Logic Challenge #16: Composite Board

[orion]

I really think 6 does not have a correct answer, and I explain why below in more detail, as part of my solutions. I believe I've found counterexamples all the choices (as well as general rules explaining the counterexamples). For A, B, C, and D, I have sequences which are allowed by the original rule but not by the new rule, and for E I have a sequence that's allowed by the new rule and not by the original rule. I've checked them all backward and forward against the game rules and I can't find anything wrong with any of them, however Noah claims that I've missed something and that there is indeed a correct answer. Can somebody find the flaw in my reasoning?

--------

In any case, here are my solutions:

We have eight elements: Fi, Gi, Hi, Km, Lm, Pw, Ow, and Sw. The first (capital) letter in each of these symbols is the specific name of the element, the second (lower-case) letter is its type.

We have 6 or 7 layers, which means either 2 elements or 1 element will be left out, respectively.

There must be at least one element of each type selected, so if we have 6 layers we cannot leave out both m's.

We have the following rules:

I. The sequence ii (insulator followed by insulator) is forbidden.
II. An i may not be first or last.
III. The sequences FL (fleece followed by lead) and LF (lead followed by fleece) are forbidden.
IV. The sequences KF or FK may occur only as iKF or FKi.
V. The sequences mw and wm are forbidden.
VI. Wood must be both first and last or neither first nor last.

We can also make the following deductions:

1. Combining II and VI tells us that the first and last elements must either be both w or both m.
2. Combining VII and V tells us that we have the following possible block arrangements, where slots in parentheses are present in the 7-element case:
(a) mi_(_)_im
(b) wi_(_)_iw
(c) wwi_(_)iw, wi_(_)iww

Examining these block arrangments in the light of I, V, and the fact that there are only two m's gives the following possibilities:
(a) miwwim, miwiwim, miwwwim
(b) wimmiw, wimimiw, wimiwiw, wiwimiw
(c) wwimiw, wwimmiw, wimiww, wimmiww

Since F is so restricted, it's worth noting that we can also make one more deduction about these sequences:

3. The sequences miwwim, miwwwim, wimmiw, wwimmiw, and wimmiww may not include F, as rules III and IV tell us that F (an i) may only be next to either w or an m which has an i on the other side, and this is not possible in these sequences.

Now for the actual problems:

1. (E)
This is a simple exercise in checking the rules against the answer choices. My preferred method of attack is to pick the rule that seems easiest to check, apply it to all the choices, eliminate those that fail, select the next-easiest rule, and repeat until all but one choice has been eliminated.
II and VI simply require examination of the first and last elements rather than a search through each entire list, so they're good starting points. II doesn't help, but VI allows us to eliminate (B).
III is very simple and specific and so is a good next choice. It eliminates (A).
IV is nearly as specific, so it's next. It eliminates (D).
I and V are next, and V eliminates (C).
(E) is therefore the answer.

2. (C)
If F is second, the allowed sequences are those with an insulator second, which includes all type (a) and type (b) sequences, and the type (c) sequences wimiww and wimmiww. By deduction 3 the only type (a) sequence which may contain an F is miwiwim, the only type (c) sequence which may contain F is wimiww, and all type (b) sequences except wimmiw may contain F. Checking all the answer choices against these permitted sequences, we see that only (C) is forbidden, so this is our answer.

3. (B)
If S is fourth and there are 7 layers, the only possible sequence is the type (a) sequence miwwwim. We already know this sequence cannot contain F, so (B) is impossible and is therefore our answer.

4. (C)
We must have one of the sequences GLK or KLG. Since K is already next to L (the only other m) and may not touch wood (by rule V), it must be next to an i. By rule IV this i may not be F, so since G is already used it must be H. So we have GLKH or HKLG.
Looking at our sequence library, we see that the only sequences containing mm are wimmiw, wwimmiw, and wimmiww. However, since P is out, we only have 2 w's available, so our sequence must be wimmiw. There are no restrictions on which w is O and which is S, so we can put O first or S first, and we can put GLKH in the middle or HKLG in the middle, giving the four possible arrangments OGLKHS, OHKLGS, SGLKHO, and SHKLGO.
Therefore the answer is (C).

5. (A)
In this case we have little choice but to check the answers one by one.
(A) fails because the only sequence with m first and i fifth is miwwim, which cannot contain fleece by deduction 3. So this will be our answer, but I will confirm the others briefly anyway.
(B) L is allowed to touch G and H individually, and we have several type (b) and (c) sequences which contain imi, so GLH or HLG is possible.
(C) We have the type (a) sequence miwwwim, so OPS or SPO are possible.
(D) The type (c) sequence wwimmiw permits O (a w) to be second and G (an i) to be sixth.
(E) Several sequences permit mm, so KL or LK is permitted.

6. (no correct answer)
We can confirm that there is no correct answer by examining each answer choice and asking if there are any sequences permitted under the old rule which are not permitted under the new rule or vice versa.
(A) fails because the new rule does not permit the previously allowed sequences miwwim, miwiwim, wimmiw, and wimimiw. Sample arrangement that is allowed by the original rule but violates the new rule: OGKLHP.
(B) fails because the new rule does not permit the previously allowed sequences miwwim, miwwwim, wimmiw, wwimiw, wwimmiw, wimiww, and wimmiww. Sample arrangement that is allowed by the original rule but violates the new rule: OGKLHP.
(C) fails because the new rule does not permit the previously allowed sequences wimiwiw, wiwimiw, wwimiw, and wimiww. Sample arrangement that is allowed by the original rule but violates the new rule: OFKGPS.
(D) fails because the new rule does not permit the previously allowed sequences miwiwim, wimiwiw, and wiwimiw. Sample arrangement that is allowed by the original rule but violates the new rule: KGOFPHL.
Choice (E) isn't violated by any of our arrangements, but it allows the arrangements wmimiww and wmimiw (and mirror reflections), which violate the original condition that w does not touch m. Sample arrangement allowed by the new rule which violates the original rule: OLGKFP.
[Please prove me wrong on this one!]

7. (E)
We need to have ww and mm both in the same sequence, which means we have one of the type (c) sequences wwimmiw and wimmiww. Note that by deduction 3 neither of these may have F, so our i's must be G and H. Also, all w's and m's must be included. Since F is absent, we may arrange our w's, m's and, i's however we like in the appropriate slots. The farthest distance between any w and any i is between the first and sixth slots in wwimmiw or between the second and seventh in wimmiww. There are four layers between the elements of each of these pairs, so our answer is (E).

[rx211]

alrite i will give this messy game a try~~
1. Which one of the following could be a list of the layers used in a board, in order from bottom to top?

(A) pine, oak, Gore-Tex, lead, fleece, spruce (FL can't be together)
(B) krypton, Gore-Tex, spruce, fleece, oak, pine (no wood in position 1)
(C) spruce, hay, krypton, Gore-Tex, lead, pine (lead/metal can't be with pine/wood)
(D) pine, fleece, krypton, lead, Gore-Tex, oak (KL-> IkI)
(E) oak, spruce, fleece, krypton, Gore-Tex, pine


2. If fleece is second, each of the following could be true EXCEPT:

(A) A layer of insulator is sixth. (Possible)
(B) A layer of wood is seventh. (Possible)
(C) A layer of metal is sixth. (Possible)
(D) A layer of insulator is fourth. (Possible)
(E) A layer of wood is sixth.

W I W/M I W/M I W
1 2 3 4 5 6 7

3. If spruce is the fourth of seven layers, each of the following could be true, EXCEPT:

(A) Oak touches Gore-Tex. (Possible)
(B) Fleece touches spruce. (Possible)
(C) Pine touches hay. (Possible)
(D) Pine touches spruce. (Possible)
(E) Lead touches hay.

__ __ __ W __ __ __
__ __ __ S __ __ __
I ~M ~M I

4. If lead touches krypton and Gore-Tex, and pine is not used in the construction of a given board, how many different arrangements of layers can be used?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

__ G L K __ __
W I M M I W

__ __ K L G __
W I M M I W

Two different I's to exchange between F & H two types X two choices = 4

5. Each of the following could be true EXCEPT:

(A) Lead is the first layer and fleece is the fifth. (Possible)
(B) Lead touches Gore-Tex and hay. (Possible)
(C) Pine touches oak and spruce. (Possible)
(D) Oak is the second layer and Gore-Tex is the sixth.
(E) Krypton and lead touch each other. (Possible)

__ O __ __ __ G __
__ W __ __ __ I W/M
Not enough to fill the 3 in the middle without breaking rules

6. Which one of the following, if substituted for the condition that a layer of wood cannot touch one of metal, would have the same effect on determining the arrangement of layers?

(A) All three layers of wood are used.
(B) All three layers of insulator are used.
(C) Both layers of metal are used.
(D) Each layer of insulator must touch both a layer of wood and one of metal.
(E) Each layer of metal touches at least one layer of insulator.

M/W I M/W I M/W I M/W

7. If two wooden layers touch each other and two metallic layers touch each other, what is the greatest possible number of layers in between Gore-Tex and spruce?

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

_ G _ _ _ _ S
W I M M I W W

[orion]

rx211: We got very different answers so presumably one of us has really messed up our solutions, but I don't know which. Maybe you can help me sort it out.

For problems 1 and 7 I agree with both your answers and your reasoning.
For problem 2 I think the sequence OFKGPS is a counterexample to your answer choice.
For problem 3 I think the sequence KGOSPHL is a counterexample.
For problem 4 I think your answer is right but not your explanation. Note that F can't be next to either K or L in either of those arrangements, and don't forget to consider the ordering of the wood elements too.
For problem 5 I think POHKLGS is a possible sequence (counterexample to your answer choice).
For problem 6 I think there is no correct answer. As I pointed out in my solutions above, I think there are previously allowed answer choices that become forbidden when we switch to rule B, for example OGKLHP.

Anyway, let me know what you think.

[rippinradio]

6. (no correct answer)
We can confirm that there is no correct answer by examining each answer choice and asking if there are any sequences permitted under the old rule which are not permitted under the new rule or vice versa.
(A) fails because the new rule does not permit the previously allowed sequences miwwim, miwiwim, wimmiw, and wimimiw. Sample arrangement that is allowed by the original rule but violates the new rule: OGKLHP.
(B) fails because the new rule does not permit the previously allowed sequences miwwim, miwwwim, wimmiw, wwimiw, wwimmiw, wimiww, and wimmiww. Sample arrangement that is allowed by the original rule but violates the new rule: OGKLHP.
(C) fails because the new rule does not permit the previously allowed sequences wimiwiw, wiwimiw, wwimiw, and wimiww. Sample arrangement that is allowed by the original rule but violates the new rule: OFKGPS.
(D) fails because the new rule does not permit the previously allowed sequences miwiwim, wimiwiw, and wiwimiw. Sample arrangement that is allowed by the original rule but violates the new rule: KGOFPHL.
Choice (E) isn't violated by any of our arrangements, but it allows the arrangements wmimiww and wmimiw (and mirror reflections), which violate the original condition that w does not touch m. Sample arrangement allowed by the new rule which violates the original rule: OLGKFP.
[Please prove me wrong on this one!]

It sounds like you've misread the wording of the question. You're reading it as "Which one of the following rules, if substituted for the original, in conjunction with the other rules, results in the same allowable sequences of materials?" The question is simply asking for an answer choice that creates the effect of that particular rule (in isolation). When we use all three insulators, there must be 7 materials in the boards, and wood cannot touch metal:
M I W I W I M or W I M I M I W

I suppose you could argue that using three insulators does have an additional effect that the original rule doesn't: it forces the boards to have seven materials. If "determining the arrangement of layers" includes determining the number of layers, you may have yourself a case, since the wood/metal rule doesn't preclude six layers.

[orion]

I see your point, rippinradio, and I think you're right that that's what the question intended. It seems kind of poorly worded to me, though. It might have been better stated as, "Which one of the following rules would be an alternate method of prohibiting wood from touching metal?" or something like that. Is this type of ambiguity something one might see on an actual LSAT?

[rippinradio]

Rule replacement questions (in which a new rule has the same effect as one of the original rules) first showed up on the June LSAT. That particular rule replacement didn't result in any changes to the possible sequences allowed by the rules. I haven't seen the September test yet, but I've heard that there were rule replacement questions on it as well.

[orion]

Hm. It would be interesting to know what the official guidelines are on how the rule replacements work. I have seen rule replacements on practice exams where you were supposed to simply determine the effects of a new rule, but I've not seen a previous one where you were supposed to determine which rules would have the "same effect". So knowing the LSAT definition of a "same effect", and how such questions are likely to be phrased would be very helpful.

[rippinradio]

I'm guessing the standards to which LSAT questions have to live up to dictate that a credited replacement rule would need to have the exact same effect as the original, no more, no less. I'll have to see the September test to say for sure though.

[orion]

I certainly hope that's the case. I am probably too literal-minded to be able to figure out any question that's otherwise (as I demonstrated with this problem).

[rippinradio]

I just confirmed that the answer to the rule substitution effect question on September's test created the exact same limitations as the original rule did, so unless things change, you should be good on that front.

[noah]

Nice catch!

[orion]

Nifty. Thanks for checking that, rippinradio. Are the September tests already available for purchase somewhere, or do you have the inside scoop?

[noah]

Hi Guys, We dug a bit deeper into the rule change issue, and we agree with you. I'm going to re-word that question to make it more exact. Again, nice catch!

Here's the change:

Original
6. Which one of the following, if substituted for the condition that a layer of wood cannot touch one of metal, would have the same effect on determining the arrangement of layers?

(A) All three layers of wood are used.
(B) All three layers of insulator are used.
(C) Both layers of metal are used.
(D) Each layer of insulator must touch both a layer of wood and one of metal.
(E) Each layer of metal touches at least one layer of insulator.

Re-wording:
6. Which one of the following, if substituted for the condition that a layer of wood cannot touch one of metal, would also prevent a layer of wood and one of metal from touching each other?

(A) All three layers of wood are used.
(B) All three layers of insulator are used.
(C) Both layers of metal are used.
(D) Each layer of insulator must touch both a layer of wood and one of metal.
(E) Each layer of metal touches at least one layer of insulator.

- Noah

[interestedintacos]

When you guys created this game did you have in mind that the best solution would involve working out the possibilities (like the first person did)?

I did the game without working out the possibilities in advance and found a couple questions difficult, but overall I was able to get through. The important thing was noting that the composite would be topped and bottomed with either M or W, and that if it were done so with M it would be extra limiting because no other M would be available. Those are the main things I had in mind as I went to the questions. My decision not to work out each possibility was I suppose just because there would still be a ton of variation (especially with the 6/7 element). But looking at what the first person worked out, it looks like there actually weren't that many potential solutions, so I'm not sure what you guys had in mind and what would have been the best way.

[noah]

I'm the jerk that wrote this game!

I don't write games with a specific strategy style in mind - in some ways the game has a mind of its own.

I just resolved this, and I have to say I found it really tough! But, I didn't write out all the solutions. That's very rarely the move I make. I focused much more, as you say, on the material types and the potential ends and beginnings. It was a bit of a messy process indeed.

I don't think I've said much, but hopefully that keeps you from defaulting to "exhausting the possibilities." It's cool that that strategy worked this time, but it so rarely is an efficient approach.

[MarkE967]

regarding problem 3, the answer is B. There is no mistake. All others are possible. The rule crucial in this problem is:
"The top layer IS wooden if... the bottom is as well"
If spruce is in position 4 out of 7 and touches fleece, then the top and bottom layers cannot be the same. This is because the fleece at either the 3 or 5 position forces one end (1/2 and 6/7 respectively) to have two consecutive of the same type of material. If the other two remaining wood materials are used on one end, then that means a metal must be on the other end. Alternatively, if both metals are used consecutively on one end, then clearly, only wooden materials remain for the other end.

[kates44]

I hated this game!

On number 6, I am still not understanding why C is the answer. Under the new condition, wouldn't
Km - Pw - Gi - Ow - Hi - Sw - Lm be allowed? Wood and metal are next to either other in this situation but I can't figure out any other rules it violates.
-Each category of layers is used (i, m and w)
-No individual layer is used twice
-"i" is touching exactly 2 layers, neither of which is another "i"
-"L" and "F" aren't touching
-"K" is not touching "F"
-Neither the top nor the bottom layer is "w"
What am I missing??

Question:
6. Which one of the following, if substituted for the condition that a layer of wood cannot touch one of metal, would also prevent a layer of wood and one of metal from touching each other?

(A) All three layers of wood are used.
(B) All three layers of insulator are used.
(C) Both layers of metal are used.
(D) Each layer of insulator must touch both a layer of wood and one of metal.
(E) Each layer of metal touches at least one layer of insulator.