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This is a Grouping Game: Defined-Fixed, Unbalanced: Overloaded.
This game features a fixed group of four selections, with an overloaded group of seven variables available to fill those four spaces. Thus, four variables are always selected and three variables are not selected.
The game contains only four rules. The first two rules are quite powerful and “reserve” two of the four available spaces. These rules are represented directly on the game diagram with dual-options. The third and fourth rules are both simple conditional rules, and are represented with arrow diagrams in our setup:
Of course, there are also inferences that can be made in the game:
- Because J, K, N, and P will collectively occupy exactly two of the spaces, the remaining two spaces are occupied by the group of L, M, and Q. Thus, we can infer that two people from the group of L, M, and Q must always be selected. This will be represented directly on the diagram using a parenthetical notation. Note that any time one of the members of the group of L, M, and Q is not selected, the other two must be selected.
From the last rule, when Q is selected then K must be selected, and from the first rule when K is selected J cannot be selected. Thus, Q and J cannot be selected together. This rule is shown with a double-not arrow.
Note that some students attempt to draw an inference between P and L by combining the second and third rules. There is no usable inference that can be drawn from these two rules (an inference is present, “Some Ls are not Ps” but that inference has no value in this game).
With the information above, we are ready to attack the questions.