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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:
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:
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 YS
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 ZS
Rule #3. This is a conditional rule with two necessary conditions:
- YR XS
- 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.
- Z Y
- Y Z
- Y Z
The other rules and inferences can be diagrammed on the side, creating the following master setup:
- XS YS
- XS ZS
- YR XS