This is a logic problem about tennis.

[Terrell]

Homework Statement

Pat beat Stacy in a set of tennis, winning six games to Stacy’s three. Five games were won by the player who did not serve. Who served first?
source: https://ocw.mit.edu/courses/mathematics/18-s34-problem-solving-seminar-fall-2007/assignments/hw8.pdf

Homework Equations

N/A

The Attempt at a Solution

Since Pat won 6 games which is at least 5 games then she is the player who did not serve. Since Pat never served then she could not be the one who served first. Therefore, Stacy. Is my line of reasoning correct?

 

[SammyS]

Homework Statement

Pat beat Stacy in a set of tennis, winning six games to Stacy’s three. Five games were won by the player who did not serve. Who served first?
source: https://ocw.mit.edu/courses/mathematics/18-s34-problem-solving-seminar-fall-2007/assignments/hw8.pdf

Homework Equations

N/A

The Attempt at a Solution

Since Pat won 6 games which is at least 5 games then she is the player who did not serve. Since Pat never served then she could not be the one who served first. Therefore, Stacy. Is my line of reasoning correct?

No. Your line of reasoning has at least one error and could use a bit more explanation.

In tennis, the serve alternates in a given set, from one game to the next.

There were 9 games, so one of the players served 5 games, the other 4.

Therefore, it's wrong to say that "Pat never served".

 

[haruspex]

Five games were won by the player who did not serve.

This could have been clearer: five games were won by the player who was not the server in that game.

 

[Kaura]

Since the server alternates and either one could have been the first server the only possible set of servers can be PSPSPSPSP or SPSPSPSPS

 

[scottdave]

One thing that you can deduce is that Pat won the last game (ending the set). It is sort of obvious, but should help in determining who served on that game.

 

[scottdave]

I was able to work out a scenario for one of the players starting the serve, but not yet with the other one. I am guessing that this is what they were looking for.

 

[haruspex]

Pat beat Stacy in a set of tennis, winning six games to Stacy’s three.

@Terrell, it looks like you are unfamiliar with the rules of tennis. You need to know that the set ends if one player has won six games and the other fewer than 5.
So who won the last game?

 

[Terrell]

@Terrell, it looks like you are unfamiliar with the rules of tennis. You need to know that the set ends if one player has won six games and the other fewer than 5.
So who won the last game?

let me clarify. Do I need to win a set first so I can win each remaining set with only 5 games? I'm totally unfamiliar to the rules except I need to be the first one to win 6 games within a set to win that set.

 

[SammyS]

let me clarify. Do I need to win a set first so I can win each remaining set with only 5 games? I'm totally unfamiliar to the rules except I need to be the first one to win 6 games within a set to win that set.

Did you read each of the replies ?

 

[Terrell]

Therefore, it's wrong to say that "Pat never served".

But the problem directly stated "Five games were won by the player who did not serve". What does the problem mean by that?

 

[Terrell]

One thing that you can deduce is that Pat won the last game (ending the set). It is sort of obvious, but should help in determining who served on that game.

If my understanding of the problem is correct. The player who won is the player winning each of the 5 games she did not serve? So it must be the scenario: SPSPSPSPS. Therefore, Stacy served first is the answer.

 

[Terrell]

This could have been clearer: five games were won by the player who was not the server in that game.

that is a hundred times clearer for me. thanks!

 

[SammyS]

If my understanding of the problem is correct. The player who won is the player winning each of the 5 games she did not serve? So it must be the scenario: SPSPSPSPS. Therefore, Stacy served first is the answer.

Yes, and Pat won each of the 5 games in which Stacy served, plus one more - one of the games in which Pat served.

Added in Edit:
Ignore this post of mine. Obviously it is incorrect.

 

[Terrell]

Yes, and Pat won each of the 5 games in which Stacy served, plus one more - one of the games in which Pat served.

thank you!

 

[haruspex]

The player who won is the player winning each of the 5 games she did not serve?

No. In each game, one player serves, the other receives. Five of the games were won by the receiver in that game. Some of these may have been won by Pat, the rest by Stacy.

 

[haruspex]

Yes, and Pat won each of the 5 games in which Stacy served, plus one more - one of the games in which Pat served.

Please see post #15.

 

[haruspex]

Do I need to win a set first so I can win each remaining set with only 5 games

That makes no sense.
There is only one set here. It consists of a sequence of games.

I need to be the first one to win 6 games within a set to win that set.

Yes. (There is a bit more to it, but not that's relevant to this question.)
And of course the set ends as soon as someone has won it. So who won the last game?

 

[rcgldr]

let me clarify. Do I need to win a set first so I can win each remaining set with only 5 games? I'm totally unfamiliar to the rules except I need to be the first one to win 6 games within a set to win that set.

Even if you don't know the rules of tennis, the problem states that Pat won 6 games, Stacy won 3 games. There were no other games played. As posted previously, the set ended when Pat won for the 6th time.

 

[Terrell]

No. In each game, one player serves, the other receives. Five of the games were won by the receiver in that game. Some of these may have been won by Pat, the rest by Stacy.

okay. that complicates things. hmmm

 

[scottdave]

If my understanding of the problem is correct. The player who won is the player winning each of the 5 games she did not serve? So it must be the scenario: SPSPSPSPS. Therefore, Stacy served first is the answer.

Not necessarily. There were a total of 9 games played. Each of those games was won by a person. They tell you that 5 of those 9 games were won by the person who received (did not serve for that game). So you know that 4 of the games were won by a person that served.

It may be that Pat won 5 games that she received in, or maybe some other combination that can be figured out. It should be obvious that Pat achieved the 6th win on the last game, ending the set.

We have two serve possibilities: PSPSPSPSP or SPSPSPSPS. Take one of these, and look at Pat winning the last game. Then see if you can fit the other criteria. Then look at the other possibility, see if you can fit the criteria into that one.

 

[haruspex]

okay. that complicates things. hmmm

You need to create some variables so that you can write some equations.
Let the number of games where Pat served and won be one of them, and similarly three more.
What equations can you then write?

 

[rcgldr]

It's been a few days, should we offer more hints?

 

[haruspex]

It's been a few days, should we offer more hints?

I vote not. I gave @Terrell plenty to work with in post #21. Looks like he/she has just lost interest.

 

[SammyS]

I vote not. I gave @Terrell plenty to work with in post #21. Looks like he/she has just lost interest.

I agree.

 

[rcgldr]

It's been a few days, should we offer more hints?

I vote not. I gave @Terrell plenty to work with in post #21. Looks like he/she has just lost interest.

I agree.

I was thinking of other people that might be curious about the solution, although anyone that know tennis well shouldn't have an issue with this problem.

 

[Terrell]

I vote not. I gave @Terrell plenty to work with in post #21. Looks like he/she has just lost interest.

sorry i have had just plenty of school work to prioritize. I'm still interested!

 

[Terrell]

i'll work on this after finals week! how could i lose interest in this lol

 

[Terrell]

We have two serve possibilities: PSPSPSPSP or SPSPSPSPS. Take one of these, and look at Pat winning the last game. Then see if you can fit the other criteria. Then look at the other possibility, see if you can fit the criteria into that one.

thanks. i understand the idea.. will work on this. Sometimes schoolwork becomes so uninteresting so I needed to find something interesting :)

 

[haruspex]

sorry i have had just plenty of school work to prioritize. I'm still interested!

Welcome back.

 

[Terrell]

let:
$$P_s$$ be the no. of games won by Pat where she served.
$$P_r$$ be the no. of games won by Pat where she received.
$$S_s$$ be the no. of games won by Pat where she served.
$$S_r$$ be the no. of games won by Pat where she received.

$P_r$ + $S_r$ = $5$
6 - $P_r$ = $P_r$
3 - $S_s$ = $S_r$
$S_s$ + $S_r$ = 3

are these the equations @haruspex ?
still working on it :D
sorry for the terrible formatting.

 

[haruspex]

$P_r+ S_r = 5$

Yes

$6 - S_r= P_r$

No. I don't think that is what you meant to write.

$S_s+ S_r = 3$

Yes.
I fixed up the latex by changing your single dollar signs to double hash signs (#).

 

[rcgldr]

... Ss be the no. of games won by Pat where she served. ... S_r be the no. of games won by Pat where she received.

That should be Ss = # games won by Stacy when she served ... Sr # games won by Stacy when she received.

6 - P_r = P_r

That should be 6 - Pr = Ps, or rewriten, Ps + Pr = 6 (number of games won by Pat). You already have Ss + Sr = 3 (number of games won by Stacy).

As scottdave posted, there are only two possible orderings for who served in a set PSPSPSPSP or SPSPSPSPS, which you responded to previously, but it's not clear if you've followed up on this.

 

[Terrell]

That should be Ss = # games won by Stacy when she served ... Sr # games won by Stacy when she received.

That should be 6 - Pr = Ps, or rewriten, Ps + Pr = 6 (number of games won by Pat). You already have Ss + Sr = 3 (number of games won by Stacy).

As scottdave posted, there are only two possible orderings for who served in a set PSPSPSPSP or SPSPSPSPS, which you responded to previously, but it's not clear if you've followed up on this.

yeah those were typos. i got a bit lazy typing it. i'll work on this if i still have the brain power since schoolwork deadlines are near.

 

[Terrell]

I almost forgot about this after burning out on finals week, anyway here is what I got:

$P_s + S_s = 4$
$P_r + S_r = 5$
$P_r + P_s = 6$
$S_r + S_s = 3$

then we get the following clues to deduce from the case1: PSPSPSPSP or case2: SPSPSPSPS:

$P_r - S_s = 2$
$P_s - S_r = 1$

Observing case 2 first, we must have $P_r + S_s = 5$ and due to $P_r - S_s = 2$, it must be that $P_r > S_s$. Furthermore, $P_r$ cannot be 5 or 4 since $5 - 0 \neq 2$ and $4 - 1 \neq 2$. It is reasonable that $P_r be \leq 3$, but it won't satisfy $P_r + S_s = 5$. However, for case 1 we can set $P_r = P_s = 3$ and $S_s = 1$ and $S_r = 2$ to satisfy the clues given above. Therefore, Pat served first. Did get it right?

 

[haruspex]

Yes, you got it right, but having reached this point for case 2

we must have $P_r + S_s = 5$ and due to $P_r - S_s = 2$,

you can solve for those two variables and discover an impossibility.

 

[Terrell]

Yes, you got it right, but having reached this point for case 2

you can solve for those two variables and discover an impossibility.

because $P_r$ cannot be 7/2 ? Also, in retrospect, I never thought that I need to set up an equation to work on this problem so are there problems that does not need setting up equations? thank you!

 

[haruspex]

because $P_r$ cannot be 7/2 ? Also, in retrospect, I never thought that I need to set up an equation to work on this problem so are there problems that does not need setting up equations? thank you!

Equations are just a convenient way of expressing reasoning. The Ancient Greeks managed without. Whether it is worth encoding a problem as equations depends on how succinct the notation is and how complex the problem.

 

[rcgldr]

Also, in retrospect, I never thought that I need to set up an equation to work on this problem so are there problems that does not need setting up equations? thank you!

This can be done logically. If Pat wins on Stacy's serve and Stacy win's on Pat serve, then the set's game score is the same as if Pat won on her serve and Stacy won on her serve. The key factor is how many times Pat won on Stacy's serve minus how many times Stacy won on Pat's serve. Assume game score is even and equal to the number of times Stacy won at some point in the set. Consider the case where the game score is Pat 4, Stacy 4, and that Pat wins the last two games, the set ends up Pat 6, Stacy 4 regardless of who served first. Now consider the case where the game score is Pat 3, Stacy 3, and that Pat wins the last three games. If Stacy served first, then Pat won both times Stacy served in the last 3 games with Pat winning on Stacy's serve 2 more times than Stacy won on Pat's serve, but 5 can't be split up so that the difference equals 2. If Pat served first, then Pat only won once on Stacy's serve in the last 3 games, Pat winning on Stacy's serve 1 more time than Stacy won on Pat's serve, and 5 can be split up so the difference equals 1, specifically 3 - 2 = 1.

In tennis when a player wins when the other serves it's called a "break".
A list of possible outcomes for a set won when the winner wins their 6th game:

6 4 => +1 break, doesn't mater who served first
6 3 => +1 break if winner served first, +2 breaks if loser served first
6 2 => +2 breaks, doesn't matter who served first
6 1 => +2 breaks if winner served first, +3 breaks if loser served first
6 0 => +3 breaks, doesn't matter who served first