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This is a Grouping/Linear Combination Game.
This game requires you to focus on two separate functions: establishing the group and then ordering the group:
Any variable that is in the group of five must then be ordered, and we have shown that relationship with an arrow from the group to the order.
As the first three rules are applied, we can begin to fill in the diagram:
The last three rules are more complex, and must be discussed in detail:
The fourth rule can be diagrammed as follows:
- GH H G
If we link the fourth rule with the second rule, we can make the inference that if H is tested then F must be tested first. Here is why: if H is selected, then if G is selected we know from the fourth rule that H would be tested ahead of G; according to the second rule, either F or G must be ranked first, so if H is tested, then G could never be first and we can infer that F would have to be first:
- H F1
- FK K F
If we combine the fifth rule with the second rule, we can make the inference that if K is tested then G must be tested first. Here is why: if K is selected, then when F is selected (and we will discover below that F must be selected) we know from the fourth rule that K must be tested ahead of F; according to the second rule, either F or G must be ranked first, so if K is tested, then F could never be first and we can infer that G would have to be first:
- K G1
This rule is extremely restrictive because the selection of M automatically adds two other members of the group. And, because the group already includes L and I, the selection of M yields only one possible group of five cold medications: M-F-H-L-I. When this group is ordered, F must be ranked first, and, of course, L must be second.
One final inference remains, and this inference is tricky indeed. An examination of the rules reveals that F is a critical variable: F appears in three of the six rules while no other variable appears in more than two of the rules. In fact, F is so critical that F must be one of the five cold medications that are tested. Consider the following: if F is not selected to be tested, then by the contrapositive of the last rule we know that M cannot be selected for testing. This situation forces the remaining five cold medications—G, H, I, K, and L—to comprise the entire testing group. Because both G and H are included in this group, the fourth rule is enacted, and we know H must rank better than G. This causes a violation of the second rule, which requires either F or G to be ranked first. Hence, because an acceptable group cannot be selected when F is not included, we can infer that F must be included in the group of five cold medications tested in the study. The group of cold medications is therefore as follows:
The group above is not given in order; although F can be first, if G is also tested then G could be first instead of F. Please note also that because F must be selected, from the action of the fifth rule we can deduce that K can never be tested last.
Of additional note is that there are no randoms in this game.