Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping: Partially Defined game.

The game scenario presents a situation where a student chooses at least three courses from among seven sessions. Because there must be at least three courses but an exact number is not established, this game is Partially Defined. The initial setup appears as follows:

PT58-Sept2009_LGE-G4_srd1.png

The game contains only three rules, but each of the three rules has a compound necessary condition in which both terms are negative. These rules must be handled carefully, so let’s examine the first rule and determine the best choice for diagramming each rule.

Rule #1. In complete conditional form, this rule is diagrammed as:

PT58-Sept2009_LGE-G4_srd2.png

Functionally, this statement means that if H is taken, then S and M are not taken. Thus, from a component level, H and S cannot both be taken, and H and M cannot both be taken, which can be represented in separate diagrams as:

PT58-Sept2009_LGE-G4_srd3.png

Because there are only three rules, and because there is minimal overlap between the three rules (M is the only variable that appears as a sufficient condition in one rule and a necessary condition in another rule), the second representation is how we will choose to diagram each rule.

If you choose to diagram the rule as shown in the first representation, you can still successfully complete this game. We have chosen the second representation for its ease of use, especially since the variables in each relationship are positive.

Rule #2. This rule should be diagrammed as:

PT58-Sept2009_LGE-G4_srd4.png

Rule #3. This rule should be diagrammed as:

PT58-Sept2009_LGE-G4_srd5.png

Some students simply use these six rule representations to attack the game, and in doing so they are quite successful. Others choose to examine the restrictions present for each variable, as follows:

PT58-Sept2009_LGE-G4_srd6.png

The variables are written out horizontally in order to save space.

This list highlights that some variables—such as L and T—have few or no restrictions, whereas other variables—notably M—have several restrictions. In essence, the list above is simply a different way of reflecting the information contained in the three rules.

Regardless of your understanding of the game, one of the keys is remembering that at least three courses, but possibly more, must be taken.

The information above leads to the final setup for the game, save for some numerical information we will discuss later:

PT58-Sept2009_LGE-G4_srd7.png

[Jordan]

I can't seem to figure out this partially defined grouping game. After trying to solve this game under untimed conditions, I still can't figure out #23 and it seemed to be a cumbersome and slow process of getting 20, 21, and 22.

I diagrammed the three conditional rules and their contrapositives (ensuring to switch the conjuncts to disjuncts - and's to or's) and also added to the list conditionals of: S--> ~H & ~W ; H v W--> ~S (AND) P--> ~M & ~W ; M v W--> ~P
Because I wanted a list of powerful variables - ones controlling 2 or more variables (being H, M, W, S, P). From there I figured that the numerical distribution must be 3-4 or 4-3 (taken to not taken)... and that was it for inferences.

Am I missing any inferences? or doing something inefficiently?

Thanks.

[Nikki Siclunov]

Hi Jordan,

First, let's simplify your set-up. There is no reason to work with multiple sufficient/necessary conditions when you can simply use the double-not arrow to signify the negative relationships between variables:

H <--|--> S

H <--|--> M

M <--|--> P

M <--|--> T

W <--|--> P

W <--|--> S

As you mentioned, the maximum number of courses to be taken is 4 (the minimum is 3, as indicated in the scenario). Based on these rules, you should have identified M as the power variable, it appears in three of the rules:

M --> no P, no T, no H

Taking M leaves us only with L, S, and W. Based on the last rule, however, it is impossible to take both S and W. Since the minimum number of courses to be taken is 3, taking M requires us to take L along with either S or W (but not both):

M --> L + S/W

See if the questions get easier now that you have that inference.

[Jordan]

That definitely helps with #23.

Thanks

[Patrick.a.anderson]

Hi, I was confused on what inferences to make for this question, so I had to do each of 6 questions locally. What inferences do you think are key to solving the game? Question #22 gave me particular trouble. To find out this answer, would you make multiple scenarios and just see which solution fits? Also, in question #23, would you just test out each solution if time permits?

[Dave Killoran]

Hi Patrick,

Thanks for the question. This is a classic Grouping game, and one that really tests your conditional reasoning abilities.

The first step is to look at each of the rules as being two rules in one. Because of the way each rule is worded, it produces two diagrams. For example, the first rule is really two separate statements:

H S
and
H M

The problem with all these rules is that they are all negative, and so you can't link them together to make a series of power inferences. So, the good news is that you weren't as confused as you might have thought. There aren't any links to be made, so diagramming the rules is the most you can do as far as inferences.

However, a lack of inferences always makes me suspicious (I don't trust the test makers anyway, but something like this puts me on guard even more). Since diagramming the rules takes no time, and there aren't any inferences to be made, you should take another step during the setup, and that is to look at what happens when each course is taken. That appears as follows:

H S, M

M P, T, H

P M, W

S H, W

T M

W P, S


This process helps clarify all of the relationships, and really crystallizes what can occur. For example, it's easy now to see that selecting M causes the most issues, whereas T is the course that causes the least number of issues. Then, at that point, you are ready to attack the questions.

I'll answer question #23 in just a moment, with a second post.

Please let me know if that helps with the main setup. Thanks!

[briana4b]

I'm still confused on how you got the numerical distribution. Is there a quick way of determining that there is a max of 4 options or did you just make hypotheticals?

[Francis O'Rourke]

Hi Briana,

Since this game tells us a minimum number of variables, but no maximum, the number of possible variables included should be on your mind. "Making hypotheticals" is a good way of describing what I am doing after I see the rules, but you want to have a goal to creating a hypothetical. For me, that goal was to try to get as many included as possible.

As Dave said above, it is clear that selecting M causes the most problems. Since we are exploring to how select the greatest number of variables, I will say to myself "let's take M out." A lot of variables have two other variables exclude two other variables, but T and L are different. Selecting L gets one variable included for free, and selecting T only forces M out (which was already out). So at this point you should recognize that it would be worth it to select T and L.

You now have 2 variables selected (L and T), and only 4 left over to possibly select: H,W,S, and P. We can't get them all in, because some of those four force some other ones out. Then if you narrow your focus to just these four, you will see that selecting either H or W would force both S and P out. So out of these four, we can at most select 2.

This may seem difficult right now. That is because it is a logically complicated game. With enough practice though, you will be able to see this in what feels like no time.

[beeke]

I am almost positive that I have a rule wrong.

I eliminated every answer on number 18.
Here are the rules as I understood them:
H S
H M
M P
M T
W P
W S

and inferences:
H P because H M P
H T because H M T

and eliminated because
A.) H S
b.) H M
C.) H P
D.) W P
E.) M T

[Emily Haney-Caron]

Hi beeke,

Thanks for posting! What is giving you trouble here is those inferences. Since all these rules are negative, you can't connect them together; there are no inferences that involve chaining together the rules. So, take out the inferences you listed, and instead for inferences start thinking about how every variable impacts every other. You might find it helpful to try the strategy Dave used, above, and list for yourself every variable and next to it which variables are knocked out if that variable is in. From there, go back to the questions, and that should help a lot!