Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

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

The game scenario establishes that each of five artifacts originated in one of three countries—Iceland, Norway, and Sweden:

PT72_Game_#3_setup_diagram 1.png

Since exactly five artifacts will be assigned to the three countries, but it is unknown how many artifacts originated in each country, the game is Defined-Moving. This type of uncertainty is sure to increase the difficulty of a game. In addition, each of the five artifacts originated in one of the three countries, and thus the game is Balanced.

When you create the setup, it is critical that the correct base be selected. There are two choices: the five artifacts or the three countries. Since each artifact originates in exactly one country, it may be tempting to use the artifacts as the base in order to reduce the level of uncertainty inherent in each group. Notice, however, the numerical restriction placed by the third rule: more of the artifacts originated in Iceland than in Norway. (Always read the rules thoroughly before deciding on a setup!) Furthermore, it is entirely possible that some of the countries have no artifacts originating in them, because the scenario does not rule this possibility out. A base of five artifacts would be a poor choice for the purpose of tracking that uncertainty.

• When in doubt, take a look at the wording of the answer choices to the List question, which is invariably the first question in the game. Here, each answer choice assigns the five artifacts to the three country groups, reaffirming our decision to use the countries as the base.

At this point, many test takers would probably wonder if it’s worth examining the Numerical Distributions that govern the assignment of the artifacts to each of the three countries. Unfortunately, the 5-into-3 distribution is almost entirely Unfixed. Indeed, the only relevant rule in this regard is the third rule, which requires Iceland to have more artifacts than Norway. However, since the minimum number of artifacts per country is zero, it is possible that Iceland has only one artifact, while Norway has none.

Our inability to establish a minimum number of variables per group leaves open the possibility of an exceptionally large number of Fixed distributions, making the task of examining them counterproductive. Instead, let’s focus on the rules in the hopes of establishing the minimum and the maximum number of variables per group.

The first rule states that W and Y originated in the same country, creating a WY block:

PT72_Game_#3_setup_diagram 2.png

The second rule stipulates that X originated in Norway or Sweden, creating a Split Dual-Option along with an X Not-Law:

PT72_Game_#3_setup_diagram 3.png

The third rule establishes the only numerically-relevant rule in this game: I > N.

PT72_Game_#3_setup_diagram 4.png

The fourth rule establishes the following conditional relationship:

PT72_Game_#3_setup_diagram 5.png

Examine the contrapositive carefully. If Z did not originate in Sweden, then V did not originate in Iceland. Consequently, V must have originated in either Norway or Sweden:

PT72_Game_#3_setup_diagram 6.png

Now let’s return to our Numerical Distributions. As discussed earlier, creating an exhaustive list of every possible Fixed distribution in this game would be counterproductive. However, you should at least analyze the minimum and the maximum number of variables that can be placed in each of the three groups. Uncertainty in group sizes is never a good thing, which is why you need to attempt to confine it as much as possible. Examining the min/max limits of each group would be a good place to start.

Thanks to the third rule, we know that I > N, which means that at least one artifact must have originated in Iceland. Since there is no stipulation to the contrary, the minimum number of artifacts in either of the two remaining groups is zero:

PT72_Game_#3_setup_diagram 7.png

What about the maximum number of artifacts per country?

• 1. Iceland. Although there is no explicit limit to how many artifacts originated in Iceland, clearly that number cannot be 5, because X originated in either Norway or Sweden, not in Iceland (second rule). Furthermore, the last rule prohibits us from having both V and Z originate in Iceland, because if V originated in Iceland, then Z must have originated in Sweden. Therefore, the maximum number of artifacts that could have originated in Iceland is 3.
  
  2. Norway. The number of artifacts originating in Norway cannot exceed 2. This is because there are only 5 artifacts available, and fewer of them originated in Norway than in Iceland.
  
  3. Sweden. We can infer that the maximum number of artifacts originating in Sweden is 4. At least one artifact must have originated in Iceland (third rule), so Sweden cannot have all five artifacts. However, no other rule places any restrictions on how many artifacts could have originated in Sweden.

Thus, our final analysis of the minimum and the maximum number of variables per group should look like this:

PT72_Game_#3_setup_diagram 8.png

Now let’s look for some inferences. As always, examine the game for specific points of restriction. Take Norway, for instance: this is the most restricted group of the three, because at most two artifacts originated there. How does that affect the selection of artifacts that could have originated in Norway? Well, if either W or Y originated in Norway, then both of them originated in Norway, because W and Y originated in the same country (first rule). To comply with the numerical distribution in the third rule, the remaining three artifacts should have originated in Iceland:

PT72_Game_#3_setup_diagram 9.png

Is this possible? Clearly not! V and Z cannot both originate in Iceland, because this would violate the last rule. Therefore, neither W nor Y could have originated in Norway—an incredibly important inference that directly answers Questions #15 and #17.

The final diagram for the game should look like this:

PT72_Game_#3_setup_diagram 10.png

[crottman21]

I had a really difficult time setting up the 3rd game of this exam. Could someone help me with the set up?

[Emily Haney-Caron]

Hi crottman,

Can you tell us a bit about what was challenging for you, where you got stuck, and how you approached the set up for these two games? That can really help us when trying to explain the games to you.

Thanks!

[crottman21]

Sure. Game 3 Questions 13-18:
-I made I N S the base, but it was so difficult to use the base because there is so much uncertainty. For instance, no one has to be on Sweden, and thus Iceland could have up to 4. I just didn't even no where to start after the diagram. What did you diagram and what did you have on your paper for the setup by the time you went to answer the questions? Additionally, when do you know your setup in general is enough? When should you draw out the options, etc.

[Nikki Siclunov]

Hi crottman,

Thanks for your question. You're correct: the 5-into-3 distribution is almost entirely Unfixed. Indeed, the only relevant rule in this regard is the third rule, which requires Iceland to have more artifacts than Norway. However, since the minimum number of artifacts per country is zero, it is possible that Iceland has only one artifact, while Norway has none.

Our inability to establish a minimum number of variables per group leaves open the possibility of an exceptionally large number of Fixed distributions, making the task of examining them counterproductive. Instead, focus on the rules in the hopes of establishing the minimum and the maximum number of variables per group. Uncertainty in group sizes is never a good thing, which is why you need to attempt to confine it as much as possible. Examining the min/max limits of each group would be a good place to start.

Thanks to the third rule, we know that I > N, which means that at least one artifact must have originated in Iceland. Since there is no stipulation to the contrary, the minimum number of artifacts in either of the two remaining groups is zero.

What about the maximum number of artifacts per country?

Iceland:

Although there is no explicit limit to how many artifacts originated in Iceland, clearly that number cannot be 5, because X originated in either Norway or Sweden, not in Iceland (second rule). Furthermore, the last rule prohibits us from having both V and Z originate in Iceland, because if V originated in Iceland, then Z must have originated in Sweden. Therefore, the maximum number of artifacts that could have originated in Iceland is 3.

Norway:

The number of artifacts originating in Norway cannot exceed 2. This is because there are only 5 artifacts available, and fewer of them originated in Norway than in Iceland.

Sweden:

We can infer that the maximum number of artifacts originating in Sweden is 4. At least one artifact must have originated in Iceland (third rule), so Sweden cannot have all five artifacts. However, no other rule places any restrictions on how many artifacts could have originated in Sweden.

Thus, our final analysis of the min/max number of variables per group should look like this:

Iceland: 1-3
Norway: 0-2
Sweden: 0-4

We can also make a few conditional deductions:

Norway (1) Iceland (3)
Iceland (1) Norway (0)

This should help you examine the implication of the various rules in the game. Let me know if it all makes sense

Thanks!