Logic Game | Grouping: Selection Defined Diagram | Explanation

This week, I'm doing a blog post to explain my approach to solving last week's Free LSAT Logic Game | Grouping: Selection (Defined).

Again, here are the categories and topics:

P - ABE
R - HLO
W - STY


I'd put a "1-" next to each category because of the rule that we have to have at least one from each category. "1-" indicates that there's "at least one" from each, but we don't know the maximum number of variables we can have from each, so I'm not "closing" the range.

P - ABE - 1-
R - HLO - 1-
W - STY - 1-


Main Diagram
I'd draw it on paper like this:



I put a P, R, and W in parentheses to the side of three of the "in" slots because we know "at least one from each category is in." It just makes that rule a bit more concrete and visual by reserving a slot for everything we know so far.

However, it's just easier to type horizontally (and some of you read this on mobiles that can't see picture files, so whenever you see a diagram I've typed, just think of it being draw vertically):

6 In: _ _ _ _ _ _
3 Out: _ _ _


1st Indented Rule:

If S is in, E and L are out.

Contrapositive: If E or L is in, S is out.


Summary:

S <--+--> E
S <--+--> L

That's a "double-not" arrow. It means that at least one of the things on either side of the arrow CANNOT be in. Looking at the S-E double-not arrow (in isolation), it means that:

(1) S is out

OR

(2) E is out

OR

(3) S and E are both out.


Same goes for the S-L double-not arrow.

We can summarize the two double-not arrows as:

S <--+--> E, L



2nd Indented Rule:

If L is out, Y is in.

Contrapositive: If Y is out, L is in.

Summary:

(1) Y is in

OR

(2) L is in

OR

(3) Y and L are both in.

See Conditional Reasoning for an explanation of this type or rule (and the double-not arrow, as well).


By combining the 1st and 2nd indented rules, we can form the chain:



At this point, we should think something along the lines of, "Wow, that S is pretty powerful stuff. Let's see what happens when I place it 'in'. "

Putting S "in" means E and L are "out," and Y is "in" (we inferred from that conditional chain I just made, above):

6 In: S Y _ _ _ _
3 Out: E L _

That's all we can do with S right now, but let's look at the...


3rd indented rule:

At least one topic on war will not be selected.

This means we'll always have either 1 or 2 war topics "in", but never all 3. At this point, I'd draw a "(W)" to the side of one "out" slot, just to reserve it for a War topic.

(Remember: we can't have 0 war topics "in" because we always have to have at least one topic from each category "in.")

Here are the categories and topics from before, but I'd now add in the number of topics we can have from each category:

P - ABE - 1-3
R - HLO - 1-3
W - STY - 1-2

***

If we JUMP back to that little "S-in" diagram we made just a moment ago...

6 In: S Y _ _ _ _
3 Out: E L _

we'll see that we already have 2 Ws "in" (S and Y are both from the War category).

Because we can't have all 3 Wars from this category, T will have to be "out".

6 In: S Y _ _ _ _
3 Out: E L T

If we now have 3 "out", and 6 have to be "in" total, this means that everyone but E, L, and T has to be "in." Our "S-in" diagram is now complete, giving us:

6 In: S Y A B H O
3 Out: E L T

I'd write this off to the side and circle "S" to indicate that "S-in" is what started everything.

I'd then make another main diagram with S "out" and leave all the other slots blank.

***
I'm not going to explain ALL the questions because that just spoils the fun...but I will give a hint for this type of game in general.

The 1st question is a typical "List / Acceptability" question regarding specific topics. You can solve this simply by applying the rules.

However, the 2nd, 3rd, and 4th questions refer to the general categories as well.

Once you hit the 2nd question, the reference to the religion category should make you start thinking about numerical distributions (the various combinations of numbers of variables you can select from each category).

If you're decent or better with numbers, or you think you can get better by practicing, try to list all the combinations quickly.

The War category is the most limited because you can only have 1 or 2 from that category.

If we have 1 from war, that leaves 5 remaining that we'll need to pick (6 total - 1 from War = 5).

How can we make 5?

5+0, 4+1, and 3+2

However, there's no category with 4 topics, and we need at least one topic from each category (which means having 0 "in" isn't okay), so it's either going to be 3 from Politics and 2 from Religion or 2 from Politics and 3 from Religion.

This gives us the distributions:

3 Politics, 2 Religion, 1 War

2 Politics, 3 Religion, 1 War


BUT

If we have 2 from War, that leaves 4 remaining that we'll need to pick to be "in" (6 total - 2 from War = 4)

How can we make 4?

4+0, 3+1, and 2+2.

Again, there's no category with 4 (and again, we also can't have 0 from any category "in"), so it's either going to be 3 from Politics and 1 from Religion, or 1 from Politics and 3 from Religion, or we'll have 2 from Politics and 2 from Religion:

3 Politics, 1 Religion, 2 War

1 Politics, 3 Religion, 2 War

2 Politics, 2 Religion, 2 War


Of course, you don't want to take up all that space by writing so much.

Instead, I'd write the numerical distributions like this:

P-R-W
3-2-1
2-3-1
3-1-2
1-3-2
2-2-2

(I'd write them spaced-apart without a dash, but I don't want the formatting to get messed-up here.)

Recognizing numerical distributions is good stuff, but I'm not getting a kick out of it just because it's fun.

It also helps us answer the question because it becomes obvious that only the 3P-1R-2W distribution has 1R, so we know that we'll have to have 3P and 2W.

We know that the variable S will be "out" because we already have the complete "S-in" scenario (which is 2-2-2, by the way).

If S is "out", the 2Ws "in" will be T and Y, and all 3 Ps "in" means ABE are "in"... (you can take it from here)

***

What if you're not good with numbers and don't think you can get better at them?

Then, instead of trying to list all the combinations, just start with the 2nd question's limitation, that we have exactly 1 Religion topic and figure out that specific distribution instead of taking the time to list them all.

However, it makes the game much easier if you know all of them, so it's worth practicing how to do this.

I hear you saying, "but it's not fair that I have to be good at math..."

I feel your pain. I couldn't stand precalculus or calculus, but this is just addition. It's about recognizing all the ways to get to 4 and 5.

You can do addition, can't you?

The LSAT is all about thinking of all the possibilities, not just the obvious ones.

So make your 2nd grade math teacher proud and beat everyone at craps.

Photo by wisse / CC BY-NC-ND 2.0

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Free LSAT Logic Games | Grouping: Selection (Defined)

I posted my Grouping: Selection / In-Out Logic Game on the blog a month ago, but that one was "Undefined," meaning that we didn't know how many variables were in (voted for) and how many were out (voted against).

The game I've written for this week, on the other hand, is "Defined," which means that we know exactly how many variables are selected and how many are not selected.

Please post your thoughts and questions in the comments!

***

Here's this week's Logic Game:

From among nine topics, a student will select six to debate at a tournament. The topics are organized into three categories. Of the topics, three-A, B, and E-are on politics, three-H, L, and O-are on religion, and three-S, T, and Y-are on war. At least one topic will be selected from each category. The student selects the topics according to the following conditions:

If S is selected, neither E nor L is selected.
If Y is not selected, L is selected.
At least one topic on war will not be selected.

1. Which one of the following could be a complete and accurate list of the topics that the student selects?

(A) A, B, E, H, L, O
(B) A, B, E, H, L, T
(C) A, B, E, H, O, T
(D) A, B, E, H, S, Y
(E) A, B, H, S, T, Y


2. If exactly one of the topics on religion is selected, then which one of the following is a complete and accurate list of the other topics that must also be selected?

(A) A, B, E, T
(B) A, B, E, H, Y
(C) A, B, L, T, Y
(D) A, B, E, S, T
(E) A, B, E, T, Y


3. Which one of the following must be true?

(A) A is selected.
(B) E is selected.
(C) T is selected.
(D) Of at least one of the three categories of topics, exactly two topics are selected.
(E) Of at least one of the three categories of topics, exactly three topics are selected.


4. If exactly two topics from each category are selected, then which one of the following must be true?

(A) B is selected.
(B) H is selected.
(C) Y is selected.
(D) L is not selected.
(E) T is not selected.


5. Each one of the following is a pair of topics that could be among the topics selected EXCEPT:

(A) A, Y
(B) B, H
(C) B, O
(D) S, T
(E) S, Y


6. Each of the following, if known, would fully determine the selection of the six topics EXCEPT:

(A) B and E are not selected.
(B) B and T are not selected.
(C) E and O are not selected.
(D) L and T are not selected.
(E) O and Y are not selected.


The text below contains the answers to the above Logic Game.


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

***

Can't figure out how to do the game? Not to worry - leave a comment!

See my tips on how to solve this game in Logic Game | Grouping: Selection Defined Diagram | Explanation.

***

For some LSAC-written Logic Games similar to this one, check out:

PrepTest 24, Section 4, Game 4 (page 213 in 10 More)
PrepTest 33, Section 4, Game 3 (page 178 in Next 10)
PrepTest 40, Section 2, Game 2

Photo by ennuiislife / CC BY-NC 2.0

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Logic Game Grouping: Matching Templates Diagram | Explanation

As promised, I'm doing a blog post to demonstrate the templates (multiple main diagrams) for last week's Grouping: Matching Templates Logic Game, so here goes.

So, in the game, we have 7 projects. 2 in April, 3 in May, and 2 in June, giving us this layout of slots for each month:

Depending on whether you choose to make your templates refer to specific variables or simply to the categories of variables, you might want to put a "D" with a slash through it under June.

Our variables are:

Classifieds: C C
Electronics: A B
Weapons: D L R

So we can diagram our rules like this:

The options for the game are more limited than they might appear at first. The key is figuring out which rule or block to use to create your templates. You want to go from the most restricted variables and work your way towards the least restricted variables.

The vertical WW block and the vertical WE block could go on any of the 3 months, so they're actually pretty "loose," so we don't want to start there. We also don't want to make those the first variables we work with because there are 3 "W" variables: DLR, and 2 "E" variables: AB.

However, there are only 3 main possibilities for the Classifieds.

They could be on April and May:


on May and June:


or on April and June:


These are our 3 main possibilities, and every possible scenario will fall into one of these 3. You could leave off here, but I prefer to get more specific.

Let's start with the first - the 2 Classifieds on April and May.

In this possibility, we could have EW on May, and WW on June:


OR we could have WW on May and EW on June:


Now the 2nd main possibility - the 2 Classifieds on May and June.

In this possibility, we could have EW on April and WW on May:


OR we could have WW on April and EW on May:


Finally, 3rd main possibility - the 2 Classifieds on April and June.

In this possibility, we're forced to put EWW on May together (which satisfies the EW block and the WW block simultaneously).

The remaining E and the remaining W are interchangeable. One will go on April, and the other will go on June. Here's this possibility with W on April and E on June:


Here's the same possibility, only with E on April and W on June:


Of course, you can easily combine these two into one diagram with E/W on April and W/E on June:




In fact, you can also combine the 2 main possibilities for when we have 2 Classifieds on April and May. The only difference between these two possibilities is whether we have 2Ws in May and E on June or 2Ws in June and E on May. Combining them gives us:





We can also combine the 2 main possibilities for the 2 Classifieds on May and June in the same way:





These become our 3 main diagrams.


Just don't forget that D (the drone) can never be on June. Place D with a slash through it below June in each diagram.

If you prefer to diagram things more literally, you can diagram the game's main possibilities in the same way I've done throughout, only whenever you see "E", you can write "A/B", and whenever you see "W", you can write "D/L/R" (only don't forget that when "W" is on June, it's only "L/R" because D can't be there.

I actually prefer to do the game's templates (main setups/diagrams) by writing the categories rather than the specific variables. As you might imagine, diagramming the specific variables takes longer to write and looks a bit more cluttered. However, it might save you time by allowing you to avoid referring back to which variables fall within each category.

What's best? Try it both ways to determine your personal preference.

Leave your questions and thoughts in the comments!

Photo by donsolo / CC BY-NC-SA 2.0

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Free LSAT Logic Games | Grouping: Matching | Templates

Two weeks ago, I published a Grouping: Matching Logic Game I'd written. That one was a difficult type of Matching game because the number of "prayer-answerings" per deity was not fully determined.

In other Grouping: Matching games (such as the one I've written for this week - see below), the number of slots is fully determined for you. In these types of games, drawing multiple templates, possibilities, or limited options is often incredibly useful.

What are templates, possibilities, and limited options? They're all different terms describing the same technique - drawing a few main diagrams instead of only one. Each main diagram you draw will represent a different potential placement of the variables.

The key to drawing templates is to identify a variable or rule that has only a few different possibilities. Once you've found that variable or rule, draw multiple diagrams based upon every different major possibility for it, and fill in the other variables around it.

Please post your thoughts and questions in the comments!

***

Here's this week's Logic Game:


Compu-Global-Hyper-Mega-Net presents seven projects to the military over the course of three months- two in April, three in May, and two in June. Two of the projects are classified. Two others are electronics-active camouflage and a ballistics computer. The remaining three are weapons-a drone, a laser cannon, and a ray gun. The projects are presented in accordance with the following conditions:

The drone is not presented in June.
The classified projects are not both presented in the same month.
At least one of the weapons is presented in the same month as one of the electronics projects.
At least two of the weapons are presented in the same month as each other.

1. If the laser cannon is the only weapon presented in June, which one of the following must be true?

(A) A classified project, an electronics project, and a weapons project are presented in May.
(B) The ballistics computer is presented in May.
(C) A classified project is presented in May.
(D) An electronics project is presented in May.
(E) The ray gun is presented in May.


2. If two weapons are presented in April, which one of the following CANNOT be true?

(A) The drone is presented in April.
(B) The active camouflage is presented in May.
(C) A classified project is presented in May.
(D) The ballistics computer is presented in June.
(E) The laser cannon is presented in June.


3. If the electronics projects are not presented in consecutive months, which one of the following must be true?

(A) A weapon is presented in April.
(B) The laser cannon is presented in May.
(C) A classified project is presented in June.
(D) A weapon is presented in June.
(E) The classified projects are presented in consecutive months.


4. Which one of the following CANNOT be true?

(A) Both electronics projects are presented in the same month.
(B) Two weapons are presented in the same month as a classified project.
(C) Two weapons are presented in the same month as an electronics project.
(D) The active camouflage and the drone are presented in April.
(E) Two weapons are presented in June.


5. If neither of the electronics projects is presented in June, which one of the following must be true?

(A) A classified project is presented in May.
(B) The drone is presented in May.
(C) The ray gun is presented in May.
(D) A classified project is presented in June.
(E) Exactly one weapon is presented in June.


The text below contains the answers to the above Logic Game.

1. E
2. E
3. E
4. A
5. B

***
Can't figure out how to do the game? Not to worry - leave a comment!

You can also see my explanation and diagrams / templates of this game in Logic Games Grouping: Matching Templates Diagram | Explanation.

***
For some Logic Games similar to this one, check out:

PrepTest 37, Section 3, Game 3 (page 306 in Next 10)
PrepTest 38, Section 2, Game 3 (page 332 in Next 10)

Also see my Logic Games Categorization.

Photo by epicfireworks / CC BY-NC 2.0

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Logic Games Pure Sequencing Diagram | Explanation

I've received several requests for a diagram of my Pure Sequencing LSAT Logic Game, so here goes.

This blog post includes my version of the main diagram for this Logic Game. It also includes modified versions of the diagram for questions 4 and 5, which introduce new limitations to the main diagram.

***

Also see my explanation of this game's set-up in Logic Games Pure Sequencing Step-by-Step Diagram.

***

I don't normally write "earlier" and "later" on the diagram because linear/sequencing games usually involve putting things in order from earlier -> later.

For this reason, left -> right usually means earlier -> later, and it's come to feel natural. However, some of you are still learning, so I included it in the below diagrams.

Here's the main diagram (click to enlarge):


Here's the diagram for Question 4 (click to enlarge):




Here's the diagram for Question 5 (click to enlarge):




Photo by dannysullivan / CC BY 2.0

(It's from the episode "Sword of Destiny" in the TV show Arrested Development. GOB and Buster perform a magic act together, and Buster accidentally cuts off GOB's fingers. I can't believe someone actually made Lego scenes based on a TV show either. I love Arrested Development, but Lego scenes are a bit much.)

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Logic Games Matching Diagram | Explanation

I've received several requests for a diagram of my Grouping: Matching Logic Game, so here goes.

This blog post includes my version of the main diagram for this Logic Game. It also includes my diagram of the rules and some thoughts on making an effective diagram for this game (and games like it).

Here's the main diagram:



Here are the rules that I would've put to the side because they can't be placed easily on the main diagram:



Explanation of why and how I chose GHLRSV (the deities) as the base:

Many of you asked how I knew to put the deities as the "base", rather than the students.

Because the rules tell us more about specific deities than about specific students.

Deities: GHKLRSV
Students: ABJP

Let's look at the rules (which I've numbered below to reference easily):

1. Neither Lakshmi nor Rama answers Bobby's prayers, but Ganesha and Krishna both answer Bobby's.
2. Hanuman, Krishna, and Shiva answer Puja's prayers.
3. Hanuman answers at least three students' prayers, but Krishna and exactly one other deity each answer exactly two students' prayers.
4. If Vishnu answers a particular student's prayer, Hanuman does not answer that student's prayer.
5. If Ganesha answers a particular student's prayer, Rama does not answer that student's prayer.
6. Lakshmi answers at least one of the same students' prayers as Vishnu does.
7. Any deity who answers Puja's prayers also answers Bobby's but does not answer Arjun's.

Rules 1 and 2 tell us equal amounts about deities and students.
Rules 3-7 tell us more about specific deities than about specific students.

For this reason, it's much easier to use the deities as our base, rather than the students. As I mentioned in my hint in the game's initial blog post, choosing the right base can allow you to make inferences more easily.

***

What makes this Logic Game hard?

Two things:

1. Its general ambiguity.

We don't know exactly how many prayers each deity will answer, so this game is "loose." In other words, there are many possibilities (in contrast to PrepTest 35, Game 2 - page 237 in Next 10 - in which 4/6 cars are fully determined).

(This is also unlike Grouping: Matching games such as PrepTest 37, Game 3 - page 306 - and PrepTest 38, Game 3 - page 332 - where the number of slots is explicitly given.)

In my Hindu deities game, only 3/7 deities "prayer-answerings" are fully determined: H, K, and V.

The other deities' "prayer-answerings" are partially, but not fully, determined. I've used a dotted line (----------) to indicate ambiguity in the number of prayer-answerings.

Any letters below the dotted line are fixed. This means they will always be there in any valid scenario.

The letters above the dotted line might be there, but they also might not be there.

On the diagram, this means:

G might answer B only, but G could also answer the prayers of one or two other students as well.

L definitely answers A, but L might also answer J's prayers as well.

R will answer at least A or J, but R might also answer the other one of those two as well. This is why I wrote A/J with the potential of the other (J/A) as well.

S definitely answers B and P, but there's no reason S couldn't also answer J.


2. The third rule

Particularly, this part of it:

"Krishna and exactly one other deity each answer exactly two students' prayers."

The entire game revolves around which "other deity" answers exactly two.

That "other deity" could be G, L, R, or S.

If G answers 2, then L answers 1, R answers 1, and S answers 3.

If L answers 2, G answers 1 or 3, R answers 1, and S answers 3.

If R answers 2, G answers 1, L answers 1, and S answers 3.

If S answers 2, G answers 1 or 3, L answers 1, and R answers 1.


***

Are all Grouping: Matching games like this?

No. You can breathe a sigh of relief. Some are much easier and can be solved by using templates / possibilities / limited options (whatever you call it, it's the same thing).

PrepTest 37, Game 3, and PrepTest 38, Game 3 (both mentioned above) can be solved efficiently by using templates. See the Logic Game I wrote this week for a (difficult) example of this type of game.

Photo by iskcondesiretree
("Krishna kills the bird demon, Bakasura." He's the one standing in its mouth.)

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Logic Games Tips | Conditional Reasoning

The following two rules are extremely common in Grouping: In-Out / Selection games, but they give test-takers a lot of trouble.

These rules come up in other types of Logic Games, too. Make sure you can recognize them at a glance and instantly know what they mean.

Please note: "~" is a symbol meaning "not." I'd use a slash ("/" ) through a letter in the below examples to indicate "not."

However, it doesn't come out well when typed, so I'm not using any slashes in this blog post. If you prefer the slash (I do), use it instead.


Rule #1:

Positive Variable -> Negative Variable = + --> -



Original: X -> ~Y
Contrapositive: Y-> ~X

Meaning: at least one does NOT occur, and maybe both will not.


Example:

If I eat cookies, then I don't eat donuts.

If I eat donuts, then I don't eat cookies.

Therefore, I cannot eat at least one of them, but perhaps I'll eat neither.


Whenever you see a positive sufficient condition (the one before the arrow), and a negative necessary condition (the one AFTER the arrow), this means you can never select both, so at least one will not be selected.

(See LSAT Logic: Necessary vs. Sufficient Conditions)

In other words, you must always lack at least one of the two. However, there's no reason you can't lack both.


LSAT Examples:

PrepTest 33, Game 2 - birds in the forest (December 2000 - in Next 10 Actual, page 177):

Original: Harriers -> ~Grosbeaks
Contrapositive: Grosbeaks-> ~Harriers

Meaning: The forest cannot contain both Harriers and Grosbeaks - it will always lack at least one of the two, and maybe it will lack both.


PrepTest 36, Game 1 - fruit stand (December 2001 - in Next 10 Actual, page 278):

Original: Kiwis -> ~Pears
Contrapositive: Pears -> ~Kiwis

Meaning: The fruit stand cannot carry both Kiwis and Pears - it will always lack at least one of the two, and maybe it will lack both.


Rule #2:

Negative Variable -> Positive Variable = - --> +

Original: ~X -> Y
Contrapositive: ~Y-> X

Meaning: at least one MUST occur, and maybe both will occur.

Example:

If I don't eat peas, then I must eat carrots.

If I don't eat carrots, then I must eat peas.

Therefore, I must always eat at least one of the two, but there's no reason I can't have both.

At this point, students often ask, "Why is it possible to have both?"

Answer: Because the rule has no policy for what happens when you already have one of the two.

This rule only has a policy for what happens if I don't eat one (I must eat the other). It has no policy for what happens when I already ate one (or am going to eat). This is why nothing stops me from eating both. There's simply no rule against it.

The only thing this rule means is I can't LACK both.

LSAT Examples:

PrepTest 33, Game 2 - birds in the forest (December 2000 - in Next 10 Actual, page 177):

Original: ~Jays -> Shrikes
Contrapositive: ~Shrikes -> Jays

Meaning: The forest cannot lack both Jays and Shrikes - it will always have at least one of the two, and maybe it will have both.



PrepTest 36, Game 1 - fruit stand (December 2000 - in Next 10 Actual, page 278):

Original: ~Tangerines -> Kiwis
Contrapositive: ~Kiwis -> Tangerines

Meaning: The fruit stand cannot lack both Kiwis and Tangerines - it will always have at least one of the two, and maybe it will have both.


(Summarized by the Logic Games Bible as ~X <--/--> ~Y, which is an overly-complicated way of saying if one is out, the other is not out)

For the birds in the forest game, I would summarize the rule as:

J
S
JS

Every valid list of birds in the forest will fall into one of those three categories - one is in, the other is in, or both are in.


For the fruit stand game, you might summarize the rule as:

K
T
KT

Every valid list of fruits in the fruit stand will fall into one of those three categories - one is in, the other is in, or both are in.

***

You can also see my approach to Birds in the Forest Logic Game.

Photo by _sk / CC BY-NC-SA 2.0

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