Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a [bGrouping: Defined-Fixed, Unbalanced: Overloaded game.[/b]

The initial scenario, rules, and inferences can be diagrammed as follows:

J03_Game_#4_setup_diagram 1.png

In the setup, some students create a second row within the group to show the type of animal—monkey, panda, or raccoon. While there is nothing wrong with this decision, operationally it has little effect as the questions focus more on the individual animals than on their type, and also, the questions can easily be answered without adding a second row.

There are more inferences in this game than might appear at first. Let us take a moment to examine the inferences that can be drawn.

First, at the group level, we can deduce from the first rule that if two monkeys are selected, then G must be one of those monkeys (we know this by Hurdling the Uncertainty—since F and H can never be selected together, if two monkeys are selected then G must always be selected). As a corollary, we can conclude that at most two monkeys will be selected (and, from the second rule, we can conclude that at least one monkey must be selected: since N and T cannot be selected together, we cannot form a viable group from just the six pandas and raccoons, and therefore at least one monkey will always be selected).

Second, at the variable level, several more inferences can be drawn. By connecting the second and fourth rules we can infer that K and T can never be selected together. By recycling that inference and combining it with the third rule, we can also infer that H and T can never be selected together.


The last two rules form a chain linking H, K, and N. Consequently, H is an important variable because if H is selected then K and N must also be selected, and once H, K, and N are selected we can conclude from the first two rules that F and T would not be selected. At that point, any of the remaining animals could be selected because all four rules in the scenario would have been satisfied (and, as you might expect, the remaining variables—G, L, V, and Z—are randoms). Hence, the selection of H yields a number of ready-made hypothetical solutions:

J03_Game_#4_setup_diagram 2.png

As with any conditional chain sequence, you should also consider what occurs if the final necessary condition variable is not selected. In this case, if N is not selected, via the contrapositive you know that K and H cannot be selected. At first, this may appear unremarkable, but remember that this is a “9 into 6” grouping scenario, and if H, K, and N are all eliminated, then there are no “extra” variables, and all the remaining variables must be used. This creates the following hypothetical:

J03_Game_#4_setup_diagram 3.png

Note that you should not just simply assume that such a hypothetical is valid; instead, quickly check the rules to make sure there are no violations (in the above hypothetical, there are no violations). If there was a violation, you would then know that the necessary condition—in this case, N—would have to be selected in every scenario and you would have gained a valuable piece of information (as it stands, since the hypothetical scenario above is workable, N does not have to be selected).

The other variable of note is T. Because T and N cannot be selected together, when T is selected there can be only one solution to the game (because N will not be selected, leading to the hypothetical discussed above).