times and the winner 17.
Let a be the number of points won by the winner while serving.
Then the loser wins 17-a points while the winner is serving.
So the loser, who only got 16 points won the other 16 - (17-a) = a - 1 points while he was serving.
Total points won by the server = a + (a - 1) = 2a - 1, which is an odd number.
So the loser did not serve first.

Just to check, if the winner serves first, then the winner serves 20 times and the loser 17.
Let a be the number of points won by the winner while serving.
Then the loser wins 20-a points while the winner is serving.
So the loser, who only got 16 points won the other 16 - (20-a) = a - 4 points while he was serving.
Total points won by the server = a + (a - 4) = 2a - 4, which is an even number.
So the winner (Smith) served first.

24 = 2a - 4, so a = 14.  
Smith won 14 points while Smith was serving and 7 while Jones was serving
Jones won 10 points while Jones was serving and 6 points while Smith was serving.

Ady's example, on the first post, was actually an example of the winner serving.  And the 24 was not totally irrelevant.  The relevant information is that it was even.

  Posted by Steve Herman on 2013-07-17 12:58:38
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