Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

This is a Defined-Fixed, Unbalanced: Overloaded Grouping game.

The game scenario establishes that we must select five out of eight foods, which are broken down into desserts, main courses, and side dishes. Additionally, we are told that three of the foods—F, N and T—are hot.

PT65_D11 LG Explanations_game_#3_setup_diagram 1.png

To keep track of the variables that are not selected, it is advisable to create an “out” or an “unassigned” (“U”) group. The latter is smaller and therefore more restricted than the group of variables that are selected, making our decision to represent both groups particularly advantageous. How you designate the fact that F, N and T are “hot foods” is a matter of personal preference. This is clearly a secondary attribute, as only one of the rules is directly concerned with it. You can use an (h) subscript next to each variable, or a simple notation, as shown above.

You can decide to create a second row within the group to show the type of food (desserts, main courses, or side dishes), and even a third row to show whether the food is “hot” or “not.” While there is nothing wrong with this decision, operationally it has little effect as the questions focus more on the individual foods than on their type (or temperature). The questions can easily be answered without adding those extra rows, so keep your diagram as simple as possible. Clutter is both time-consuming and distracting.

The first rule establishes that at least one food from each type must be selected.

min. 1/type

Since there are only two desserts to choose from, at least one of F or G must always be selected:

PT65_D11 LG Explanations_game_#3_setup_diagram 2.png

The second rule states that at least one hot food must be selected. It is best to represent this rule at the variable level, right next to your notation:

PT65_D11 LG Explanations_game_#3_setup_diagram 3.png

The third rule stipulates that if either P or W is selected, both must be selected. Since the selection of either variable requires the selection of the other, this is a bi-conditional rule that must be represented using a Double arrow:

PT65_D11 LG Explanations_game_#3_setup_diagram 4.png

By the contrapositive, if either of these two variables is not selected, then the other one cannot be selected either:

PT65_D11 LG Explanations_game_#3_setup_diagram 5.png

Take a moment to understand precisely what this rule means: either P and W are both selected, or else neither of them is selected. This is a powerful rule that is likely to have a significant impact on the selection of variables in the game.

The fourth rule establishes the following conditional relationship between G and O:

PT65_D11 LG Explanations_game_#3_setup_diagram 6.png

The last rule tells us that if N is selected, V cannot be selected. In other words, N and V cannot be selected together, making a Double Not-arrow an appropriate way to represent this relationship:

PT65_D11 LG Explanations_game_#3_setup_diagram 7.png

We should also represent the implication of this rule “internally.” Given that N and V cannot be selected together, at least one of them is never selected. Applying the Hurdle the Uncertainty principle, we should notate this inference using a Dual-option in the “unassigned” group:

PT65_D11 LG Explanations_game_#3_setup_diagram 8.png

There are more inferences in this game than the rules suggest, even if the conditional rules do not lend themselves to the formation of chain relationships. When working with Defined games, it is critical to take advantage of the fact that the size of each group is determined in advance. The “unassigned” group is particularly restricted: after accounting for the N/V Dual-option, we are left with only two vacancies in it. So, if we knew which two variables, other than N or V, are not selected, then we would immediately know which five variables must be selected. The variables most likely to produce such a restricted scenario are O, P and W, because each of them is a necessary condition for the selection of some other variable. So, if any of them were not selected, then we would know which other variable is also not selected, maxing out the “unassigned” group and forcing the remaining variables in. Let’s take a closer look:

If O is not selected, by the contrapositive of the fourth rule we know that G cannot be selected. And, since N and V cannot both be selected, the “unassigned” group is maxed out and the remaining variables must all be selected:

PT65_D11 LG Explanations_game_#3_setup_diagram 9.png

Do not 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 were a violation, you would then know that O must always be selected (as it stands, since the hypothetical scenario above is workable, O does not have to be selected).

We can apply the same logic to the other two variables of note, P and W. Thanks to the contrapositive of the third rule, we know that if either of them is not selected, the other one cannot be selected either. This—along with the N/V dual option in the “unassigned” group—completes that group, forcing all remaining variables into the “selected” group:

PT65_D11 LG Explanations_game_#3_setup_diagram 10.png

With these deductions in place, your final diagram should look like this:

PT65_D11 LG Explanations_game_#3_setup_diagram 11.png

PT65_D11 LG Explanations_game_#3_setup_diagram 12.png

[lusk2006]

I am working back through my Dec. 2011 test and re-working it. I am SO stuck on Game 3 (foods). Please help!

[Dave Killoran]

Because although the game is a classic Defined Grouping game, the test makers leave you in a bind with the selection of at least one food type each (dessert, main, side) and at least one hot food dish (F, N, T, which cuts across all three food types). Most students realize that the three food types occupy three discrete spaces and the hot food only occupies one, so they choose to show the group of 5 with D, M, and S under one space each (to reflect the minimums). Still, the game feels a bit indistinct and, because the hot food minimum can overlap with the food minimums, there is some uncertainty there. Thus, we have to look at some other aspect in our setup to get a grip on this game.

When approaching Overloaded Grouping games, I always consider the number of variables in the pool and the number of variables to be selected, and the margin between the two. In this game we have 8 foods available for 5 spaces ( 8 --> 5). That margin of 3 foods seems like a lot, but, as is always the case, there are restrictions in the game that tighten things up considerably, In this case, N and V cannot be selected together, which narrows the margin to 7 --> 5. Plus, P and W must be selected together, further limiting the options to:

F, G, N/V, O, PW, T

7 --> 5 may not seem like an especially important or interesting relationship of variables-to-spaces, but that margin of 2 variables is critical for two important reasons:

1. If you remove any two variables (except for N/V), the remaining group must be the selection pool (with the choice of either N or V).

2. If there are any conditional relationships between the 7 variables, if you remove the necessary condition you will also remove the sufficient condition, leaving you with just a 5 --> 5 relationship.


Point 1 above is the broader concept that underlies point 2, so let's focus on point 2. In this game, there are several "positive" conditional relationships:

G --> O
P --> W
W --> P

So, for example, if you remove O, then G is out (leaving F, N/V, PW, T as the food selections).

The relationships above help explain a question like #13 (if you remove both O and P, you also remove G and W, leaving an insufficient number of foods available).

The same logic works in question #16. In this question, T and V are the only side dishes selected. This means that W is not selected, and thus P is not selected. That leaves T, V, F, G, and O (remember, N is out because V was selected). That leads to answer choice (A).

Some of the other questions, like #14, could be said to fall under the points above, but they also are straight rule connection tests (if O is the only main course selected, then N and P are not selected, and from the third rule when P is not selected then W is not selected).

In summary, this isn't an easy game because the test makers actively worked to avoid having a setup to this game that was concrete and that captured all the information smoothly. That happens, and in those cases, you have to think about what Overloaded Grouping is fundamentally about: creating a workable group, which gets harder and harder the smaller the margin between available variables and open spaces. Focus on that aspect of these games, and you will be way ahead of the curve.

Please let me know if that helps. Thanks!

[lusk2006]

Thank you, Dave!

[crottman21]

Can someone explain how to do this section for me?

It took me a really long time and I'm not sure I did it properly.

[Emily Haney-Caron]

Hi Crottman,

If you can reply and let me know how you approached the game and what your set-up looked like, that will help me tailor my answer more directly to your experience and explain the game in a way that might make it easier for you to understand.

Thanks!

[crottman21]

I set it up as _ _ _ _ _ I _ _ _..... 5 in, 3 out.

It was just difficult to organize them with the additional information of some needing to be hot and some needing to me 1 of 3 courses, etc.

What is the best way to attack a game like that?

[Nikki Siclunov]

Hi crottman21,

What you need to do is figure out the number of variables from each subgroup (desserts, main courses, side dishes) that must be selected. The resulting Numerical Distributions can also lead to the creation of several Templates. As far as the hot foods are concerned, just notate with a little star or subscript which foods are hot, and make sure to always have one of those in the "in" group.

Here's what your setup should look like:

Desserts: F, G
Main: N, O, P
Side: T, V, W

Hot: F, N, T

__ __ __ __ __ | __ __ __
(D)(M)(S)

P W
G O
N V

Given that we must have at least variable from each group, there are only five possible ways to select them:

D M S
2-2-1
2-1-2
1-3-1
1-1-3
1-2-2

These five distributions are extremely powerful, and you can probably see how they can lead to the creation of five templates. For instance:

D M S
2-2-1 (F, G, O, N/P, ...)
2-1-2 (F, G, O, T, V)
1-3-1 (F/G, N, O, P, W)
1-1-3 (F/G, P, T, V, W)
1-2-2

The last distribution is the most complex, and might benefit from several templates based on whether P and W are both selected, or else neither of them is selected.

A very similar game was given in June 2003, Game 4. FYI

Hope this gives you a good place to start!

Thanks,

Nikki

[LSAT student]

Question about rules 3 and rule 5. For rule 3, I diagrammed them as 2 separate conditionals as the biconditional, and realize I've been doing that for most to all either/or conditional statements. Is that always okay, or incorrect? Next question is about the last rule - is it possible for both N and V to not be selected?

[Robert Carroll]

L,

A biconditional is just a combination of two conditionals, so it's a matter of style and taste how you represent them, I'd say. The prime consideration should be how readily you can use the representation in inferences, both on the main diagram and in specific questions.

It is possible that neither N nor V is selected. Any Double Not-Arrow allows neither element to be selected - it just prevents both from being selected. Sometimes outside restrictions will make it so you have to select one of the elements, but I don't see any such restrictions here. So, it should be possible to select neither N nor V.

Robert Carroll