Tennis puzzle

plutoman · 2008-01-21 21:29:26

Hi,

I was lucky enough last week to be in Melbourne (what an amazing place) and see some of the incredible tennis there. So here's a tennis question:
Unlike in most sports, you can win a tennis match despite winning fewer points than your opponent. What's the *biggest* negative points difference you can have, and still win the match?
I realise there is more than one answer here, but that makes the question more interesting, doesn't it?

JaneFairfax · 2008-01-22 00:29:13

Assuming its a mens Grand Slam match, you can win a match with your opponent having won 71 points you none at all. Your opponent wins the first two sets 6-0 6-0 and the first five games of the third set without dropping a single point, but while on forty-love up in the sixth game of the third set, he concedes the match.

mathsyperson · 2008-01-22 02:00:20

If the last set doesn't have a tie-break at 6-6 (and so the match keeps going until someone is two games ahead), then the difference could be potentially infinite.

6-6
Opponent gets four straight points and wins the game.

6-7
Game gets to 30-30, then our player wins two points and wins the game.

7-7

This is essentially the same situation as the 6-6 was (both players still need two games in a row to win) but in those two games, the opponent scored two more points than our player. Keep repeating that cycle for as long as you want, and then eventually make your player win.
By those rules, there's no limit to what the point difference could be.

NullRoot · 2008-01-22 03:00:17

THAT would be an interesting game. Let it run for infinite games and the scores are both equal and infinitely different....
2p - p = d

As p -> ∞, d -> ∞
And as p -> ∞, 2p -> ∞

It's like P and R all over again.

Ricky · 2008-01-22 04:12:33

If the last set doesn't have a tie-break at 6-6 (and so the match keeps going until someone is two games ahead), then the difference could be potentially infinite.

I believe that's only in the French open, no? At least, I'm fairly certain the US open has a 5th set tiebreaker.

If such were the case, you lose every game with 0 points, win every other game by 2. That's 12 points per set, making it a 60 point difference. Now in the tiebreaker, 3 sets you win by 2, 2 sets you lose by 7. So that's +6 - 14 = -8. Thus, the total is a -68 difference.

This is out of 4*6*5 + 6*6*5 + 7*2 + 12*3 = 120+180+14+36 = 350 points, which is rather significant.

mathsyperson · 2008-01-22 05:50:59

Ricky wrote:

If the last set doesn't have a tie-break at 6-6 (and so the match keeps going until someone is two games ahead), then the difference could be potentially infinite.
I believe that's only in the French open, no? At least, I'm fairly certain the US open has a 5th set tiebreaker.

The only one I'm certain of is Wimbledon, but that follows the rules I described. US Open rules makes for a less trivial puzzle though.

I think your number can be beaten by making it so that our player only just scrapes wins (letting it get to 6-6 and winning the tie-breaker by a nose), but when the opponent wins a set, he does so with a clean sweep.

As you worked out, for the sets that our player wins the opponent will be 12 points up when it gets to 6-6. He'll then gain two points back at the tie-break and so each of those three sets will have a -10 point difference.

As Jane worked out, if the opponent steams through a set without conceding a single point, that would put him 24 points up.

Therefore, the total is -30 - 48 = -78. If we allow Jane's idea of the opponent conceding as he's about to win the match, that difference could probably be boosted higher still.

Also, what's P and R?

JaneFairfax · 2008-01-22 13:03:51

All right, lets suppose it goes to five sets, and that the tiebreak rule applies to the fifth set as well. In order to take the match to five sets, you gotta win two sets and lose two sets. You can lose the two sets 0-6, conceding 24 points in each set. The other two sets must be won on tiebreak. In each, you must win 7 games (including the tiebreak) by 2 points and lose 6 games by 4 points, thus scoring 6×4−7×2 = 10 points fewer than your opponent. Then you will have scored 68 points fewer than your opponent by the time the fifth set arrives. In the fifth set, your opponent can win 23 consecutive points before conceding the match. In the end, you will have scored 91 points fewer than your opponent.

plutoman · 2008-01-22 19:00:06

Some great responses here. In fact the US Open is the only Grand Slam that does play a tie-break in the fifth set - the others all require a two-game margin when the score reaches 6-6. Indeed Federer was taken to 10-8 in the final set in the Aussie Open a few days ago.

You've pretty much covered all bases now. If there's no tie-break in the fifth, the points difference can be potentially infinite. I like the way NullRoot put it: "scores are both equal and infinitely different".

And if there is a fifth-set tie-break, -78 is the magic number as Mathsyperson explained. Retiring from the match did cross my mind, but retiring at 0-5, 0-40 in the fifth set to give a difference of -91 never occurred to me, so well done Jane!

It might also be interesting to look at the lowest percentage of points you can win while still winning the match. I guess (for the 5th-set tie-break case) you'd do that by making the total number of points as small as possible while keeping the difference at -78. So all the games our player wins (except the tie-breaks) would have 6 points, all the opponent's games would have 4, and all the tie-breaks would have 12. I make that a total of 264 points, of which our player wins just 93, or 35.23%.

For the 'no tie-break' rule, I reckon you'd need a 6-4 final set, meaning our player would win 86 out of 244 points, or 35.25%.

Anyone want to check these figures or my reasoning? The fact that it's possible (though extremely unlikely) to win a tennis match with barely a third of the points shows that not all points are equal. I suppose that's part of the beauty of the tennis scoring system.

And NullRoot, what's P and R ?

NullRoot · 2008-01-22 22:00:03

What's P and R ?

Couldn't remember the LaTeX codes for them at the time. They would have made more sense if they weren't just bold.

I was referring to the set of prime numbers, P, and the real numbers R. Both contain infinite elements (points?) and if you remove the elements of P from R, R still remains infinite.

A better example might have been the set of all whole numbers and the set of all even whole numbers, but Prime and Real was what sprang to mind at the time.

mathsyperson · 2008-01-23 05:19:30

plutoman wrote:

It might also be interesting to look at the lowest percentage of points you can win while still winning the match.

Potentially 0% if the opponent concedes.
Otherwise, I think you're spot on.

Edit: For future reference, NullRoot, you get the nifty double-lined letter thingies by using the \mathbb command whilst writing LaTeX.

Math Is Fun Forum

#1 2008-01-21 21:29:26

Tennis puzzle

#2 2008-01-22 00:29:13

Re: Tennis puzzle

#3 2008-01-22 02:00:20

Re: Tennis puzzle

#4 2008-01-22 03:00:17

Re: Tennis puzzle

#5 2008-01-22 04:12:33

Re: Tennis puzzle

#6 2008-01-22 05:50:59

Re: Tennis puzzle

#7 2008-01-22 13:03:51

Re: Tennis puzzle

#8 2008-01-22 19:00:06

Re: Tennis puzzle

#9 2008-01-22 22:00:03

Re: Tennis puzzle

#10 2008-01-23 05:19:30

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