Logic Game Challenge #36: The Trainee Game (Hard)

[ohthatpatrick]

Our latest Logic Game Challenge, The Trainee Game, is live. Post your answer explanations (or questions) questions of the 'hard' version here. Good luck!

[zagreus77]

I'm going to take a stab at this. My typing is crappy so please forgive me. Also, I hope to get the technical problem with the diagram sorted out -- the system is collapsing my diagram for some odd reason.

Four different trainees"”S, T, U, and W"”work together on five different projects"”M, N, O, P, and R. At least one trainee works on each project. The following conditions apply:

1 U works on O but S does not.
2 Everyone must work on either N or P, but no one works on both.
3 More people work on M than on any other project.
4 Either S or U is the only person who does not work on project R.
5 Anyone who works on N must work on exactly four projects.


Let's symbolize and diagram using STUW as the elements and MNOPR (5 positions for placement).


S Num. Dist: N P 3-1 1-3 2-2
T T #N and #O Max 3 Min 1
U W N --->#4 (also must be in O)
W U S S/U N <-/->P
M N O P R ~N -->P
\_/_/ (~P -->N)
~s ~s

Symbolized rule 1 by placing U in the O slot and a ~S under the O slot. And since according to rule 2, no one can be in both N and P, this mean that S can be in 3 projects at most since O is excluded and M and R are the only ones left. From this we can infer from rule 5, that S can not go in N, nut since ~N-->P derived from rule 2, S must go in P. So S is placed in P and a ~ S under N.

We have a further constraint from rule 3 that M must have all the elements STUW, since we know from rule 4 that three must work on the project R and thus M must have 4 elements, which would include all 4 projects STUW. We can also infer the maximum and the minimums for N and P. We know that N (or interchangeably P) can only have at most 3 elements or it would violate rule 3. We also know therefore that since any element not in N is in P and vice versa that each must have a minimum of one in the slot. In the case of N, that element must appear in slot O as there are only three remaining positions, P being excluded, so all including O must have that element.

We also can infer that if U appears four times that U not S must appear in slot R, because whether U is in P or N (and in can be in either as rule five is a N--> 4, not 4 --->N) there are only three remaining slots other than R for U to go. U, thus, must be in R, and S must be out since one, and only one of S and U must be out.

As for TUW if any or all appear in N, because they must appear 4 times, they must appear in O ( U already had to appear as a given rule) as P is ruled out they must fill W O and R which are the only remaining positions. Interestingly, T, U or W can appear in P and still go 4 times, but again must go in O. Also for T and W it is also a sufficient condition for appearing in 4 if they appear in O since we know 1 appearance between N and p and the must appear in M and R. So adding O makes 4.

If anyone is interested, I'll be happy to go into this game in greater depth and apply the above to each question.

[ohthatpatrick]

Great work!

It certainly seemed like you figured out every nook and cranny of the deductions/limitations of the game.

Did you figure out all of those before you went to the questions, or did some of those insights come to you as you worked through the questions? (For me, it was about 75% the former and 25% the latter)

It is very annoying to try to draw games in this typing environment; all empty spaces are deleted except for the ones between characters, so it's impossible to space things the way you want to see them and get it to appear that way.

You sometimes have to use "..." or "___" to force your way to the right.

Let me try to clean up your diagram a little so that others can behold its glory.

N <---> ~P
~N <--> P
Num. Dist:
N P
3-1
1-3
2-2

N --->#4 (also must be in O and R)
#N and #O Max 3 Min 1

S _/ _/ _/ _/
T __ __ __ T
U __ __ __ W
W __ U .S S/U
M .N .O .P .R
..~s ~s

If you or anyone would else would like to try elucidating your approach to each question, please do.

And if anyone else would like to give us your own version of the setup/deductions, feel free.

Keep up the good work!

[sjs]

There's an ambiguity with the phrasing of Question #6:
"6. If fewer trainees work on project P than on any other project, how many projects must be assigned the same number and combination of people?"

You could read the sentence as being completed with [as project P] or [as each other]. The former points to answer A, whereas the latter points to answer C. You might think that the former is too obvious to be the actual question, but it's consistent with the language.

Item removed from scoring!

[ohthatpatrick]

Ahhh, I see what you're saying. (although Zagreus77 had to clarify it for me)

I thought you were saying the first half of the comparison was ambiguous, but you're saying the 2nd half of the question leaves open the possibility that we're asking

How many projects have the same number and combination of trainees as each other
vs.
How many project have the same number and combination of trainees as P does

I like your critical eye! (and even more, I just like the victorious glee of "Item Removed from Scoring!")

For no other reason than for sport, I will attempt to defend the question as unambiguous.

Let's say we have a game that includes 4 kids who are each assigned a given number of gumballs.

And let's just say there's one rule that says, "If Roger has 5 gumballs, then everyone else has 2."

If I had a question that said,
"If Roger has 5 gumballs, how many kids have the same number of gumballs?"

then this would be a question that really contains the ambiguity you were discussing.

Are we asking how many kids ALSO have 5 gumballs? (in which case the answer is 'none')

Or are we asking how many kids have matching # of gumballs? (in which case the answer is '3').

The grammatical reason for this ambiguity is that we actually specified a quantity of gumballs in the first half of the question.

Does "same" mean = same as the quantity Roger has
or does "same" mean = matching quantity (same as someone else)

I would argue that for the question for the Trainee Game, we don't really suffer from the same ambiguity.

It says
"If fewer trainees work on project P than on any other project, how many projects must be assigned the same number and combination of people?"

Grammatically, this sentence doesn't allow for a comparison between P's number/combination of people and that of others, because it never specified P's number/combination of people.

The first half of this question is just a limitation that relates to P's number of people, but it doesn't define P's number/combination of people, so we can't read in that comparison.

The other interpretation, meanwhile, stands alone.

"How many projects must be assigned the same number and combination of people?" doesn't need a "as each other" tacked onto the end in order to be understood.

On its own, it means "how many projects match each other".

So we would only have to add the 'as each other' for clarification if we were grammatically creating ambiguity by putting a specific number/combination of people into the first half of the question.

Whaddyouthink?

As I said, I don't know if I'm 100% persuaded by my own logic, but that's lawyering for you. Make the best case you can given the evidence you got.

[ptraye]

i answered all the questions in the game correctly, except #4. could someone explain how the answer is B?

also, on question #5, could someone explain the language in answer choices C, D and E? with the Could Be True EXCEPT (Must Be False) question stem, plus the exactly (one, two, four) projects, i was not certain how i should understand those answer choices.

thanks.

[yhzhangbj]

Is the competition still on? May I submit the explanation as PDF attachment?

[yhzhangbj]

Do you still offer the chance to win a prize for posting explanation of this game?
The webpage here http://www.manhattanlsat.com/Logic_C_vi ... ChallID=46 , calls this game an archived one.

[ohthatpatrick]

Yup, the contest for this one is still up and running and PDF form is fine.

Thanks for letting us know about the game having the wrong header info at the top (saying that it's archived and therefore not contest-eligible). We'll try to fix that.

Good luck.

[anjelica.grace]

I was just wondering if it would be more appropriate for Question 1 to read "Which one of the following is a complete and accurate list of the projects ANY OF which S could work on?". Otherwise, as it is currently written, it sort of reads as an acceptability question, in which two choices would be viable.

The reason I ask this is because after making many mistakes, I noticed this distinction in the question stem: "any of which" asks for a list of all possible placements (which are not necessarily used simultaneously) vs. without "any of which" asks for an acceptable list, which is necessarily simultaneous.

Sorry if this doesn't make sense.

[ohthatpatrick]

Hey, Anjelica.

Great question.

While I've definitely seen the language you're referring to in LSAT questions, I don't think the way it is currently written is ambiguous (because I've also seen LSAT write a question the current way as well).

Basically, when it comes to these "complete and accurate" list question stems, the key difference is whether the word "could" comes before or after "complete and accurate" list.

Had our question #1 read:
Which one of the following could be a complete and accurate list of the projects on which S works?

Then, it would have that unintended meaning you were suggesting. (It would mean that every project in the answer choice was something S was simultaneously working on ).

When 'could be' comes BEFORE 'complete and accurate', the answer choices refer to ONE possible scenario.

But it actually says:
Which one of the following is a complete and accurate list of the projects on which S could work?

When 'could' comes AFTER 'complete and accurate', the answer choices refer to ALL scenarios.

Here's the clearest way to argue for the current wording:

Is M a project that S could work on? If yes, it goes in the list. If no, it doesn't.

Is N a project that S could work on? etc. ......

So, ultimately, while agree that "any of which" would be a very helpful clarification, I really want to make sure going forward you understand how the test uses the placement of "could" to distinguish between a list of things that are simultaneously true and a list of things that could ever separately be true.

Hope this helps. Let me know if you have unresolved qualms.

[maithu.raman]

Wondering .. Do we really get such tough games in actual LSAT or its just for the geeks !!!!

[kenyamlee]

Quick lame question. Rule #4 says either S or Unis the only,person out. It doesnt say but not both. So is this implied? If a rule says either X or Y are out, does that mean they both cant be out?

[ohthatpatrick]

No, but if it says "X or Y is the only person out", then it implies they can't both be out.