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Setup and Rule Diagram Explanation

This is a Grouping Game: Defined-Fixed, Underfunded, Unbalanced, Numerical Distribution.

This is the most difficult game of the section. At first, this appears to be a standard Grouping game, but the test makers use the Numerical Distribution to raise the level of difficulty.

The game scenario establishes that three nations—X, Y, and Z—each export exactly two crops from a group of five:
D04_Game_#4_setup_diagram 1.png
D04_Game_#4_setup_diagram 1.png (1.5 KiB) Viewed 903 times
Since there are only five crops (and each crop must be exported by at least one nation), but there are six exporting slots to be filled, exactly one of the crops must exported twice, and there is a 2-1-1-1-1 Numerical Distribution present. This distribution is critical, because any time one of the crops is known to be exported twice, then no other crop can be exported twice. Since some of the rules result in a crop being exported twice, this distribution comes up in many of the questions. As we analyze the rules, we will discuss how each rule affects the distribution, where applicable.

Note that we have chosen the nations as the base because they establish a fixed 2-2-2 spread; if we chose the crops as the base, then we would have to deal with the uncertainty of which crop is doubled in our diagram.

Rule #1. This is the least complex of the four rules, and the correct diagram is:
D04_Game_#4_setup_diagram 2.png
D04_Game_#4_setup_diagram 2.png (1.02 KiB) Viewed 903 times
This rule does not allow us to make any powerful inferences, but can infer that if O is the crop exported twice, then W would have to be exported by the country that does not export O; if W is the crop exported twice, then O would have to be exported by the country that does not export W.

Rule #2. This can be a tricky rule to diagram if you are not familiar with the phrasing used by the test makers. The phrase “if but only if” is identical to the phrase “if and only,” which produces a double-arrow diagram. The proper diagram for “if but only if” is also a double arrow:
  • ..... ..... ..... ..... ..... ..... XS :dbl: YS
Functionally, this rule means that if X exports S, then Y exports S; and if Y exports S, then X exports S. A horizontal block could be used to show this relationship, but a block would not show the conditional aspect of this rule, which is that if one of the two countries exports S, then the other must do so as well (a block might imply to someone that both countries must always export S, which is not true).

From a numerical standpoint, this is a powerful rule because if either X or Y exports S, then the other country must do so as well, and that means that S would be the one and only crop that is exported twice. The other four crops would then be exported a single time.

We can also infer that if either X or Y exports S, then Z cannot export S:
  • ..... ..... ..... ..... ..... ..... XS :dblline: ZS

    ..... and

    ..... ..... ..... ..... ..... ..... YS :dblline: ZS
This occurs because if once X or Y exports S, then the other must also export S, and if Z also attempts to export S, then each nation would export S. With all three nations exporting S, the Numerical Distribution is violated (S would occupy three of the six available exportation slots, leaving an insufficient number of spaces to export each of O, R, T, and W).

Rule #3. This is a conditional rule with two necessary conditions:

  • ..... ..... ..... ..... ..... ..... ..... XT

    ..... ..... ..... ..... YR ..... :arrow: ..... +

    ..... ..... ..... ..... ..... ..... ..... ZT
Like the previous rule, this rule also results in one of the crops being exported twice, in this case T. So, the immediate inference we can draw is that the second and third rules pose a conflict because both result in different crops being exported twice, and if one rule is enacted, then the other one cannot be enacted:

  • ..... ..... ..... ..... ..... ..... XS

    ..... ..... ..... ..... ..... ..... or ..... :arrow: ..... YR

    ..... ..... ..... ..... ..... ..... YS

    ..... and

    ..... ..... ..... ..... ..... ..... ..... ..... ..... XS

    ..... ..... ..... ..... ..... ..... YR ..... :arrow: ..... +

    ..... ..... ..... ..... ..... ..... ..... ..... ..... YS
The two diagrams above, although perfectly accurate, are a bit unwieldy. A simplified representation would be these two double-not arrow diagrams:

  • ..... ..... ..... ..... ..... ..... YR :dblline: XS
    ..... and

    ..... ..... ..... ..... ..... ..... YR :dblline: YS
The inferences above are easily missed, so let’s cover the reasoning behind them again:
  • The game scenario establishes that the five exported crops must fill a total of six exportation slots (two per nation). Thus, exactly one of the crops must be exported twice (this balances the crops to export slots at six to six, eliminating the underfunded aspect of the game). Because only one crop can be exported twice, any rule that results in a crop being exported twice automatically prohibits any other rule from being enacted that results in a crop being exported twice. Since both the second and third rules result in different crops being exported twice, those two rules cannot both occur in a viable solution to the game. Therefore, if one of those two rules occurs, the other will not occur. This can be directly stated as if X or Y exports S (the second rule), then Y cannot export R (the sufficient condition of the third rule), and if Y exports R (the sufficient condition of the third rule), then neither X nor Y cannot export S.
Rule #4. This rule can be diagrammed as:

  • ..... ..... ..... ..... ..... ..... Z :arrow: Y
The contrapositive is:
  • ..... ..... ..... ..... ..... ..... Y :arrow: Z
The combination of those two diagrams creates a double-not arrow relationship, where Y and Z cannot have any exports in common:
  • ..... ..... ..... ..... ..... ..... Y :dblline: Z
However, if you diagram this relationship off to the side, you might forget the rule, so a better approach is to diagram the rule internally by placing it right into the main diagram:
D04_Game_#4_setup_diagram 3.png
D04_Game_#4_setup_diagram 3.png (1.72 KiB) Viewed 903 times
The other rules and inferences can be diagrammed on the side, creating the following master setup:
D04_Game_#4_setup_diagram 4.png
D04_Game_#4_setup_diagram 4.png (2.71 KiB) Viewed 903 times
  • XS :dbl: YS
  • XS :dblline: ZS
    YS :dblline: ZS
  • ..... ..... ..... XT

    YR ..... :arrow: ..... +

    ..... ..... ..... ZT
  • YR :dblline: XS
    YR :dblline: YS
Using the setup above and keeping the Numerical Distribution firmly in mind, we are ready to attack the questions.
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Good morning:
I am working my way through the Logic Games Bible and PrepTest practice problems; however, I continue to have problems with the set up and inferences that can be derived from the rules of Grouping games. For this particular game, can you please assist me with what inferences should be made and the set up for the game? I think that I am taking way too much time working through each individual question without having the proper set up and inferences.
Thanks so much!!
 Emily Haney-Caron
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Thanks for the question!

For this game, there are only a few inferences to be made:
First, it is important to note that there will only be one repeating crop; four crops will be used once, one will be used twice.
Second, because there are only three nations and each crop must be used, if soybeans don't go to X and Y then they MUST go to Z (and only Z). By the same token, if Z does not get soybeans, then X and Y both have to.
Next, a minor inference, but when you put the second and third rules together, if Y exports rice then X cannot export soybeans.
Other than that, the important thing for this game is just being constantly aware of ALL of the rules and how they interact - not specific inferences, but rather keeping them all in mind and considering how they work together.

Does that help at all?
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I have two distributions for this game: 3-1-1 and 2-2-1. So I don't know how only one crop can be used twice and the rest only once. A full set up would clarify what I'm not getting.
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Never mind I found my problem.
 Kristina Moen
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That's always our goal - that you can identify and understand your own errors or omissions. On test day, you'll be flying solo, and you're doing a great job so far! :)
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I am wondering- why does rule #2 not leave the possibility of S being in Nation Y but not in Nation X? Thank you in advance!
 Adam Tyson
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"If, but only if" creates a double arrow, lsat.bea, also known as a biconditional. It means that either both of those things happen, or else neither of them does. Think of it as two rules:

X exports soybeans if Y does: YS :arrow: XS

X exports soybeans only if Y does: XS :arrow: YS

If either of those two exports soybeans, the other one must, so if either does not, the other also cannot (and then soybeans would have to go to Z and nowhere else).

You would also get that result from rules that say any of the following:

X exports soybeans if, and only if, Y does.

X exports soybeans if Y does; otherwise, not.

X exports soybeans if Y does, and vice versa.

All of these create that biconditional, or double arrow, and result in either both things occurring or else neither of them occurring.

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