Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

This is a Grouping: Balanced, Partially-Defined game.

The game scenario establishes that five kinds of materials are recycled at three recycling centers,
with each center recycling at least two, but no more than three, kinds of materials:

june07_game_4_diagram_1.png

Each kind of material will be recycled in at least one center, making this game Balanced. Notice,
however, that each variable can repeat as many as three times, because the same material can be
recycled at more than one of the centers. The size of each group, is equally unclear, because each
center can recycle as few as two, and as many as three, kinds of material. Thus, the game is
Partially-Defined.

At this point, many test takers would probably wonder if it’s worth examining the Numerical
Distributions that govern the assignment of the five materials to each of the three centers.
Unfortunately, the 5-into-3 distribution is fairly open-ended, because we know neither the total
number of variables to be distributed, nor the precise number of available spaces in each group. A
numerical analysis would therefore be counterproductive.

The first rule states that any center that recycles wood also recycles newsprint:

june07_game_4_diagram_2.png

You should immediately notice that the contrapositive leaves only three kinds of material to be
recycled by any center that does not recycle N:

june07_game_4_diagram_3.png

The second rule stipulates that every kind of material that Center 2 recycles is also recycled at
Center 1. It is best to represent this rule internally, as shown below:

june07_game_4_diagram_4.png

You should closely examine the implications of this rule, and make at least some of the following
inferences:

• 1. If Center 1 does not recycle a given material, then Center 2 cannot recycle that
  material either. Consequently, the material must be recycled at Center 3. And,
  by the contrapositive, if a given material is not recycled at Center 3, then it
  must be recycled at Center 1:

june07_game_4_diagram_5.png

• 2. Center 1 must recycle at least as many kinds of material as Center 2:

• Center 1 ≥ Center 2

• 3. If Center 2 recycles three kinds of material, then Centers 1 and 2 will recycle
  exactly the same kinds of material as each other, because Center 1 cannot
  recycle more than three kinds of material.

Moving onto the third rule, it establishes that only one of the centers recycles plastic, and that
center does not recycle glass. We get two rules for the price of one here, so it is best to consider
them individually:

The fact that only one of the centers recycles plastic means we can only have one P in our setup,
which is something we can notate using a subscript as shown below:

• G N P1 T W5

Notice the importance of the second rule here. Since every kind of material that Center 2 recycles
is also recycled at Center 1, we can conclude that Center 2 cannot recycle plastic, an inference
that is tested directly in Question #19. Consequently, the center that recycles plastic must be
either Center 1 or 3:

june07_game_4_diagram_6.png

The second part of the third rule prohibits plastic and glass from being recycled in the same
center, a prohibition we can represent using either a vertical Not Block or a Double-Not arrow, as
shown below:

june07_game_4_diagram_7.png

With all the rules diagrammed and some inferences already made, let’s consider the game from a
purely grouping perspective, that is, a perspective that does not dwell on the distinctions between
Centers 1, 2, and 3. Although the three groups are not functionally identical (the second rule
ensures that Centers 1 and 2 are two functionally different groups), ultimately our task will
always be to select either two or three variables out of the available five. The first and the third
rules will be key in this selection process. Here’s why:

As discussed earlier, thanks to the contrapositive of the first rule, if a center does not recycle N,
then it cannot recycle W. Such a center is left with only three kinds of material – P, T, and G –
two of which – P and G – cannot be recycled in the same center (third rule). Thus, if a center does
not recycle N, then it must recycle T, along with either P or G (but not both):

june07_game_4_diagram_8.png

Several additional inferences can be made:

• 1. From the contrapositive of the inference made above, if a center recycles not two but
  three kinds of materials, then it would be impossible for that center not to recycle N.
  So, N must be recycled at any center that recycles three kinds of materials:

• 3 materials N

• 2. If a center does not recycle N, then it must recycle T, and vice versa: if it does not
  recycle T, then it must recycle N. Clearly, then, each center must recycle neither N or
  T (or both), an inference we can plug into our main diagram:

june07_game_4_diagram_9.png

The final diagram for the game should look like this:

june07_game_4_diagram_10a.png

june07_game_4_diagram_10b.png

[vdoshi1016]

How should I diagram game 4 from this exam? I would love some advice. Thanks.

[Dave Killoran]

Hi V,

Thanks for the question. This is a Grouping game, so you want to use the three recycling centers as the base:

G N P T W


__ __ __
1 2 3


Each center has either 2 or 3 slots above it for the materials, to conform with the distribution possibilities mentioned in the game scenario.

The three rules in this game are very interesting, and in many ways represent a classic trio of Grouping rules. Let's briefly look at each.

Rule #1. This rule, diagrammed as W N, is important because of what the contrapositive tells us: if N is not selected, then W cannot be selected, which would leave only G, P, and T available for recycling. So, N becomes very important, because if it is removed we have a very restricted situation. This plays a big role in question #22.

Rule #2. This is a rule that should be diagrammed right on the diagram, with an arrow from Center 2 to Center 1.

Rule #3. This is two-rules-in-one, with the first part establishing that P is recycled at just one center (which immediately allows for the inference that, per rule #2, P cannot be one of the materials recycled at Center 2). The second part establishes that P and G cannot be recycled at the same center. When that rule is combined with the first rule, the restriction created by the removal of N becomes even more critical: if a Center recycles three materials, N must be among the three, and if N is not recycled at a Center, then only two materials can be recycled at that center , and T must be one of the two (the remainder being the choice of P or G).

Ok, that's some basic setup information.

Please let me know if you have any questions on the above. Thanks!

[vdoshi1016]

thanks for your help dave!

[lathlee]

Hi this game killed me for one reason. I immediately found out why when i came to the forum. Where in this game or paper location or a passage containing paper state at least once each materials must be recycled ? I looked and looked and found no such CHARACTERISTICS. If it doesnt state such rule, what authorizes you GUYS CAN ASSUME THAT CHARACTERISTICS exist in this game? Super important

[lathlee]

Hi. can one of you guys please answer me on this. i am actually losing sleep over it

[Dave Killoran]

Hi Lathlee,

It's the part of the scenario that says: "Exactly five kinds of material are recycled..." That's saying in a strict sense that five kinds of materials (namely G N P T W) are used in the game. As we know, these kinds of rules and statements re critical to determining what's going on in a game, so always examine the scenario closely to make sure you see or don't see references like this.

Now, get some sleep Thanks!