In a game of tennis, the player who puts in the most effort in a match, and wins the majority of points, does not necessarily win the match as a whole.

Imagine two tennis players compete in a 5-set match, with each set following the scoring system of tennis, and a first to 7 point tie-break takes place if the score in a set is 6 games each. Let the total number of points won by the person who wins the match be represented by W, and let the total number of points won by the person who loses the match be represented by L.

If by the end of the match L-W is equal to a POSITIVE integer, then what is the maximum value this integer can be? Furthermore, develop an equation to determine the integer formed from L-W for a match of x number of sets.

Note: Enough information regarding the scoring system in tennis required to solve the problem, can be found at http://tennis.about.com/cs/beginners/a/beginnerscore.htm

(In reply to re: Solution, Standing on the shoulders of giants. by Hugo)

Ah, good call, I must've read it too quickly.  It shouldn't be too hard to convert the formula to calculate points instead of games, it's the same idea.

The loser will score 24 points in each of the (x-1)/2 sets that he wins 6-0.  In the sets he loses 7-6, he will score 24 points in the 6 games he wins, 18 points in the games he loses, and 6 points in the tiebreak, for a total of 48 points in each of the (x+1)/2 sets he loses.

The winner will score 38 points in each of the (x+1)/2 sets he wins, and none in the rest.

Therefore, L - W = 24(x-1)/2 + 48(x+1)/2 - 38(x+1)/2

This simplifies to L - W = 12(x-1) + 5(x+1) = 17x - 7

For a 5-set match, the loser will score 48 + 30 = 78 more points than the winner.

 

Just to clarify where those numbers came from:

When the "loser" wins a set, he wins 6 games to 0, and in each of those 6 games he wins 4 points to 0.

When the "winner" wins a set, he wins 7 games to 6 (the 7th being the tiebreaker).  In each of the 6 games he loses, he is outscored 4 points to 0.  In each of the 6 games he wins, he wins 5 points to 3.  In the tiebreaker, he wins 8 points to 6 (as I believe in tennis, all games/tiebreakers must be won by 2 points).

I think that's right, but I hope somebody checks it. :)

Posted by tomarken on 2006-03-27 09:51:18