Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping: Partially Defined game.

In an everyday scenario such as this one, where food is being matched with people, it makes sense to use the people as the base since most test takers think of giving the food to a person, not the other way around. Because this approach is more intuitive, it generally makes the game easier to understand. In general, if a base selection choice seems intuitive to you, go ahead and use that as the base.

The basic scenario and rules can be diagrammed as follows:

O94_Game_#3_setup_diagram 1.png

Let’s briefly discuss each rule:

• The first rule simply means that multiples of each food are not available to each person. This serves to reduce the number of possibilities in the game.
  
  The second rule is represented by a vertical HS not-block, which is the most powerful visual representation of this rule.
  
  The third rule indicates that at least one hot dog and at least one pretzel are purchased.
  
  The fourth rule is represented with an S above Mendel in the diagram. When this rule is combined with the second rule, we can infer that Mendel does not buy a hot dog, which is shown with an H Not Law under Mendel.
  
  The fifth rule is represented with an F above Nastassia in the diagram.
  
  The sixth rule is shown with P Not Laws under Lara and Nastassia. This rule, when combined with the third rule, allows us to infer that Mendel must buy a pretzel, and so a P appears above Mendel.
  
  The seventh rule is shown by internal diagramming, which means that a double-not arrow is placed within the diagram between M and N. This makes the rule easier to remember and work with. Also, because Mendel buys a shish kebab and a pretzel, from this rule we know Nastassia cannot buy a shish kebab and a pretzel, and those inferences are represented with S and P Not Laws under Nastassia.

In addition, although part of the third rule is now satisfied (at least one person buys a pretzel), the portion stipulating that at least one person buys a hot dog is not yet fulfilled. Because Mendel cannot buy a hot dog, either Lara or Nastassia must buy a hot dog. And, because from the second rule a person who buys a shish kebab cannot also buy a hot dog, we can infer that if Lara has a shish kebab, Nastassia must have a hot dog:

• LS NH

Normally we could infer the other side of this dual-option inference (if Nastassia buys a shish kebab, then Lara buys a hot dog), but Nastassia can never buy a shish kebab so that side of the inference is non-applicable.

Finally, the last rule, which states that Mendel and Nastassia do not buy any similar foods, is the key to questions #14 and #17. As there are only four available foods, and since neither can select a food that overlaps with the other’s selection, the maximum number of foods the two can jointly select is four. This inference is especially crucial for question #17, where some restrictions are lifted.

[kmpaez]

What is the best way to diagram rules 1, 3, and 7?

[Adam Tyson]

Thanks for the question, kmpaez - this is a game that we haven't yet created an "official" explanation for, but we are getting to it. Meanwhile, here's my advice:

Rule 1, which limits the instances of any given variable to no more than once in any one group, is the type of rule that I actually would NOT diagram. There are many cases of this, where it happens that some rules are just the kind of thing you need to internalize and carry with you as a basic understanding in the game. This is similar to a game scenario telling you that each variable is to appear just once and that no two variables occupy the same space. You don't diagram that, you just put it in your memory as an over-arching guideline - everything goes exactly once without any ties. In this case, you cannot repeat a variable within a group - nobody gets, for example, two hot dogs. It's when those rules are NOT present that you really need to think about them, and perhaps diagram something.

Rule 3 can be diagrammed, but it doesn't require much complexity. Maybe just jot down "H & P are in" to make sure they are never left out. Another approach I have seen, and used myself on occasion, is to circle or underline the variables in the list of variables to indicate that they MUST be included. I do that more often when it is an inference instead of a rule. If, for example, there is a rule that I must always have two of X, Y and Z, but that X and Z cannot both be selected, there is an inference to be made that Y is always selected (either with X or else with Z). I might circle the Y in my variable list as a shorthand way of indicating that Y will be found in every single solution. I would also expect a question in a game like that about which variable is always in, and I will have prephrased that answer right from the start. (This is exactly what happens in a certain Grouping/Linear Combination Game from February 1994, btw). You can also use this rule coupled with Rule 6, that neither L nor N gets a pretzel, to make the inference that M gets a pretzel, because someone has to and M is the only one left. Putting a "P" in Group M, and "not-laws" for P under Groups L and N will take care of diagramming the pretzel portion of Rule 3.

For Rule 7, the way I most commonly see that one diagrammed is with a double-not arrow in the base between Groups M and N, to show that the two groups can never have anything in common. That will also give you the immediate inference, thanks to Rule , that M does NOT buy a fruit cup.

If you like, share with us your diagram for this game, and we will give you additional feedback on what appears to be working well and where things could be improved! We find it much easier to help students when they share their thoughts and approaches with us, so we can understand what path they are on and make gentle corrections as needed, rather than just having us draw our own path and hope they find it useful.

Looking forward to hearing more from you about this one!

[kmpaez]

Thank you! That explanation was very helpful.

I didn't diagram rule 1 and used a double not arrow in the base for rule 7.

I simply put a mark next to rule 3 so I would be able to reference it more easily if needed. I appreciate you sharing your approach to underline/circle the variable in the list, I will definitely try it.

Thanks again!

[lathlee]

Just making sure, "None of the three buys more than one portion of each kind of food" meaning a total of 4 foods, if one buys one food, then other cannot? as in Lara buys a hot dog then no one else can buy a hot dog?

or

if Lara buys a hot dog, then she cannot buy another hot dog again. ...

It is the second option, correct/Right?

[Claire Horan]

Hi lathlee,

Your second interpretation is correct. If L buys a hot dog, she does not buy a second hot dog.