Logic Challenge #28 - Toss, Toss, Toss

[kshircliff]

1. E
2. E
3. D
4. C
5. B
6. A

1.
Not A: P cannot toss to K
Not B: N can only receive a toss from M
Not C: K can only toss to P
Not D: L cannot pass to N
*E: L can pass to M; M can pass to L; L can pass to M

2. K must pass to P, P must pass to O, so: KPO_
Not A: K can only toss to P
Not B: K can only toss to P
Not C: M can only receive from L
Not D: N can only receive from K
*E: P can receive from O

3. _ _ _ O
A: If N starts, order must be NKPO
B: Could be MLPO
C: Could be LMPO
*D: If K starts, order must be KPO, but O can't pass to O
E: Could be POPO

4. If N wins, order must be _LMN
K must pass to P - not KLMN
L can't pass to L- not LLMN
*M can pass to L - MLMN
N can't pass to L - not NLMN
*O can pass to L - OLMN
P can't pass to L - not PLMN
**2 orders
*C

5. O _ _ _
K can't receive from O
*L can receive from O
M can't receive from O
N can't receive from O
*P can receive from O
O can't receive from O
** 2 children
*B

6.
*A:N passes to K, K to P, P to O, but N can't receive from O
B:M can pass to O, O to L, L to M
C:L can pass to P, P to O, O can pass to L
D:K passes to P, P to O, O can pass to P
E:K passes to P, P to O, O can pass to L

[ManhattanPrepLSAT2]

Thanks kshircliff!

Do u mind sharing with us what work, if any, you did before diving into the questions? In particular, I'm sure others would be interested in knowing what type of diagram you set up, etc.

BTW -- make sure to double check that you read all the questions correctly.

[tiourina]

Agree with kshircliff except # 5.

1E, 2E, 3D, 4C, 5C, 6A

5.
O - P - O
O - L - M
O - L - P

[sirromlehcar]

1. E
2. E
3. D
4. C
5. C
6. A

1.
Option A: M>L>P is possible, but P can only toss to O. So option A cannot work.
Option B: M>P>O is possible, but N can only receive a toss from M. So option B cannot work.
Option C: M>N>K is possible, but K can toss to P. So option C cannot work.
Option D: L can only toss to P or M, so Option D does not work from the first pass.
Option E: L can only pass to M or P, so L>M works. M is unrestricted in who it passes to other than K, but M passes to L, so that is possible. L must pass to M or P once again, so L>P works
For question 1, option E is the correct answer.

2.
Option A: K must toss to P and P must toss to O, so L cannot receive the second toss. Option A is not possible
Option B: K must toss to P and P must toss to O, so M cannot receive the second toss. Option B is not possible
Option C: K must toss to P and P must toss to O. M can only receive from L, it cannot receive from O. Option C is not possible.
Option D: K must toss to P and P must toss to O. N can only receive the ball from M, M can only receive the ball from L. N cannot receive the ball from O. Option D is not possible.
Option E: K must toss to P and P must toss to O. O can then toss to P because P has no restrictions on who it can receive from and O has no restrictions on who it can toss to. Option E is the correct answer.

3.
Option A: N starts. N must toss to K, K must toss to P, P must toss to O. (Possible)
Option B: M starts. M can pass to anyone but K and M.
M>N>K>P (not possible, K must toss to P, it cannot pass to O)
M>L>P>O (possible)(can stop here, but I will show other options)
M>L>M>O (possible)
M>O>P>O (possible)
M>P>O>O (not possible, O cannot pass to O)
Option C: L starts. L must toss to P or M.
L>P>O>O (not possible, O cannot pass to O)
L>M>P>O (possible)
No other options starting with L are possible.
Option D: K starts. K must toss to P, P must toss to O.O cannot pass to O. Option D is not possible. (At this point D is established as the answer, but I will also show the reasoning eliminating E as an option next)
Option E: P starts. P can only toss to O. O can toss to all but N, K, M, and O.
P>O>L>O (not possible, L can only toss to P or M)
P>O>P>O (possible)
D is the correct answer.

4. For this question you should work backwards. N can only receive a toss from M, and M can only receive a toss from L. So we know _>_>M>N from the start. Based on the explanation only L can pass to M. So we have _>L>M>N. Based on the explanation K, P, N and L cannot pass to L. This leaves M and O as the only possibilities. So, M>L>M>N and O>L>M>N are the only possible orders. Option C (2) is the correct answer.

5.Make a list of possibilities... O can only toss to L or P (as demonstrated in option E of question 3)
O>L>P (L can only toss to P or M)
O>L>M (" ")
O>P>O (P can only toss to O)
Only these three options can work. Answer C (3) is correct.

6. For this question it is best to write down N_ _ N, M_ _M... and so on for the other 3 options. Then go one-by-one to see if they are possible.
Option A: N>_>_>M, N can only toss to K, K can only toss to P, P can only toss to O. This yields N>K>P>O. Option A is not possible and is therefore the correct answer. (For the sake of explanation I will explain the other options).
Option B: M>_>_>M, M can only receive from L and L can receive from anyone but K, P, N, L. So it can receive from O or M, but the order cannot go M>M. So L must receive from O. The order would be M>O>L>M. (Possible)
Option C: L>_>_>L, L must toss to P or M. So we have L>P>_>L and L>M>_>L. P must toss to O making L>P>O>L (possible). M can toss to O which makes L>M>O>L (possible).
Option D: K>_>_>P, K must toss to P, P must toss to O. O can toss to P, so K>P>O>P works (possible).
Option E: K>_>_>L, K must toss to P, P must toss to O. O can toss to L, so K>P>O>L works (possible).
Option A was the only not possible solution, so it is the correct answer.

[poldon]

To begin, the introduction tells use there are six children gathered in a circle to play a game. Next, we're told that there are three "tosses" between the children and that children can receive the ball twice (but can't toss to themselves). So, while the circular arrangement could have meant something, in this game it turns out to be a red herring. We have a sequence of four children (chosen from six) with repeats allowed (but not immediately following each other). This game picture is confirmed by a quick glance at the first question.

To the rules, then.

1> K can only toss to P means that if K tosses then P is immediately next. It's important to notice what the rule doesn't say, as well: there's nothing to stop K from being 4th in line and thus tossing to no one. A simple summary: If K -> K-P (or K is last)
2> Similarly, If P -> P-O (or P is last)
3> Again, though we're told N can only receive a toss from M, there's nothing to stop N from being first. So, If N -> M-N (or N first)
4> A two-part rule. The first part is straight-forward: If L -> L-P or L-M (or L is last). Next, M can only receive a toss from L, like rule 3: If M -> L-M (or M is first)
5> Similarly, If N -> N-K (or N is last), and If K -> N-K (or K is first).

Some of these could be combined, but since all the rules are conditional it's probably not worth it until a question asks demands a narrower situation.

On to the questions ...

1. Which of the following could be the order of tosses, from the child who starts to the child who wins?
Just go through the rules to eliminate violators.
Rule 1 eliminates C.
Rule 2 eliminates A
Rule 3 eliminates D and B.
That leaves us E.

X(A) M to L to P to K
X(B) M to P to O to N
X(C) M to N to K to L
X(D) L to N to K to P
***(E) L to M to L to P

2. If K starts the game, it could be true that
If K starts the game, we know (R1) that P follows and then (R2) that O follows P, so our setup is K-P-O-?. This eliminates A an B easily enough, but we need to solve for that last spot. Rules 3, 4, and 5 have the recipient rules.
N must receive from M -- D is OUT.
M can only receive from L -- C is OUT.
We're left with E. Since P has no receiving rules, we're good.

X(A) L receives the second toss
X(B) M receives the second toss
X(C) M is the winner
X(D) N is the winner
***(E) P is the winner

3. If O is the winner, each of the following could be true except:
This question looks like it could involve a lot of work, but it doesn't because of the our previous work. We know we're ending with O, and a quick look at the answers tells us all we need is the first child in the game. This is easy because we just solved this with question 2. If we start with K, we get K-P-O-?. Since O can't follow itself, it's not possible to start with K and end with O. Answer D it is.

(A) N started the game
(B) M started the game
(C) L started the game
***(D) K started the game
(E) P started the game

No need to go in order, and #5 looks more promising, so it's next.
5. If O starts the game, how many different children can receive the second toss?
The tricky thing here is that second toss can easily mistaken for second child. No good! There are no sending rules for O, so we need to look at the recipient rules in 3, 4, 5. R3: No N, R4: No M, R5: No K. We know from the paragraph that O can't follow itself, so that leaves L or P for the second child. So ... who can receive the second toss, then? L -> P or M, and P -> O. So that's three possible children for the second toss.

(A) 1
(B) 2
***(C) 3
(D) 4
(E) 5

4 and 6 look to be possibly more irritatingly open. Oh well.
4. How many different orders of tosses would result in N being declared the winner?
Let's work backwards.
N can receive only from M (R3). M can only receive a toss from L (R4). L can receive from anyone. That's ?-L-M-N, so far. Who can toss to L, then? Not K (R1), L (R4), N (R5), or P (R2). M is fine, O is fine. That's 2.

(A) 0
(B) 1
***(C) 2
(D) 3
(E) 4

6. Which of the following can’t happen in the same game?
It would be irritating to test each to find the reject, so let's see what previous questions can do for is.
Q2 K-P-O-P is OK so eliminate D.
Q4 tells us a game ending with N must start with M or O, which makes A impossible. That's it then.

***(A) N starts and N wins
(B) M starts and M wins
(C) L starts and L wins
X(D) K starts and P wins
(E) K starts and L wins

I think that's it, anyway.

[ReadingNation]

To me, the trickiest part of the game was setting it up ... the throwing and receiving elements had plenty of headache potential so I knew the set-up was going to be absolutely critical here ... I struggled for a little bit but eventually I came up with the following system

I decided to have the subscripts "˜t’ and "˜r’ respectively stand for throwing and receiving

So, take the first rule for example, "K can only toss it to P" ... I decided to represent this as

Kt → P

On the flip side, I symbolized the third rule, "N can only receive a toss from M'" as follows

Nr → Mt

In this fashion, I created the two columns you’ll see below ...

Kt → Pr ............ Nr → Mt

Pt → Or .............Mr → Lt

Lt → Pr or Mr .......Kr → Nt

Nt → Kr

As soon as I was done reading the rules I strongly suspected that some questions would test how comfortable you were with the throwing restrictions exclusively, some questions would test how comfortable you were with the receiving restrictions exclusively and some questions would test a combination of both ...

I needed a system that would allow me to do two things:

1) Separate the throwing restrictions from the receiving restrictions so I could easily scan both sets of rules

and

2) I needed a system that would allow me to quickly scan for the explicit or implicit variable being tested ...

I felt the rules as they were written did not allow me to do that so once I had this set-up, I felt confident enough to tackle the questions ...

Opps ... I forgot to talk about the opening paragraph ... By far the most important sentence in the opening paragraph was the second sentence ... It wasn’t hard to figure out but it wasn’t exactly a "˜gimme’ either ... I had a feeling that if people rushed through that sentence they would be coming back to it shortly ... So the two most important things to take away from the opening paragraph ...

1) The game involves a total of three throws amongst 4 out of 6 people

and

2) No one person could throw to themselves

So, finally on to the questions ...

Edit note ... I edited the columns up top ... originally, they did not come out the way I wanted them to

[ReadingNation]

The first question asks ....

"Which one of the following could be the order of tosses, from the child who starts to the child who wins?"

(A) M to L to P to K
(B) M to P to O to N
(C) M to N to K to L
(D) L to N to K to P
(E) L to M to L to P

Answer choice (A) violates the 2nd rule "P can only toss it to O" by having P toss it to K

(B) violates the 3rd rule, "N can only receive a toss from M", by having N receive a toss from O

(C) violates the 1st rule, "K can only toss it to P" , by having K toss it to L

(D) violates both the first clause of the 4th rule, "L can only toss to P or M" and the 3rd rule, "N can only receive a toss from M" by having L toss it to N

(E) is perfectly fine and the correct answer ...

"¢L can only toss to P or M, so L to M is admissible

"¢M is not restricted by who he can throw to so M to L is admissible

"¢And again, L can only toss to P or M so L to P is admissible

Question # 2

If you tackled this question correctly, you started off by exclusively considering the rules regarding throwing and ended by exclusively considering the rules regarding receiving ... in this sense, this question was a good question to gauge how well you were juggling the two main sets of information

If K starts the game, it sets off the chain reaction K-P-O were K has to throw to P and P has to throw to O ... So the first three "˜slots’ are not up for discussion but the 4th one is ...

O is not limited to who he can throw to so it would be tempting to stop here and start tackling the answer choices ... however, there are three variables, children N, M and K, who are limited in terms of whom they can receive a throw from and O is not allowed to throw to any of them ... so, in conjunction with the rule that a person cannot throw a ball to their self we should keep in mind that the only "˜end’ options for this scenario are "˜O throwing to L’ or "˜O throwing to P’

If K starts the game, it could be true that

(A) L received the second toss
(B) M receives the second toss
(C) M is the winner
(D) N is the winner
(E) P is the winner

Both (A) and (B) are wrong because in this particularly scenario, we have already established that "O" could be the only child to receive the second toss

Both (C) and (D) are wrong because L and P are the only children who could potentially win the game

Question #3

If O is the winner, each of the following could be true except:

(A) N started the game
(B) M started the game
(C) L started the game
(D) K started the game
(E) P started the game

Because "˜O’ is not limited by who he can receive a throw from, the key to # 3 is realizing that you have to start engaging the answer choices right away

(a) ... If "˜n’ starts the game you know he can only throw to "˜k’, so "˜k’ has to receive the ball in the second slot ... Once "˜k’ receives it, it’s her turn to throw and she can only throw it to "˜p’, so "˜p’ receives it in the third slot ... Finally, "˜p’ can only throw to "˜o’, so if "˜o’ is the winner of the game, N starting the game is permissible

(b) There are no restrictions as to who "˜m’ could throw to ... so for now, it’s o’k to overlook (B) ...

(c) L does have throwing restrictions, but one of those restrictions is "˜m’, which affords us some leeway on how we could develop and end the game ... So I would also skip (C) for now

(d) This answer choice is a good candidate worth pursuing ... Not only is K’s throwing restricted but she is also tied up to other variables whose throwing is also restricted ... From our set-up we should know that if "˜k’ throws and there are enough "˜slots’ available, the k-p-o chain reaction is set off ... in this scenario, "˜k’ starting the game would dictate that "˜o’ must receive it in the third slot ... But if "˜o’ receives it in the 3rd slot, she cannot throw it to herself in the 4th slot to end the game, but in this scenario "˜o’ must win the game ... Hence, given the rules, there is no way "˜k’ can start a game in which "˜o’ wins

[ReadingNation]

Question #4

How many different orders of tosses would result in N being declared the winner?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

# 4 is a great "˜think backwards’ sort of question ... The question stem is implicitly keying in on the receiving element of the game ...

So, "˜n’ wins the game but "˜n’ can only receive from "˜m’ ... that means nobody but "˜m’ could be in the third slot

"˜M’ can only receive from "˜l’, so nobody but "˜l’ could be in the second slot

Here’s where the question could get semi-tricky ... "˜l’ does not have any restrictions as to who she can receive a throw from but other variables do have restrictions as to who they can throw to ... those variables would be k, p and n ... neither k, p nor n can throw to "˜l’ and since "˜l’ cannot throw to herself, that leaves only 2 viable candidates for starting the game when "˜n’ wins, "˜m’ and "˜o’ ... since the rest of the slots are filled out there are only 2 different order of tosses if "˜n’ is declared the winner

1) M L M N
2) O L M N

Question # 5

If O starts the game, how many different children can receive the second toss?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

"˜O’ does not have any explicit restrictions on who she can throw to but there are explicit rules that state that there are 3 variables who can only receive a throw from variables other than "˜o’

So, ostensibly, in this scenario, "˜o’ could only throw to "˜l’ or "˜p’ ... since there are no restrictions on who can win the game in this scenario it would be a pretty safe bet that "˜l’ or "˜p’ receiving the lst toss would not lead to any indirect violations down the road

Now, let’s consider both scenarios ...

If "˜l’ receives the lst toss, then "˜m’ or "˜p’ are the only variables that are eligible to receive the second toss ...

1) O L M ( N,P,L and O )

2) O L P (O)

If "˜p’ receives the lst toss, then there is only one variable that is eligible to receive the second toss ...

1) O P O (L or P)

Since there is no overlap between the three examples in regards to who receives the second toss, there are three children who can receive the 2nd toss

Question # 6

Which of the following can’t happen in the same game?

(A) N starts and N wins
(B) M starts and M wins
(C) L starts and L wins
(D) K starts and P wins
(E) K starts and L wins

Question #6, like question #3, was a "˜get your hands’ dirty sort of question where there was a good probability you would have to engage each and every answer-choice on its own ... luckily (A) was the correct answer choice ...

If N starts and wins the game ... N -- -- N, then N K M N would have to be true, since "˜n’ could only throw to "˜k’ and "˜n’ could only receive from "˜m’ ... however, "˜k’ could only throw to "˜p’, so according to the rules, N -- -- N would lead to a violation of either the lst rule in N K M N or the 2nd rule in N K P N

Below are acceptable versions of the answer choices ...

(B) ... M O L M
(C) ... L P O L
(D) ... K P O P
(E) ... K P O L

[sharmyn.blake]

Go ahead and post your explanations here!

We'll select the best explanation posted here, and also randomly select someone with the correct answer (but for those, simply go ahead and submit the answers through the game page). Winners receive either $200 off a live course or a free strategy guide. You can win up to 3 times, after which you are officially good at these.

We imagine that this might be a game that inspires a wide variety of breathtakingly brilliant solutions, so give it your best shot!

Hello Mike - Is possible to provide the level of difficulty for the LSAT Games?

Thank you

[ManhattanPrepLSAT2]

These challenge games are supposed to be super tough -- in designing them we try to stretch to the outer limits (and possibly beyond!) in terms of showing you the challenges the LSAT might present on test day.

This game, in particular, seems to be one that is proving to be quite brain-bending (in a good way) for many students --

I wouldn't worry about trying to do this at an "LSAT" pace -- I think the sum of the challenges presented in the set-up and questions is probably greater than anything you should expect on test day --

But I think this is a great game for --

1) Thinking about the decisions you make in terms of how you diagram. This game tests typical LSAT skills in an atypical way. Are your diagramming skills intuitive enough and flexible enough to help you adjust?

2) Testing your ability to read and utilize your diagram. The questions require some clever inferences -- you have to be able to read your diagram correctly and use it to your full advantage. Does your diagram help you connect information and answer questions more easily? If not, why not?

[sharmyn.blake]

These challenge games are supposed to be super tough -- in designing them we try to stretch to the outer limits (and possibly beyond!) in terms of showing you the challenges the LSAT might present on test day.

This game, in particular, seems to be one that is proving to be quite brain-bending (in a good way) for many students --

I wouldn't worry about trying to do this at an "LSAT" pace -- I think the sum of the challenges presented in the set-up and questions is probably greater than anything you should expect on test day --

But I think this is a great game for --

1) Thinking about the decisions you make in terms of how you diagram. This game tests typical LSAT skills in an atypical way. Are your diagramming skills intuitive enough and flexible enough to help you adjust?

2) Testing your ability to read and utilize your diagram. The questions require some clever inferences -- you have to be able to read your diagram correctly and use it to your full advantage. Does your diagram help you connect information and answer questions more easily? If not, why not?

Thanks!! I was quite surprised that some of my answers were similiar to others who have posted (hope that we are right) but the time it took to get those answers.....(sighs) was ridiculous

[ManhattanPrepLSAT2]

Last call for explanations! We've gotten some terrific responses so far, but there are many ways to skin this cat! (If anyone has a less disturbing analogy I can use here please let me know.) Do you have a creative way you solved this game? Share it with the world and (perhaps) win free stuff and (more importantly) gain eternal fame!

[sgorginian]

I created a Grid Chart.

Please see attachment. (5 pages) thank you!

I made a grid of Who can pass to who. From - To

this was fun

[pracillafelicia]

I divided the solution into a two part description:

Those that can only be tossed to:

K -->P
P -->O
L -->P
L -->M
N-->K
K-->N

Those that can only receive a toss from:

N-->M
M-->L

Then I counted the spaces for the winner, three tosses means 4th person wins: __ __ __ __

I used this set-up for each question by posting the restrictions and came up with appropriate answers for each. For example:

Can N start and win?

N __ __ N

N can only toss to K and K can only toss to N but no one can toss to themselves. So if N tosses to K, K tosses to N that would mean N would have to toss to himself in order to win: N K N N This cannot be correct.

I used this for the process of elimination.

[gptchamdjou]

The entities: K,L,M,N,O & P
Action: toss the ball to one another
Limitations: #1: 3 tosses in each game in order to have a winner.
#2: None can toss to him/herself and any can handle the ball more than once, it follows that at least 2 children and at most 4 children must be involved in the 3-toss-game.
THE RULES: First I wish I cld draw my sketch here, anyway I know everybody got a right sketch. No need to explain the rules and the conclusion to draw it will be too long. But my answers to the questions will explain some of the rules.

Answers:
1E
2E
3D
4C
5C
6A
(just as everyone made it)

Explanations:
#1
A. P>K impossible coz only P>O possible, ie R2
B. O>N impossible coz only N<M possible, ie R3
C. K>L impossible coz only K>P possible, R1
D. L>N impossible coz only L>M/P possible, R4
*E. From the stated rules nothing prevent L from receiving a toss from M.

#2
A. K>P>L must be false coz though K can only toss to P, there is noway P can toss to L since he can only toss to O, ie R2
B. K>P>M must be false coz only P>O can be true.
C. K>P>O>M must be false coz though P can toss to O, there is noway O can toss to M directly since M receives only from L (M<L), ie R4
D. K>P>O>N must be false coz N can only receive from M, ie R3
*E. K>P>O>P cld effectively be true since nothing from the stated rules prevents P from receiving from O.

#3
A. N>K>P>O cld be true based only on the stated rules.
B. M>O>P>O or M>L>P>O cld as well be true since none of the rules prevents M>O or O>P and none also prevents M>L.
C. L>M>P>O may indeed be true since no rule prevents M>P
*D. K>P>O>O must with no doubt be false since O cannot toss himself as we know from the beginning.
E. P>O>P>O is well possible coz O>P is possible as explained above and one can handle the ball more than once.

#4
Looking at the sketch, the different possible orders are only N<M<L<O and N<M<L<M. ie 2 different orders. So *C is the correct answer. Again we know that on the one hand only N<M<L is possible and either L<O or L<M is possible on the other hand since no rule prevents O>L or M>L from being true.
So therefore answer choices A, B, C & E must be incorrect.

#5
Order 1: O>P>O... possible since O>P is as well possible
Order 2: O>L>P... possible since O>L is not invalid
Order 3: O>L>M...
These are the only 3 different orders in which the children receiving the 2nd toss are different when O starts. We know from the implied rules that only L&P cld receive the 1st toss from O, and we just proceeded with wt else cld be possible based only on the set of rules. As such, answer choice *C is the correct answer whereas answer choices A, B, D & E are just not true in this case.

#6
*A. N>K>P>N is just not possible at this stage, we cld not prevent O from winning in this case, N cld never, since P cld only toss to O.
B. M>O>L>M is in fact possible. No need to remind that M>O & O>L are possible.
C. L>P>O>L cld be true.
D. K>P>O>P cld be true. Im just having fun on this question!
E. K>P>O>L not invalid. Same reasoning!

Good Luck to u all!! Great to relax with LG whenever LR gives me headaches!!

[ManhattanPrepLSAT2]

Thanks to everyone for playing! We've gotten some fantastic answers to this game, and hopefully it's been fun for you to read through them. Unfortunately, there can only be one winner, and that winner is....

SGORGINIAN!

We're honored by your color-coding. Great work.

These solutions are getting better and better -- we're excited to see how you all take apart the next game.

- MK

[mikburger44]

Agree with EEDCCA

KMN limited from whom they can receive (NLM respectively)
NKP L limited to whom they can throw (KPO P&M)

OPL Open to receive from any other child
OM Can pass to any other child

L has option to pass to P or M

The answers follow from diagram and above limits

L - M - N -K -P -O L - (P or M) -O this later allows "looping" - for question 5 - thus O -P -O or O -L -M or O -L-P

[tamwaiman]

Can someone tell me where the question is?
Thank you.