Setup and Rule Diagrams

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Setup and Rule Diagram Explanation

This is a Grouping: Undefined, Identify the Templates game.

The game scenario establishes that each of six books will be published in one of two seasons—fall or spring.

PT76_O15 LG Explanations_Game #4_setup_diagram 1.png

The size of either group is unknown, making this an Undefined Grouping Game. Nevertheless, powerful inferences can be drawn from the fact that each variable must be placed in exactly one of two groups: when a book is not in the fall, it must be in the spring (and vice versa)! These inferences often involve the contrapositive of a conditional rule—an extremely common type of rule in Undefined Grouping Games.

The first rule states that M and P must be in different groups from each other: if one is in the fall, the other must be in the spring (and vice versa). Therefore relationship between M and P is one of a not-block:

PT76_O15 LG Explanations_Game #4_setup_diagram 2.png

The clear implication of this rule is a dual-option for M and P, which we can represent in our main setup:

PT76_O15 LG Explanations_Game #4_setup_diagram 3.png

The second rule stipulates that K and N must be in the same group as each other. In other words, K and N are a block:

PT76_O15 LG Explanations_Game #4_setup_diagram 4.png

This rule, when combined with the fact that there are only two groups, is an early suggestion that the game might be best attacked with Templates.

The third rule establishes the following conditional relationship:

PT76_O15 LG Explanations_Game #4_setup_diagram 5.png

Note that this rule, unlike the previous one, does not require the two variables to be in the same group as each other at all times. It is entirely possible, for instance, that K is published in the spring, and O—in the fall. To conclude otherwise would be a Mistaken Negation.

The last rule establishes another conditional relationship:

PT76_O15 LG Explanations_Game #4_setup_diagram 6.png

Note that this rule, unlike the first rule, does not require the two variables to be in different groups at all times. It is entirely possible that M and N are both published in the spring. To conclude otherwise would be a Mistaken Negation of the original rule.

The second rule and the interconnection of the variables suggest that an exploration of the two basic possibilities would be worth a glance here. That’s unusual in Undefined Grouping Games, which are often too open-ended to provide value from templates. Let’s look at the placement of N and K, which must be in the same group as each other:

• Template 1: NK published in the fall
  
  If N and K are published in the fall, then from the third rule O must also be published in the fall. Next, the contrapositive of the fourth rule forces M to be published in the spring. Last, applying the first rule forces P to be published in the fall. That places all of the variables except L, which as a random can unsurprisingly be in either group:
  
  PT76_O15 LG Explanations_Game #4_setup_diagram 7.png
  Template 2: NK published in the spring
  
  If N and K are published in the spring, none of the remaining rules are triggered. Thus, the following template will govern the assignment of variables to each group:
  
  PT76_O15 LG Explanations_Game #4_setup_diagram 8.png
  This template is quite open-ended, which means you will likely see it more often than not in the questions.

Note: templates were not necessary to answer any of the questions in this game, as you could instead use the linkage of the rules. The templates do reveal, however, that it is possible to have a 5-1 distribution of variables in either group, and certainly speed up the solution to Questions #20, 21, and 22. Since most students would at least explore the two basic directions of the game (based on the NK block), we’ll use them as well in our main setup.

The final diagram for the game should appear as follows (with L labelled as a random variable):

PT76_O15 LG Explanations_Game #4_setup_diagram 9.png

[srcline697@gmail.com]

Hello,

these templates helped me when I was going over this game again, but would it be incorrect to just flip the spaces for temp 1? If NK were in spring .....wouldn't you be able to fill in the spots for the spring position? So it would be:

Fall: M L
Spring: K N P O

B/c of the contra positve rules regarding O and K and M ans N?

Thankyou
Sarah

[James Finch]

Hi Sarah,

Unfortunately no, that would be a Mistaken Negation of the rules. The templates begin only with knowledge of which group K-N are in, so if all we know is that the K-N block is in spring, what does it tell us? We know that when we have M in Fall, that the K-N block must go in Spring, as well as that rule's contrapositive, that when the K-N block is in Fall, M must go in Spring, but when K-N are in Spring we cannot determine where M will go (unless we know P), as the rules are silent on this possibility. The same is true for O, as it is only forced into the same group as the K-N block when we know that O is in Spring or that the K-N block is in Fall. So the K-N in Fall template is almost completely determined (only L can be in either group), but the K-N in Spring template doesn't give us the locations of any of the other variables.

Hope this clears things up!