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(4): Solution, Standing on the shoulders of about.com by tomarken)

tomarken, you CAN win a tie-breaker 7-6.  The margin of 2 only applies to ending a set before 12 games.  That aside...

You are correct in saying 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...".

 

But in the games he loses, and for the loser to make the maximum number of points in the remaining 3 sets but still lose, he must win half (6 games) of the set (4-0), lose half  (6 games) of the set, and lose the tie-breaker of each set only by 1.  Thus, for each set he will win 6 (4-0)games, lose 6 (3-4) games, and lose the tie-breaker (3-4).  The loser may therefore win only 21 points in the 7 games he loses (the 6 games and the tie-breaker), while the winner will win a minimum of 28 points in each of the 7 games.

 

 

Therefore, L – W = 24(x-1)/2 + (24+21)(x+1)/2 – 28(x+1)/2

 

…and simplifying, L - W = (41x -7)/2

 

For a 5-set match, the loser will score (41*5 - 7)/2 = 99 more points than the winner. 

Edited on March 27, 2006, 2:58 pm

Posted by Dej Mar on 2006-03-27 13:24:59