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- Mon Oct 02, 2017 2:03 pm
#40228
Setup and Rule Diagram Explanation
This is a Grouping: Balanced, Defined-Moving, Numerical Distributions, Identify the Templates game.
The game scenario establishes that each of five buildings were owned by one of three families—the Trents, the Williamses, and the Yandells, and that each family owned at least one of the buildings:
Since exactly five buildings are being assigned to the three families, but it is yet unknown how many buildings were owned by each family, the game is Defined-Moving. In addition, since each of the buildings was owned by exactly one family, the game Balanced.
When you create the setup, it is critical that the correct base be selected. There are two choices: the five buildings or the three families. Since each building was owned by exactly one family, it may be tempting to use the buildings as the base in order to reduce the level of uncertainty inherent in each group. Notice, however, the numerical restriction placed by the first rule: the Williamses owned more of the buildings than the Yandells owned. (Always read the rules thoroughly before deciding on a setup!) A base of five buildings would be a poor choice for the purpose of tracking that uncertainty.
(A 3-2-0 distribution whereby the Williamses owned three of the buildings and Yandells owned two is impossible, because each family needs to have owned at least one building, and a 3-2-0 distribution leaves Trents with no buildings to own).
Since these distributions create two separate scenarios in the game, it makes sense to create basic diagrams for each option. Each diagram will likely provide a considerable amount of information, which is why a Template-based approach will probably be sufficient. Indeed, Numerical Distributions—especially Fixed distributions—often lead to setting up the game with Templates:
In both Fixed distributions the Yandells clearly owned only one building, which is an important deduction to make early on.
The second rule states that neither I nor M belonged to the owner of F. In other words, neither I nor M can be in the same group as F:
While the vertical Not-Blocks are visually appealing, you can also use conditional language to represent the relationship between F, M and I:
The last rule states that either the Trents owned S, or the Yandells owned I, or both. Since at least one of these two options must occur, we need to represent this rule conditionally: if the Trents did not own S, then the Yandells must have owned I. And vice versa: if the Yandells did not own I, then the Trents must have owned S:
(Note that if the Trents did own S, then the rule would be immediately satisfied without any necessary implications involving the Yandells. Likewise, the rule would be satisfied if the Yandells owned I, without any restrictions involving the Trents.)
Let’s examine more closely the meaning of each of the two sufficient conditions above: if the Trents did not own S, then the only families that could have owned S are the Williamses and the Yandells. Likewise, if the Yandells did not own I, then the only families that could have owned I are the Trents and the Williamses. Thus, we can rewrite the rule using positive sufficient conditions instead of negatives ones, which would greatly facilitate its application:
You should notice that if the Yandells owned S, then—judging from the first conditional rule above—they would also have to own I. However, our Numerical Distribution analysis clearly shows that the Yandells must have owned exactly one building. Therefore, the Yandells could not have owned S, and the rule can be simplified even further:
Finally, the last rule has a particularly limiting effect on the 1-3-1 distribution. For the same reason the Yandells cannot have owned S in either distribution, the Trents cannot have owned I in the 1-3-1 distribution. If they did, then neither the Trents could have owned S, nor the Yandells could have owned I, in direct violation of the last rule.
The final diagram for the game should look like this:
Observant test-takers will now notice that each of the two distributions outlined above is severely restricted by the conditional rules governing the ownership of the different buildings. For instance, if the Trents or the Williamses owned I, we immediately know who owned S—the Trents did. This leaves us with only three buildings, two of which (F and M) cannot have been owned by the same family. The most efficient way to proceed, therefore, would be to examine the assignment of I to each of the three families separately, effectively creating five Templates (remember—the Trents cannot have owned I in the 1-3-1 distribution):
Next, examine the implication of the second rule (neither I nor M can be in the same group as F) to each remaining template:
These templates reveal important inferences that would have been virtually impossible to make otherwise. For instance, in a 1-3-1 distribution, the Williamses must own G—a random variable whose limited placement in that distribution would have been difficult to predict. In the same distribution, the Yandells must own either F or I—another key inference. The game epitomizes the central proposition that is inherent in a Templates approach: you will spend a bit more time in the setup, but this time will be regained in the lightning-fast execution of the questions. It also highlights the close link between Numerical Distribution and Templates. Granted, not all distribution-driven games can be solved with Templates, and not all Template-driven games contain Numerical distributions. Nevertheless, there is a significant association between the two.
This is a Grouping: Balanced, Defined-Moving, Numerical Distributions, Identify the Templates game.
The game scenario establishes that each of five buildings were owned by one of three families—the Trents, the Williamses, and the Yandells, and that each family owned at least one of the buildings:
Since exactly five buildings are being assigned to the three families, but it is yet unknown how many buildings were owned by each family, the game is Defined-Moving. In addition, since each of the buildings was owned by exactly one family, the game Balanced.
When you create the setup, it is critical that the correct base be selected. There are two choices: the five buildings or the three families. Since each building was owned by exactly one family, it may be tempting to use the buildings as the base in order to reduce the level of uncertainty inherent in each group. Notice, however, the numerical restriction placed by the first rule: the Williamses owned more of the buildings than the Yandells owned. (Always read the rules thoroughly before deciding on a setup!) A base of five buildings would be a poor choice for the purpose of tracking that uncertainty.
- Note: 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 buildings to the three families, reaffirming our decision to use the families as the base.
(A 3-2-0 distribution whereby the Williamses owned three of the buildings and Yandells owned two is impossible, because each family needs to have owned at least one building, and a 3-2-0 distribution leaves Trents with no buildings to own).
Since these distributions create two separate scenarios in the game, it makes sense to create basic diagrams for each option. Each diagram will likely provide a considerable amount of information, which is why a Template-based approach will probably be sufficient. Indeed, Numerical Distributions—especially Fixed distributions—often lead to setting up the game with Templates:
In both Fixed distributions the Yandells clearly owned only one building, which is an important deduction to make early on.
The second rule states that neither I nor M belonged to the owner of F. In other words, neither I nor M can be in the same group as F:
While the vertical Not-Blocks are visually appealing, you can also use conditional language to represent the relationship between F, M and I:
The last rule states that either the Trents owned S, or the Yandells owned I, or both. Since at least one of these two options must occur, we need to represent this rule conditionally: if the Trents did not own S, then the Yandells must have owned I. And vice versa: if the Yandells did not own I, then the Trents must have owned S:
(Note that if the Trents did own S, then the rule would be immediately satisfied without any necessary implications involving the Yandells. Likewise, the rule would be satisfied if the Yandells owned I, without any restrictions involving the Trents.)
Let’s examine more closely the meaning of each of the two sufficient conditions above: if the Trents did not own S, then the only families that could have owned S are the Williamses and the Yandells. Likewise, if the Yandells did not own I, then the only families that could have owned I are the Trents and the Williamses. Thus, we can rewrite the rule using positive sufficient conditions instead of negatives ones, which would greatly facilitate its application:
You should notice that if the Yandells owned S, then—judging from the first conditional rule above—they would also have to own I. However, our Numerical Distribution analysis clearly shows that the Yandells must have owned exactly one building. Therefore, the Yandells could not have owned S, and the rule can be simplified even further:
Finally, the last rule has a particularly limiting effect on the 1-3-1 distribution. For the same reason the Yandells cannot have owned S in either distribution, the Trents cannot have owned I in the 1-3-1 distribution. If they did, then neither the Trents could have owned S, nor the Yandells could have owned I, in direct violation of the last rule.
The final diagram for the game should look like this:
Observant test-takers will now notice that each of the two distributions outlined above is severely restricted by the conditional rules governing the ownership of the different buildings. For instance, if the Trents or the Williamses owned I, we immediately know who owned S—the Trents did. This leaves us with only three buildings, two of which (F and M) cannot have been owned by the same family. The most efficient way to proceed, therefore, would be to examine the assignment of I to each of the three families separately, effectively creating five Templates (remember—the Trents cannot have owned I in the 1-3-1 distribution):
Next, examine the implication of the second rule (neither I nor M can be in the same group as F) to each remaining template:
These templates reveal important inferences that would have been virtually impossible to make otherwise. For instance, in a 1-3-1 distribution, the Williamses must own G—a random variable whose limited placement in that distribution would have been difficult to predict. In the same distribution, the Yandells must own either F or I—another key inference. The game epitomizes the central proposition that is inherent in a Templates approach: you will spend a bit more time in the setup, but this time will be regained in the lightning-fast execution of the questions. It also highlights the close link between Numerical Distribution and Templates. Granted, not all distribution-driven games can be solved with Templates, and not all Template-driven games contain Numerical distributions. Nevertheless, there is a significant association between the two.
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