Words Indicating Sufficient / Necessary Conditions, and Time

In conditional statements (if-then) statements, the sufficient condition is the one that goes before the arrow, and the necessary condition is the one that goes after the arrow.

Sufficient ---> Necessary

X ---> Y

is equivalent to:

if X, then Y.

X is the sufficient condition, and Y is the necessary condition.

Logic Games and Logical Reasoning contain conditional statements most prominently, but you can also find them in Reading Comp.

First, some words that indicate each type of condition.

Sufficient (before the arrow):
All, any, every, if, in order to, the only, to be, when, whenever


Necessary (after the arrow):
depends upon, must, only, only if, only when, requires, then


I like to think of the relationship between sufficient and necessary conditions in the following 2 ways:

Way #1: the sufficient condition activates the necessary condition, indicating that it will happen.

Way #2: the sufficient condition is enough to guarantee that the necessary condition happened already.


If the sufficient condition occurs, then the necessary condition must also occur (Way #1), or it must have also occurred at some point (Way #2). Yes, that's right, in Way #2, the necessary condition might occur before the sufficient condition.

I can hear some of you saying, "How could such a thing happen? If the necessary condition appears after the sufficient condition in a sentence, it must occur after the sufficient condition in real life too!"

Not so. But first, let's deal with the easier way (Way #1):

---

The sufficient condition can occur before the necessary condition (Way #1):

If A occurs, then B must occur afterwards.

For example, if I slam my head into a brick wall, then my head will hurt.
This could also be phrased, "Whenever I slam my head into a brick wall, my head hurts."

(A = slam head, B=head hurting)

If we take the contrapositive of this statement, we can say:

If my head doesn't hurt (NOT B), then I must not have slammed it into a brick wall (NOT A).


---

However, the sufficient condition can also occur after the necessary condition (Way #2):

If C occurs, then D must have already occurred.

For example, if I seduce an LSAC employee to get an advance copy of the LSAT, I must have traveled to LSAC's headquarters in Newtown, Pennsylvania.

(C= Seducing LSAT's test-writers, D=traveled to LSAC HQ in Newtown, PA)

This means that if you see me in NYC, I must not be seducing an LSAC employee at the moment.

(I know I'm going to get some emails requesting an advance copy of the LSAT, so I'll tell you right now - that sentence was a joke.)

***
Also see LSAT Logic | Necessary vs Sufficient Conditions.

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Logical Reasoning | Sufficient Assumption (Justify) Questions

Many of you dislike formal logic Logical Reasoning questions, particularly Sufficient Assumption (aka Justify) questions. In this blog post, I'll demonstrating my technique on a few of these questions.

Unfortunately, the folks at LSAC are very strict about their intellectual property and wouldn't allow me to post their LSAT questions online, so I'm only posting my explanations for the questions.

Not to worry, though.

In this blog post, I reference specific PrepTests, sections, and question numbers so you can follow along, as well as their page numbers in the books containing 10 PrepTests each (listed in Best LSAT Prep Books).

First, a quick drill:

Suppose you had an argument containing:

Evidence: A ---> B
Conclusion: C ---> B

One sufficient assumption that closes the gap would be: C ---> A

Why?

Because if all C's are A's, and all A's are B's, then all C's must be B's (C ---> A ---> B).


Another sufficient assumption would be NOT B ---> NOT C

If the contrapositive of the conclusion is true, then the conclusion itself must be true as well.

***
Side note:
Some students ask, why doesn't A ---> C work?

Well, suppose we knew A ---> C were true. What would our premises (pieces of evidence) be?

We'd now have:

A ---> B (from original evidence)
A ---> C (from new information - students' proposed answer)

This doesn't provide us with a conditional relationship between B and C. (Rather, it would simply allow us to infer that "Some Bs are Cs" and vice-versa, of course.)

It's like saying:

Apples (A) ---> Fruit (B) = If it's an apple, then it's a fruit.

Apples (A) ---> Food (C) = If it's an apple, then it's food.

Does this tell us that all food (C) is fruit (B)?

Of course not.

***

Because there are multiple ways to justify the conclusion, we can say any particular sufficient assumption does not necessarily need to be true to justify the conclusion. However, a sufficient assumption will be enough to justify the conclusion. It might involve the evidence, but it also might not. As you complete more sufficient assumption questions, you'll get a sense of which type of sufficient assumption they're looking for.


***
PrepTest 38 (October 2002 LSAT), Section 4, Question 16 - page 346 in Next 10
(the question discusses: people, distrust, confident, abilities, challenge)

Question Stem: "The conclusion above follows logically if which one of the following is assumed?"

Evidence group: People who do not believe that others distrust them
Conclusion group: People who tend to trust others


After identifying each piece of the stimulus, we can arrange them like this:

Evidence: Not Believe Others Distrust ---> Confident ---> Difficult Task as Challenge
Conclusion: Tend to Trust Others ---> Confident ---> Difficult Task as Challenge

We could diagram this as:

Evidence: NOT BOD ---> C ---> DTC
Conclusion: TTO ---> DTC

The necessary condition (regarding/thinking about each difficult task as a challenge rather than as a threat) is the same for both. Why? Because the final clause ("since this is precisely how...") supports the part of the conclusion that states "think of a difficult task as a challenge..." Since we know that "Difficult Task as Challenge" is a necessary result of "Confidence," we can put DTC after C in the evidence.

The cleanest way to close the gap is to say everyone in the conclusion's group of people ("people who tend to trust others" = "TTO" ) falls within the evidence's group of people ("people who did not believe that others distrust them" = "NOT BOD").

Choice C says this word for word.

The key is in noticing the evidence group and conclusion group are really discussing two different groups of people, and we don't know anything about how they relate to each other.

As always, try not to get overly involved with (or scared by) the topic of the argument. Instead, focus on its structure. This is especially true for the following question I explain.


***
PrepTest 35 (October 2001 LSAT), Section 1, Question 22 - page 226 of Next 10
(the question discusses: chordates, tracheophytes, Pteropsida, Hominidae)

Question Stem: "The conclusion drawn above follows logically if which one of the following is assumed?"


Explanation:

Evidence #1: No Cs are Ts = C ---> NOT T

Contrapositive: T ---> NOT C


Evidence #2: All members of P are T = P ---> T



If we connect these, we get:

Evidence #1's contrapositive and Evidence #2, combined: P ---> T ---> NOT C

Contrapositive of that combination: C ---> NOT T ---> NOT P = C ---> NOT P


Conclusion:

P ---> NOT H

Contrapositive of conclusion: H ---> NOT P


Evidence: C ---> NOT T ---> NOT P
Conclusion: H ---> NOT P


The goal of all this is to make the necessary conditions the same.

The drill at the start of this blog post already had "B" as the necessary condition of both evidence and conclusion, so there was no need to manipulate anything by taking the contrapositive.

However, this question is more difficult than the drill. We need to do some extra work by taking the contrapositive of what they gave us to get to the point where the evidence and conclusion both have "NOT P"

Now, to close the gap, I want to take the conclusion group's sufficient condition (H) and say it always falls within the evidence group's sufficient condition (C).

If all Hs are Cs, and we already know all Cs are NOT Ps, then all Hs must be NOT Ps as well, because:

H ---> C ---> NOT P

Very nice, clean, mechanical, and formulaic. No need to engage in any real thinking about the topic itself - just the pieces.


***
PrepTest 24 (December 1997 LSAT), Section 3, Question 19 - page 207 of 10 More
(the question discusses: students, school, home, lunch, part-time jobs, walk)

Question stem: "The conclusion of the argument follows logically if which one of the following is assumed?"

Explanation:

Evidence: walk to school ---> go home for lunch.
Conclusion: Some with part-time jobs don't walk to school.

E: WS ---> GHL
C: Some PTJ ---> NOT WS

I want to make evidence and conclusion have the same necessary conditions, so I'll take the contrapositive of the evidence:

Evidence: NOT GHL ---> NOT WS
Conclusion: Some PTJ ---> NOT WS


To close the gap, I want to say some students with PTJ don't go home for lunch.

I look for this, and I don't see it.

No need to go WTF, though, because we only need some overlap between the groups.

(Example: If some candy bars are delicious, then at least some delicious things are candy bars. The word "some" always allows for reversibility because the word "some" is vague.)

D indicates overlap between the groups, so it's our answer.


***

Note: this is a different "format" (logical structure) from the above questions.


PrepTest 9 (October 1993 LSAT), Section 2, Question 23 - page 64 in 10 Actual
(the question discusses: poor farmer, rich farmers, honest, dishonest)


Question stem: "The farmer's conclusion is properly drawn if the argument assumes that"

Explanation:

The sentence of the stimulus stating you're either rich or poor and either honest or dishonest is *crucial*.

This tells us the negation of rich is poor (and vice-versa) and the negation of honest is dishonest (and vice-versa).

Evidence: "All poor farmers are honest." = If Poor ---> Honest
Conclusion: "All rich farmers are dishonest." = If Rich ---> Dishonest

Evidence: P ---> H
Conclusion: R ---> DH


Contrapositive of Conclusion: NOT Dishonest ---> NOT Rich = Honest ---> Poor

Which is exactly what choice A says.

If the contrapositive of the conclusion is true (Choice A), then the conclusion itself is true also.

(As a side note, if choice A is not true, the argument is simply exhibiting a mistaken reversal, which is invalid.)


***

Also see Sufficient Assumption Questions | Tips and Categorization for more.

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LSAT Test Center Problems | How To Prepare For The Worst

Tip: Take practice LSATs at your LSAT test center.

How? Well, LSAT test centers are often within your local college, university, or law school. This means that they're often fairly easy to walk right inside, so many of you will be able to go there and take practice tests before your exam. I'd recommend doing this for at least a few Saturday mornings leading up to your test date, if you can.

Ideally, you'll be able to take it in the very room where you'll take the real thing. If not, at least try to get a look at the room so that you'll be able to take your practice tests in a similar environment.

Now going to the effort to take it in the same room seems like a lot of work (and a little weird).

Why would you want to go to all that effort?

Because rooms often have little quirks of their own. You'll hear noises in some when the heating turns on/off (especially for December / February test-takers living in parts of the world with seasons). Think about all the noises that boilers and pipes make (hissing, etc.). Unfortunately, there's no guarantee that you'll get extra time just because there's a disruption.

So do June and September test-takers have an advantage? Are you better off taking the LSAT in one of those months instead?

No, they don't get a free pass either. Some rooms might be over-air-conditioned and feel like walk-in refrigerators. Others might have no AC at all, feeling like an NYC subway platform in the summertime (trust me - it's like a furnace).

One test-taker recently wrote to me with the following:

I live in Cairo, Egypt, and I took the LSAT here in June...A woman randomly walked into the room while the exam was being administered and just started chatting with the proctor. They were whispering, but it was still audible and very distracting. Also, the electricity in the building went off for around the length of a section, so we resumed taking the exam in the dark with no AC (in Cairo in June...). Needless to say, I canceled my score on the spot.

Of course, electricity can go out anywhere, and the best you can hope for is a free retake a few months later with the original test removed from your record entirely. However, there's a very good chance that LSAC might give you nothing at all for your trouble.

Unfortunately, an uncomfortable room isn't the only obstacle you might face.

The Most Common LSAT Test Center Problems:

Time
-proctors not giving enough time per section or for the break (proctors giving too much time is not a common complaint, although it happens from time to time)
-proctors failing to give a 5-minute warning

Noise
-proctors talking to each other or talking on their phones
-proctors' cell phones ringing or beeping
-other test-takers passing out or leaving in the middle
-other test-takers making too much noise while writing
-room being too noisy
-noise from the hallway or outside

Room
-room being too hot or cold
-desks being too small

Miscellaneous
-other test-takers talking about how much they studied before the test or during the break
-proctors forcing you to eat a live spider before the start of each section

Kidding about that last one - this isn't Fear Factor.


Why I am telling you all this? Do I get a kick out of scaring already-nervous test-takers?

Nah, I'll leave that to LSAC.

I just want you to be prepared ahead of time in case any of the above happen to you on Test Day.

What's the cautious test-taker to do?

Prepare for the worst. Just about anyone would retake if the lights went out, a tornado hit, or a terrorist took everyone hostage, so I'm not going to recommend that anyone take a practice test under those conditions.


How To Prepare For The Worst

Time
-Confirm with the proctor beforehand that you'll get 35 minutes per section, a 5-minute warning before the end of each section, and a 15-minute break between the 3rd and 4th sections.

(Be very polite about this. Proctors are the police / dictators of the room. Their word is law, so respect their authority. However, this doesn't mean you can't say something if they're cutting you short on time.)

Anecdotal evidence (reports from students and blog readers) indicates that LSAC generally does nothing for people shorted on time or not given a 5-minute warning. Of course, most test-takers probably don't experience this problem, but if it happens to you, it'll be a big deal.

Noise
-Do at least one or two practice tests in somewhat noisy environments. A coffee shop or library is good for this.

Room
-Practice at least once or twice on a small desk.
-Bring layers in case the room is too hot or cold.
-Practice in rooms where the temperature is slightly uncomfortable.
-Again, try to take a few practice tests in the very room where you'll take the real thing.

Miscellaneous
-Don't talk to anyone during the break - unless you want to.

***

Check out all LSAT Test Day Tips.

***

Anything I didn't cover? Those of you who've taken the LSAT already, did any of the above happen to you? What are your tips on preparing for Test Day, and what do you wish you'd done differently?

Free Law School Personal Statement Help

*Update: Michelle is not accepting any more personal statements for review.*

However, be sure to check out these Law School Admissions | Book Recommendations.

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Logic Game | Basic and Advanced Linear | Limited Options

Limited Options, Templates, Possibilities, whatever you call them, they're ridiculously useful in several Logic Games.

This blog post covers one common, yet simple, technique for listing all the options/templates/possibilities in Basic Linear and Advanced Linear games.

Let's say we're doing an 7-slot / variable Linear game in which only one variable can go in each slot. This means we're placing the game's 7 variables (ABCDEFG) in some kind of order.

Just to keep this short and sweet, let's pretend all the other rules, inferences, and limitations have filled up (determined the placement of variables into) slots (spaces) 1, 2, 6, and 7, giving us:

D_ E_______F_G
1__2_3_4_5-_6_7

We have 3 empty slots, and we have three variables (A, B, C) we haven't placed yet.

Let's also pretend we have a rule telling us A is before B. We can diagram this as:

A-B

Because we have 3 variables remaining, 3 slots remaining, and we know 1 of these 3 (A) is before another of these 3 (B), we know there are only 3 main possibilities for the game.

How do we find these? Limited Options.

We place A and B first because we know the most about them. We don't know anything about C. C is a wild card variable.

We can place A and B into the diagram in the following 3 ways:

D E A B _ F G

D E _ A B F G

D E A _ B F G

These are the only 3 possibilities for the placement of A and B.

We can now place C into the empty slots, giving us:


D E A B C F G

D E C A B F G

D E A C B F G

I would just stack those three bolded possibilities into the diagram like this:



(I wouldn't actually number them, of course.)

LSAT Logic Games give you scenarios where this technique applies more often than you'd think. There have even been cases where it applies to the main diagram for Advanced Linear Logic Games (example: PrepTest 37, Game 2 - page 305 in Next 10). However, it more frequently comes up as something you can do for specific scenarios / hypotheticals in both Basic Linear and Advanced Linear games.

So, just keep this in mind: when there are 3 slots remaining, 3 variables remaining, and 1 of those variables most go before another, there are only 3 possibilities, and they're worth drawing out, whether it's for a main diagram or a specific scenario.

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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.

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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

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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

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

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.)

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.)

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.


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