Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping: Defined-Moving, Balanced, Numerical Distribution, Identify the Possibilities game.

The game scenario specifies that five people each participate in one of three activities. From a pure numbers standpoint, then, this game does not appear to be overly difficult at first. But, the multiple negative blocks and shifting setups produced by the two distributions combine to make this game more challenging than it might initially seem.

The information in the game scenario and the second rule combine to produce two Fixed Numerical Distributions:

S98_Game_#3_setup_diagram 1.png

Visually, these distributions produce two distinct setups:

S98_Game_#3_setup_diagram 2.png

Thus, every solution to the game will conform to one of the two setups above.

The first and third rules produce four not-blocks:

S98_Game_#3_setup_diagram 3.png

P is severely restricted by the not-blocks, and N and O also are limited significantly. More on these limitations in a moment.

The fourth rule is conditional, and can be diagrammed as:

S98_Game_#3_setup_diagram 4.png

Thus, combining all of the prior information results in the following setup:

S98_Game_#3_setup_diagram 5.png

Let’s look at the restrictions produced by the not-blocks a bit more closely. P is the most restricted variable, and thus the most important variable in the game. Because P cannot participate in the same activity as N, O, or T, if P participates in a group of 2, P must participate with V. Otherwise, P is alone. This restriction, in combination with the two numerical distributions, leads to the decision to Identify the Possibilities:

S98_Game_#3_setup_diagram 6.png

S98_Game_#3_setup_diagram 7.png

The last rule has a great impact on the six possibilities. For example, if P goes to a movie, P must go alone (otherwise the NV movie rule will apply because P would be with V). Thus, P can only go to a movie in the 1-2-2 distribution. In addition, in a number of the possibilities, when P and V participate in the same activity, then N cannot go to a movie and must instead go to a soccer game or a restaurant.

Note also that possibility #1 has some further restrictions based on the placement of the two dual-options.

[pwfquestions]

Why can't it be 3-2-0?

[Adam Tyson]

That's an inference that comes out of the first rule, pwfquestions. N, O, and P cannot ever go together, so to keep them apart they must be in three different groups! A 3-2-0 distribution would require at least two of those to be in the same group as each other.

[npant120]

How did you know to use MSR as the base? I started out using NOPTV as the base since we know that each of them visit one of MSR, and I thought this was validated by the first list question in the problem which seemed to do the same. However, this made the game ultimately complicated for me at least as I tried using double not arrows instead of not blocks. When I used your approach of MSR, I found the game to be much easier. Just wondering how you knew that this was the correct base?

[Adam Tyson]

There are a few hints in the rules that drive that choice, npant120. Rules about numbers often point us towards the base, and the second rule is a good example of that. If we use the people as the base, it's hard to show that two of them will be S, but if you set up the activities as the base then it's very easy to visually represent that rule by putting two slots in that group. Also, the fact that each person has exactly one activity suggests that they won't be a good base, because that base won't capture the flexible, moving numeric relationships involved when you have two variable sets of different sizes. In other words, it's better to have groups that can be different sizes than to have variables that may or not repeat across groups, if that makes sense.

Similarly, rules about things that can and cannot go together tend to indicate which set is the base and which sets are the moving variables. The first, third, and fourth rules are all like that, telling us about people that cannot go together or, in the case of the fourth rule, who must go together in certain circumstances. Showing those "can't" rules as not-blocks is much easier, and much more visual, than showing them in something like a double-not-arrow relationship.

Watch for numeric rules and inferences to help you decide which variable set will make the better base!

Oh, and one more thing: doesn't it make more sense to put people into activities than to put activities into people? While that sort of "common sense" approach won't always work out, it usually does, because when something is more intuitive and natural in that way, it's just easier for us to manage.