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September 11, 2009

Logic Games Matching Diagram | Explanation

LSAT Blog Logic Games Matching DiagramI'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:

Matching LSAT Logic Game 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:

Matching LSAT Logic Game Rules

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.)



7 comments:

  1. what are your thoughts on making a grid to portray the information. on the x-axis have the people and on the y-axis have the gods, or vice versa. then have information relevant to whatever letter next to it, for example a 3(check) next to H and a line going from P to B with a check there, etc. I am just wondering what the reasoning is to not do it like that? To me, it seems easier...

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  2. Hi Steve, I'm wondering if there is a general formula for choosing bases? Is it always the case where the variable that you have more information on is the base.

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  3. I want to echo what anonymous from December 29 asked. For instance, on PT 35 G2 (which deals with cars and features), it seems that using the features as a base is the best choice. There are 3 different features, and I think 7 different cars. That would be the opposite way of solving your game here--where you thought it was best to line up the deities and assign the students.

    I think it comes down to the questions. In your game it was better to do it your way for the first few questions, but the later questions are easier to answer if you diagram it with the students as the base. On PT 35 G2 a few questions are very quickly answered with features as the base as opposed to cars, and using cars as the base doesn't help with any questions really.

    What do you think? In general I have been using the variable with the smaller number of options as the base, but I can see deciding on the base based on the questions. Although I haven't found a similar actual LSAT game like your game where you might want to choose the larger variable as the base, so until I get more info or see that I might just use the smaller variable on questions like this.

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  4. One thing I want to mention is to try and keep the variables in a specific order in the diagram. By doing so, you make it a little easier on both keeping track of the number of times you may have used a variable and keeping track on where a variable has been placed. For example:
    ****A B J P
    G 1 _ _ _ _
    H 2 _ _ _ _
    K 3 _ _ _ _
    L 4 _ _ _ _
    R 5 _ _ _ _
    S 6 _ _ _ _
    V 7 _ _ _ _

    If you diagram this way and apply all the rules, you can figure out the diagram as such:

    ****A B J P
    G 1 _ G _ _
    H 2 X H H H
    K 3 X K X K
    L 4 L X L X
    R 5 _ X _ _
    S 6 X S S S
    V 7 V X X X

    The "X" represents where the variable cannot be placed. I numbered the variables according to the order of how they been stated in the game. You don't have to do it that way, but I found it easier for me if I did it that way. As you may have noticed, I used the students as a base. I found this to be much easier than using the deities as a base. I used this same diagram on PT 35 G2 whereas I had the car features as a base and the cars as variables. It works wonders, but I have only done this way twice with both instances being un-timed. However, I think I may be able to do this way fairly fast after some practice. Please do tell me all what you think of it.

    P.S. I only used the "*" to help in lining up the diagram properly on this blog. Unfortunately, the diagram doesn't look perfectly lined up as I hoped it would have been. :P

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  5. G H K L R S V
    A - - X - X
    B X X X - - X
    J X -
    P X X - - X

    I kinda liked this way to do it as well. X's are where deities answered prayers. "-" are essentially Not-Laws. Hope the formatting turns out...

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  6. So, it didn't turn out great, however you get the idea. It goes with what Dan and Anonymous (Apr. 23) were talking about.

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  7. This is very interesting. When I diagrammed the game this way, it was much easier for me to answer the questions. I was having a really hard time before when diagramming with the students as a base. Geez..

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