Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

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

This game initially appears to be a fairly straightforward Defined Grouping game: two groups are presented, one fixed with three spaces and the other fixed with six spaces. However, only seven variables are available for the nine spaces, and instead of certain variables being re-used (and thus appearing multiple times), the game scenario simply states that not all of the nine positions will be filled. Therefore, because we are not certain exactly how many of the seven variables will be placed in each group (but we do know that some variables will appear in each group from the rules and from the numbers in the game), this game is Defined, but Underfunded. How, then, to handle the Underfunded aspect?

From the scenario, we know that the appellate court can have a maximum of three judges appointed, and that the trial court can have a maximum of six judges appointed. With only seven variables available for those nine spaces, there will be exactly two empty spaces total in the two groups. The easiest way to handle this Underfunded aspect is to create to placeholder variables for the empty spaces (in this case, we will use “X”). Under this option, two “X” variables are created, equalizing the game as a 9-to-9 relationship of variables to available spaces. Effectively, then, the game becomes a Defined, Balanced game.

From the game scenario only, the setup appears as follows:

PT63_Game_#1_setup_diagram 1.png

As a point of interest, the “X” variables never play a role in this game (they could have, but the test makers chose to go in a different direction). While this is not always the case, it means that if you instead chose to approach this game as a Partially Defined game (and did not create placeholder variables for the two empty spaces), the result would have been the same. Let us now examine the three rules.

The first two rules helpfully assign individual variables to specific groups:

PT63_Game_#1_setup_diagram 2.png

The third rule establishes that H and P cannot be appointed to the same court. Because the game scenario indicates that all seven candidates will be appointed to a court, the result of this rule is that H and P must be appointed to different courts. However, because we cannot ascertain exactly which court each is appointed to, the functional result is that each “occupies” a space in each group, which is best shown as an H/P dual-option:

PT63_Game_#1_setup_diagram 3.png

Note that the rule itself is shown as a vertical not-block because that representation carries the greatest visual impact in Grouping games with multiple groups.

At this point, game does not appear to be overly difficult, but regardless, let us consider the areas of interest in the setup:

• 1. The three randoms
  
  J, M, and O are all randoms, and thus three of the seven variables are randoms. When combined with the uncertainty over the exact size of each group, this means that the game likely has a large number of solutions.
  
  2. The limitation in the appellate group
  
  The appellate court group is limited in that only one more variable can be added to the group (there are initially three open spaces, but from the rules, L and either H or P must be assigned to the appellate court). Thus, only one of the randoms can be assigned to the appellate court, a fact tested in the second and third questions.
  
  3. There is only one “live” rule
  
  There are only three rules, and the first two rules are easily captured within the diagram. Thus, the third rule is the only “live” rule (that is, a rule that must still be tracked actively during the game), but even then the rule is largely captured within the diagram. This suggests that despite the uncertainty in point #1 above, this game is not likely to be difficult.

With the points above in mind, let us move on to the questions.

[asimshah]

Is there a technique to use in these grouping games? I am at a loss most of the time and it is very frustrating, it took me 2-3 weeks to learn linear games, but grouping games are very irritating. Any techniques?

Thanks.

[Rachael Wilkenfeld]

Let's start with the first game on that test (questions 1-5). You are correct that it is a partially defined grouping game. That means that we know some information about group size, but not the exact group size. Here, we know that we have two groups (the appellate court and the trial court). The appellate court will have a max of 3 new judges, and the trial court will have a maximum of 6.

We know we have seven new judges (H, J, K, L, M, O, P). Each one will be appointed to one of the two courts. L must be appointed to the appellate court, and K to the trial.

We can diagram this situation (so far) as follows

Appellate: L _ _

Trial: K _ _ _ _ _ _

The spaces represent possible appointments. There is one more rule: H and P cannot be on the same court. That means one is on appellate, one is on trial. We can represent that on the diagram as follows:

Appellate: L H/P _

Trial: K H/P _ _ _ _ _

That is it for the rules.

There is still more information we can pull from the game. We can come up with a numerical distribution---or information that tells us more about group size.

Let's start with the appellate court. We know there are a maximum of three slots. But based on our set up, there is also a minimum of 2 (L and either H or P). That means we have two distributions:

App 2, Trial 5
or
App 3, Trial 4


Hope that all helped!

[swadleyb]

Hi,

I have been having a lot of difficulty with rule substitution questions. I was wondering if there was a strategy for attacking these types of questions.

Thank you

[Steve Stein]

Thanks for your question.

You're not alone--many people find such questions quite difficult. I believe you are referring to the ones that require that an original global rule be replaced with a new rule--while still achieving all of the same effects as the original rule (as opposed to questions that suspend the rule and substitute in an entirely new and different rule).

Is that the case, and is there a particular game or question that you are looking at?

Thanks!

~Steve

[swadleyb]

Yes those are the questions I was referring to. I didn't really have a specific question in mind. I can usually figure them out but it takes me too long. I just took the June 2011 practice test and in game one the last question dealt with this sort of substitution. I worked it out but it took way too much time. Similarly with the December 2009 practice test, game two question 10 dealt with substitiuting a rule so it would still have the same effect as the originial rule. Both of these questions, for me, were very time consuming. So I guess I was wondering if there is a more efficient way of completing these types of questions in general.

[Steve Stein]

Thanks for your response. For others reading this post, for the June example, we have two sets of judge appointments, T(rial) and A(ppellate), with three and six spots respectively. Further, we are told that Judge L will be an appelate judge, and that Judge K will be a trial judge. Finally, we are told that H and P will be appointed to different types of courts. This could be diagrammed as follows:

.....___
.....___
.....___
___ ___
_H/P P/H__
_L__ _K__
A .....T

The question asks for a rule that would effectively replace the H/P rule but have the same effects. The answer choices are quite clever in this one.

Answer choice A provides that H and P cannot both be on the Appellate court--this almost replaces the original H/P separation rule, but leaves open the possibility that they could both be on the Trial court together. Since this doesn't have all of the effects of the original H/P rule, it cannot be the right answer choice. Same goes for choices B, C and D--they all allow possibilities that the original rule prohibited.

Answer choice E, however, says that no three of H, K, L, and P can be on the same type of court. This means that they must be broken down into groups of two. Since we know from the original diagram that L and K will inevitably be separated, this substitution rule has all of the same effects as the original, by separating H and P in every case.

Similarly with the December 2009 test, the wrong answer choices all made similar setups possible but didn't dictate all of the ramifications of the original rule.

With Rule Substitution questions it helps to understand how they create the tricky wrong answer choices: in this question type you can expect to see several wrong answer choices that have some of the ramifications of the original rule, but not all of them.

Also, notice that the right answers to the two questions discussed were E and D, respectively--sometimes in cases of several answer choice scenarios that need to be worked out, the test makers like to have to work through as many answer choices as possible--so it might be worth working from the bottom up when assessing the answers to this type of question.

As you have already pointed out, this is not a simple question type, which brings up another very important point--if you are confident that any given question will be particularly time consuming, you can always come back to it later, if and when you have time--if time is a constraint, which it generally is for most test takers, there may be some value in moving on to questions that are likely to be easier, more straightforward, and less time consuming.

I hope that's helpful--let me know--thanks!

~Steve

[swadleyb]

That was very helpful. Thank you for the suggestions.

[elewis10]

What is wrong with answer choice E for question 1? I see how it violates any of the rules.

[Luke Haqq]

Hi elewis10!

You actually got this one right--answer (E) doesn't violate any of the rules, so it is the correct answer.