Logic Game Challenge #35: The Toy Maker Game (Hard)

[wingedfeetxc]

The Toy Maker Game (Hard) will be live soon.

Discuss this game here.

[jabushawish4]

35. Toy Maker Game (Hard) Explanation

Summary:

Essentially this game tell of a toymaker who is painting three different toys: an airplane, a race car, and a truck using six different colors. Let call these colors F,G,I,J,K,M. Each toy is colored with exactly TWO colors one of which is used as a base, with the other color used as a highlight. The ways the toy maker colors his toys conforms to the following rules.

F will be used as a base.
G and M cannot be used on the same toy.
I will be used on the race car.
If M is used on the airplane, G will be used on the race car.
If J is used as a base, M will be used as a highlight.

So essentially this a grouping game but there are conditional elements woven throughout which makes things...actually a lot easier! (You'll see how soon) The way I set up this game was by using the three toys Airplane, Race Car, and Truck (A, RC, T) as the base of my diagram with two slots assigned (stacked) to each toy; one labeled B for the base color and another labeled H for the highlight color.


There is a fixed distribution of 2-2-2 in this game, meaning you will have three three columns (A, RC, T) with two slots in each column. There are six slots total, using six different colors, and two colors will be used on each toy. (Knowing distribution is quite helpful, especially when you move onto games that dont use every element or have more slots than elements.) In this game every color will be used and each slot will be filled which can make things easier moving forward.


As is the case for any game, the rules are essential. Having a clearly drawn out representation of each rule helped me avoid looking back up at the directions often, wasting precious minutes (I finished in about 9 minutes)


Lets move to the questions.

1.) Question 1 is a standard list question, asking which of the following choices could reflect the colors used for each toy. In this particular question we see how helpful the condition rules are in eliminating some wrong choices.

a.) gets scratched out for a quick look at rule number three which designates I to the race car. This choice has F and G in the race car slot. This is a violation of the third rule which dictates beforehand that I must be used on the race car. Eliminated!

b.) This choice is eliminated when we look at our first conditional rule that when M is assigned to the airplane, G must be assigned to the race car. In this case, the sufficient trigger M is assigned to the airplane, however the necessary condition that G be used on the race car IS NOT fulfilled. Wrong choice.

c.) This is the correct answer. When you go through your beautifully drawn checklist of rules you will find that answer choice C is not in violation of any rules. Circle this choice.

d.) Choice D is incorrect, also violating one of our handy conditional rules. This choice violates rule five which states; In the event that J is used as a base color, M will be used as a highlight. In this choice J is used as a base in the airplane, however M IS NOT used as a highlight, instead it is used as a base for the truck. Sacrilegious answer choice. Worship the rules!

e.) Answer choice E is perhaps the easiest choice to eliminate, as it violates our second rule that G and M CANNOT be used on the same car (even though goldenrod and mulberry is a hawt combo in real life) In this scenario we have both G and M used on the airplane, a clear violation. Eliminate!


2. Asks which of the following lists CANNOT be used as highlight colors.

Here we need to refer to our rules and see what we can and cant use as a highlight. Your first instinct should be to look at F which MUST, according to rule number one, be used as a base. Unfortunately LSAC/whoever designed this ego killer of a game isn't in the business of handing out easy answers, since none of the lists include F which would have been a quick answer and saved precious time. We need to dig deeper into our rulebook for this one. Lets move to some answer choices.

a.) G, I and M are used as highlights.
What do we know given this scenario? First off, we know according to rule three that the color I must be used on the race car. Secondly, we know that we have G and M left to be used as highlights for the airplane and truck respectively. Further, we know that we CANNOT put M in the airplane slot because that would force G into the race car slot as stipulated in our fourth conditional rule that when M is used on the airplane, G must be used on the race car. We already have I used on the race car. In order to adhere to the rules, M must be used on the truck and and G used on the airplane. Does this situation work? YES. If you go through the remainder of the rules you will see that when G, I and M are used as highlights it is possible to adhere to all of your rules making this a CAN be true answer choice. Eliminate it. Again we are looking for a choice that CANNOT be true.


b.) J, I and K

Is this a violation? Short response is NO. This also CAN be true. Again, we know I will be used on the race car (rule three) and J and K can switch off in between the airplane and the truck. To add confidence in this answer choice being eliminated we can look to our fifth rule and use a cleverly crafted contrapositive of this conditional rule. Rule five states that if J used as a base color, M will be used as a highlight. The contrapositive of this conditional rule is that when M is not used as a highlight J will not be used as a base OR we can word our contrapositive like this: When M is used as a base, J will be used as a highlight. Since each color can ONLY BE used as either a base or a highlight, we can arrive to solid inferences using contrapositives in this game. In this case J, I and K are highlights and you know that M, F, and G are used as bases. Does this violate our conditionals and contrapositives? NO. it CAN be true. Eliminate it! Again we are looking for a response that is a violation and CANNOT be true.


c.) G, J and K

If we use G, I and K as our highlights we also find that there is no violation of our rules. We know F, I and M will be used as bases. We can make a hypothetical set up that ensure that G and M are not used on the same car. I can be used on the race car. Sine J is a highlight, our fifth conditional rule does not trigger. This scenario CAN be true.


d.) G, I and K

This is the correct answer. If G, I and K are used as highlights, we know that F, M and J are used as bases. This scenario violates rule five. Which states that if J is used as a base, M must also be used as a highlight. In this scenario we have the sufficient trigger J used as a base however M is NOT used as a highlight and instead is used as a base. This use of highlight colors CANNOT be used and this is the correct anser.

e.) G, I and J

If G, I and J are used as highlights we know that F, M and K will be used as bases. This use of highlight colors does not violate any of our rules. A hypothetical can be created in which G and M are not used on the same toy and I is used on the race car. Our fifth condition does not activate since J is a highlight and not a base. We can also avoid placing M in the airplane slot which would result in a violation of rule four.


3. This question asks: If G is used on the Truck, for which toys will the pair of colors be determined?

This is where we start getting into some grouping action. This question is somewhat difficult because you are not told whether or not G is used as a base or a highlight for the truck. To begin, start off by ignoring the highlight or base designations, and just know that there must be two colors used on each toy.

So what do we know about F G I J K M?
-G is used on the Truck.
-M Cannot be used on the truck (rule two)
-I is used on the race car. (rule three)
-F is used somewhere as a highlight (rule one)

Here is where our conditionals come in handy.

Our primary inference in this game comes from rule four, which states that when M is used on the airplane, G must be used on the race car. Here we have G already being used on the truck so M CANNOT be used on the airplane. This would violate rule four. Since we also know M cannot be used on the truck, (rule two states G and M CANNOT be used together despite the hawtness of the goldenrod mulberry combo) We know that M must be used on the race car. So here we have I and M both being used on the race car. We do not know which will be assigned to the highlight slot or the base slot.

Before moving on for further information, take a gander at your answer choices. We can eliminate A, C, and D because none of these includes the race car. We know the pair of colors that will be used on the race car. We are left with B.) which states that it is only possible to know the contents of the race car and E.) which states that is actually possible to know the contents of both the race car and the truck. Lets see if we can in fact know what will be included in the truck.

So far we know G is in the Truck, I and M are in the race car, and we are left with F, K and J. What we actually find here is that any of F, K or J can also be included with G on the truck.

-If F is included it will be used as the base, with J and K being used on the airplane without violation.
-If J is used alongside G on the truck. F will be the airplane's base with K as its highlight. No violation.
-If K is used alongside G on the truck. F will be the airplane's base with J as its highlight. No violation.

The correct answer is B. When G is used on the truck, we can know the exact contents of ONLY the race car.

4. Question four asks: If mulberry is used as the base for the truck, which of the following represents the complete list of colors that could be used as a highlight for the truck?

Place M in its appropriate base slot on the truck. We know a lot here.

-First, we know F cannot be used as the highlight for the Truck (rule one) Eliminate answer choice D accordingly.

-Second, we know G cannot be use alongside M under any circumstances (rule two). Eliminate C and E.

Third, We know I has to be included in the race car and cannot be used on the truck. We are left with K and J as potential highlights on the truck. Both can be used without violation. Answer choice A is incomplete since it only includes K. The correct answer is B.) K and J.

5.) Question five asks: If jonquil and mulberry are used to paint the airplane, each of the following could be true EXCEPT:

Place J and M in the airplane slot. It turns out that either of these two colors can be used as the base or the highlight without violating rule five. If J is the Base M will be the highlight, however if M is the base and J is the highlight this is just an expression of rule five's contrapositive. (see explanation number 2 answer choice B)

Since M is used on the airplane, according to rule four G will be used on the race car. I must also be used on the race car. These two colors can also alternate between being used as the base or the highlight however they must be used on the race car.

This leaves F and K to decorate the Truck. We know that F will be assigned to the base slot and K will be assigned to the highlight slot (rule one)

Here's what our truths are for number 4.)

Airplane: J and M switch off between base and highlight
Race Car: I and G switch off between base and highlight
Truck: F in the base slot K in the highlight slot.

Let check out the answer choices. Remember this is a could be true EXCEPT question! We will find four response that could be true, with one response that must be false.

A.) This response could be true. Since G and I switch of in the race car this could be a true scenario.

B.) This could be true as well. Same reasons as A

C.) This also could be true. See above

D.) F being used as the base of the truck MUST be true. So this gets eliminated.

E.) K CANNOT be used as the base of the truck. Since it is paired with F, which must be used as a base (rule one) K must be used as a highlight. Answer choice E is your correct answer.

6. Question six asks: Which of the following, if substituted for the rule that if jonquil is used as a base, mulberry will be used as a highlight, would have the same effect in determining where colors can and cannot be used?

I hate these types of questions. This one actually happens to be fairly easy if we remember our sneaky contrapositive inferences . What needs to be achieved with the correct answer choice is the following: when J is used as a base, M will be used as a highlight. Lets look at our choices.


A.) If mulberry has a different role from fuchsia, jonquil will be used as a base.

a different role from F means M will be a highlight. Since F is invariable a base color. This says that when M is a highlight J will be a base. This is the reversal of what we are looking for! we want this: when J is used as a base, M will be used as a highlight. Answer choice A reverses our original conditional statement. Elimination.

B.) If jonquil has a different role from fuchsia, mulberry will be used as a base.

a different role from F means J will be a highlight. So this says if J is a highlight M will be a base. What we want is if J is a base M will be a highlight. This is incorrect as well.

C.) Jonquil and mulberry will be used for different roles

This isn't really the conditional statement we need. remember our orignal condition statement: if J is a base M is a highlight. With this conditional, both J and M can be used as highlights without violation. Remember! If M is a highlight that does not force J to be a base, it can still be a highlight as well. J being the base is the trigger. So making J and M assigned to different roles is not the same as our original conditional statement, and is more limiting.

D.) Mulberry will not be used as a base unless jonquil is used as highlight.

This translates to if J is used as the base M will be used as a highlight. This is the correct answer.

We have to parce the language of the answer choice a little bit here. "Unless" is the indicator that J will be our sufficient condition. Here is a trick I picked up when using unless. Take out unless and replace it with IF NOT.

So we have: M will not be used as a base IF NOT J is used as a highlight. Lets break this down.

Sufficient: NOT J used as a highlight= J used as a base

Necessary: M will not be used as a base= M will be used as a highlight

combine those two statements and we have:
If J is used as a base, M will be used as a highlight. Our original conditional statement. Answer choice D is correct.

E.) Unless mulberry is used as a base, jonquil will be used as a base.

This translates to IF NOT M is used as a base, J will be used as a base. Which further translates to If M is a highlight, J will be a base. This wrong for the same reasons answer choice A is wrong.

Let me know if this was helpful etc. I just took the Feb lsat, not to confident about it. There is a high chance I will be taking the June test. Currently looking for study groups as well and possibly as discounted MLSAT class Thanks!

[wsjstockstar]

Explanation for Toy Maker Game (Hard)

PREAMBLE
We know there are six possible colors: F, G, I, J, K, M
We know there are three toys: A, RC, T
We know there are two colors for each toy: Base, Highlight

Rules
Color F must be a Base
If J is used as a Base -- > M is used as a Highlight (contrapositive: If M is not used as a Highlight, then J is not used as a Base)
Color I must be on toy RC (as a Base or as a Highlight)
CANNOT have G+M (or M+G) on same toy
If M is on toy A --> G is on toy RC (contrapositive: If G is not on toy RC, then M is not on toy A)

QUESTION ANALYSIS
Question 1
a) Color I must be with RC
b) If M(A) --> G(RC)
d) If J(Base) -- > M(Highlight)
e) CANNOT have G+M

Therefore: C is the answer

Question 2
a) no rule violations, thus a possible combination
b) no rule violations, thus a possible combination
c) no rule violations, thus a possible combination
d) **RULE VIOLATION**
If J is used as a Base, then M MUST be a Highlight
However, given the arrangement, J can only be a Base, BUT the Highlight spaces are all filled. Thus, this presents a problem. The rules cannot be satisfied.
e) no rule violations, thus a possible combination

Therefore: D is the answer

Question 3
G could be a Base or a Highlight for toy T. It really doesn't matter. We know that Color I must be used for toy RC, and that Color M must not be with Color G. We also know that if we place Color M with toy A, Color G must be used with RC. But Color G is firmly planted with toy T. So Color M cannot be used on toy A. If Color M cannot be used on toy A, or on toy T, Color M must be used on toy RC. Therefore, toy RC must use Color M and Color I. It does not matter whether Color M is used for the Base on toy RC or whether Color I is used the Base on toy RC. The point here is that the Color combination for toy RC has been completely determined. There is a lot of flexibility w/r/t Colors F,K,J. However, Colors M and I must be used (not necessarily in that order) on toy RC.

Therefore: B is the answer

Question 4
We know that there are six possible colors: F, G, I, J, K, M
We know that Color M is part of the question, so we don't have to worry about it.
We know that Color I must be used with RC, so Color I cannot be in the correct answer choice.
We know that Color G cannot be with Color M on the same toy, so Color G cannot be in the correct answer choice.
We know that Color F must be a base, so Color F cannot be in the correct answer choice.
Therefore, we know that Colors F,G,I,M cannot be in the correct answer choice.
We know that Color K could possibly be next to Color M, so we don't need to test Color K.
We only need to test Color J, for Color J has a condition.
If Color J is used as a Base, then Color M must be used as a Highlight. If Color M is not used as a Highlight, then Color J is not used as a Base. If there is an arrangement where Color J is not used as a Base, then Color J can be included in the correct answer choice. There *is* a possible combination whereby Color J is not used as a Base, therefore Color J must be included in the correct answer choice. Therefore, there are two colors required in the correct answer choice, which are Colors J and K.

Therefore: B is the answer

Question 5
If J is used as the Base for toy A and M is used as a Highlight for toy A, there are only 4 other positions to fill. Since Color M is used on toy A, Color G must be used on toy RC. It does not matter whether Color I is the Base for toy RC or whether Color G is the Base for toy RC. These colors must be with toy RC given the arrangement offered by the question stem. Color F must be used as a Base, and can be used on toy T because there are no other Base positions available. So it is absolutely necessary for Color F to be used as a Base, and Color F can be used on toy T. There is no way that Color K could be used as a Base on toy T, because Color F must be a Base.

Therefore: E is the answer

Question 6
If Color J is used as a Base, then Color M is used as a Highlight.
The contrapositive is "If Color M is not used as a Highlight (Color M is used as a Base), then Color J is not used as a Base (Color J is used as a Highlight)." The only answer choice that provides the accurate contrapositive would deliver the logical statement "If Color M is used as a Base, then Color J is used as a Highlight."

Therefore: D is the answer

Well, that's a wrap. Got to Love the [Manhattan] LSAT!

[ohthatpatrick]

Great work, y'all.

Both explanations were correct and very thorough.

Nice job explaining the contrapositives, in particular the one about "if J is base, M is a highlight". When you're dealing with binary options (such as M must either be a base or a highlight), it's a GREAT habit to write the contrapositive in terms of what is, rather than in terms of what isn't.

So I definitely prefer
Mb --> Jh
to
~Mh --> ~Jb

As one, or both of you, indicated, this conditional rule does not mean that J and M have to fulfill different roles (as choice (C) in #6 suggests). After all, it's possible for J and M to both be highlights.

I thought we could use this insight to answer #2 a little faster.

Since #2 asks which could be the list of Highlights, we can certainly check to see if F is included (since it must be a base), but that would be too easy. The other rule that relates to base/highlight is this J & M rule.

Upon analyzing this rule, we know that J, M, or both always have to be a highlight. If we scan the answers looking for the foolish falsehood that would have neither J nor M, we can find (D) pretty quickly.

That was the only potentially helpful omission I saw. Great work!!

[browncoool1]

nice posts..........