Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

This is a Grouping Game, Partially Defined.

This game was widely considered the most difficult of the June 2004 exam. After three linear-based games, the test makers saved a Partially Defined Grouping game for last, but test takers do get a break because this game has only five questions.

At first, this game appears to be a straight defined Grouping game: six lunch trucks serve three office buildings. However, the game scenario does not specify that each truck serves only one building, and in fact the second rule explicitly indicates that a truck can serve more than one building (by itself, this fact opens up the game to many more possible solutions). If each lunch truck served only one building, the game would be considerably easier because the assignment of a truck to a building would eliminate that truck from further consideration. Thus, one reason test takers felt this game was more difficult was because there is much more to consider within the setup of this game compared to the prior three games (there are also twice as many rules in this game as in any of the other games on this test).

The first decision in this game is what variable set to choose as the base. Either the lunch trucks or the buildings could serve as the base, but we will use the buildings since there are fewer buildings and each of the rules references the trucks going to the buildings. There is also an intuitive element here as it is easier to see the trucks going to the buildings; if the buildings were assigned to the trucks it would be counter to how things work in the real world (trucks move, buildings don’t).

With that in mind, we can create the following basic representation of the variable sets:

pt43_j04_g4_1.png

Now, let’s examine each rule.

Rule #1. The first rule establishes that Y is served by exactly three lunch trucks, two of which are F and H:

pt43_j04_g4_2.png

Rule #2. This rule indicates that F serves two buildings, one of which is Y, and the other is X or Z:

pt43_j04_g4_3.png

Rule #3. Like the first two rules, this rule addresses a numerical relationship within the game. Given the open-ended nature of the truck assignments in the game scenario, you must look for rules that establish exact numbers, and, hopefully, a complete Numerical Distribution of trucks to buildings. More on this point later.

According to this rule, I must serve more buildings than S:

#I #S

So, at this point, I must serve either two or three buildings, and S must serve either one or two buildings (note that it is possible for I and S to serve the same building). This rule is worth tracking since other rules can (and will) impact these possibilities.

Rule #4. This rule, which states that T does not serve Y, can be added as a Not Law to our setup:

pt43_j04_g4_4.png

Rule #5. This is a powerful rule, and one whose implications can be easily overlooked. First, the diagram for this rule is as follows:

pt43_j04_g4_5.png

If F and P do not serve the same building, the obvious deduction is that P does not serve building Y. However, we already know from the second rule that F serves exactly two buildings. Since P cannot serve those two buildings and there are only three buildings, we can infer that P can serve only one building and that it must be the building not served by F. Thus, for example, if F is assigned to building X, then P would have to be assigned to building Z. There are several variations on this rule, but the gist in each case is the same: when one of F or P is assigned to building X, the other is assigned to building Z, when one of F or P is assigned to building Z, the other is assigned to building X. We can represent this with a dual F/P option on buildings X and Z:

pt43_j04_g4_6.png

Thus, numerically we have now established that P can serve only one building, and from the second rule we know that F serves exactly two buildings.

With this rule we have also eliminated several lunch trucks from serving building Y. With two trucks assigned to Y (trucks F and H), and two trucks eliminated from serving Y (trucks T and P), only two trucks remain to fill the third space at Y: truck I or S. This can also be diagrammed with a dual-option:

pt43_j04_g4_7.png

Rule #6. This rule can be diagrammed as:

pt43_j04_g4_8.png

The first part of this rule indicates that T serves two buildings. Since from rule #4 we know that T cannot serve building Y, we can infer that T serves buildings X and Z. The second part of this rule indicates that T and I serve two of the same buildings, and this means that I must also serve buildings X and Z. I could also serve building Y, but does not have to. With the information above, the diagram is:

pt43_j04_g4_9.png

Note that I can still serve the remaining building, building Y. This rule only specifies that T serves two buildings also served by I; I could serve all three buildings without violating this rule (or any other).

The setup above is the final setup for the game, but given all of the numerical rules in this game, you must examine the numerical possibilities for each variable before proceeding to the questions (remember, always examine rules about numbers!). Let’s examine the options for each lunch truck:

• F: As specified in the second rule, F serves exactly two buildings, one of which is Y.
  
  H: H is somewhat of a wild card in this game. H must serve at least building Y, but there is no other rule limiting how many buildings H must serve. Consequently, H could serve one, two, or three buildings.
  
  I: I must serve at least two buildings (X and Z), and possibly all three buildings.
  
  P: Because of the interaction of the second and fifth rules, P can only serve one building (X or Z).
  
  S: From the third rule we know that S is limited to serving either one or two buildings, but which buildings those are is undetermined.
  
  T: From the third and sixth rules we know that T serves exactly two buildings, and those buildings are X and Z.

This distribution is critical, and having command of the numerical possibilities will allow you to easily solve several of the questions.


Reviewing the game, there are three elements of uncertainty that must be tracked throughout the questions:

• 1. The F/P dual-option.
  2. How many buildings H serves.
  3. The relationship between I and S.

[catherinedf]

Hi,

I'm looking at the game about lunch trucks and office buildings. What should I use as the base for this game... the trucks or the buildings? When taking the test I used the buildings, but now I'm thinking I should have used the trucks... I got all the questions right except for number 22, where it seemed like I was grasping at straws.

Any help / advice would be very much appreciated!

Thanks!

Catherine

[Nikki Siclunov]

Hi Catherine,

Thanks for your question. You should indeed use the six trucks as the grouping base, not the 3 office buildings. Since each truck serves one or more of the three office buildings, you can easily see that each of the six groups will contain between 1 and 3 variables. This would make the process of examining the numerical distribution easier, and help narrow down the size of each group. Furthermore, the grammatical subject in each rule is a truck, not a building. If you use the trucks as your base, you can translate the rules into your set-up more easily.

Logically speaking, the game can be diagrammed both ways. However, if you use X, Y, and Z as your base, it would be more difficult to keep track of the size of each group (all you know is that three trucks serve Y, based on the first rule of the game).

By the way, check out Game 3 in December 2005: a repair facility with six technicians, each of whom repairing machines of at least one type (radios, televisions, and VCR's). Also, look at Game 2 in October 2001, where each of six cars has at least one of three options. Do you see a pattern emerging?

[cfu1]

Would it be possible to go into a bit more detail regarding the set up / initial diagram of this problem? I had a little trouble figuring out some of the questions in this game.

Thank you!

[Francis O'Rourke]

If you use the trucks as the base (as Nikki described above) your diagram will look like the following.
__ ___ ___ __ __ __
F H I P S T

Each of those trucks must serve at least one of the following three office buildings: XYZ, but the initial scenario does not tell us how many each serve.

The first rule tells us that F, H, and one other Truck serves building Y, so we can add that onto our diagram

Y Y_ _ _ _
F H I P S T

and jot down that we have exactly one more Y to place. The second rule tells us that the Falafel truck serves exactly two buildings, so we can indicate that in a few different ways, including simply writing down the number 2 below the F.

The third rule states that Ice cream serves more than Salad. So Ice cream will have 2 or variables and Salad will have 1 or 2 above it. You can notate this the same way as the second rule above.

The fourth rule states that the Taco truck does not serve Y. This can be best diagrammed by drawing Y with a strike through it underneath the T.
__
T
Y

Next, the Falafel truck and the Pita truck do not share any building in common. since the falafel truck serves 2 out of 3 buildings (Y and either X or Z), the Pita truck must serve only one building, X or Z.

Finally, the Taco truck serves 2 buildings that the Ice cream serves. This is a difficult rule to figure out, because you need to remember that there is only one more Y left to place on the diagram. Since we can't place two more Y's on the board, the Taco and Ice cream trucks both need to serve X and Z.

At the end of these rules, you should see, at the very least that

• F serves Y and either X or Z, but not both
• H serves Y, and possibly more
• I serves X and Z, and possibly Y
• P serves X or Z (whichever one F does not serve), and no other building
• S serves at most two buildings
• T serves X and Z, but not Y

[Etsevdos]

Your workbook suggests using a different base, which I thought was odd. I would suggest consistency among approaches in your materials. I believe the way outlined in this thread is better, but please confirm one universal PS approach if possible.

[James Finch]

Hi Etsevdos,

As Nikki explained, either diagram would work, which is the case with certain grouping games. Which set of variables works better as a base then comes down to factors unique to each individual--time left in the section, interpretation of rules, etc. I myself prefer to use X--Y--Z as the basis for groups, but I understand Nikki's approach as well.

Ultimately, one has to be flexible and think on one's feet to do well on this exam--these are characteristics that are important for lawyers to have, after all. Don't get bogged down in an inflexible approach, but look at each question or game with fresh, critical eyes and use the approach that seems most sensible and efficient to you.

[T.B.Justin]

There is also an intuitive element here as it is easier to see the trucks going to the buildings; if the buildings were assigned to the trucks it would be counter to how things work in the real world (trucks move, buildings don’t).

When I first was learning games I used the trucks as the base, I preferred tracking the amount of buildings to trucks, I found it more efficient to track the trucks to buildings.

[nzLSAT]

How could I possibly be included in Y if S is also chosen for Y? I thought the first rule was "F, H, & exactly 1 other truck each serve Y." So, does this rule not mean that there are only 3 trucks that serve Y? How could it mean that more than 3 can serve it?

[nzLSAT]

Also, how can S be in at most 2? I thought S can only be in 1 since it's in fewer buildings than I (rule 3) and (as I incorrectly inferred in the previous post about how I can also be in building Y), I thought S would only be allowed in 1 building.