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- Thu Aug 10, 2017 5:01 pm
#38152
Setup and Rule Diagram Explanation
This is an Advanced Linear: Balanced game.
The game scenario establishes that six books will be discussed over the course of six consecutive weeks, one book per week. The base should be the weeks, due to the inherent order of this variable set. Additionally, we need to determine which of the six books are summarized (at least one must be). Whether a book is summarized or not constitutes a second variable set, one that should be stacked above the base. This strategy makes sense, because we need to keep track of two distinctly different attributes for each week: which book is being discussed, and whether or not that book is summarized. Additionally, the order in which the books are being discussed affects whether or not they are summarized (see, e.g. first rule). This sort of relationship between the two variable sets is unique to Advanced Linear games, and is best represented using a stacked setup.
Some test-takers would understandably interpret the summarizing aspect of the game as adding a grouping dimension to it, and therefore create a two-value grouping base. This is a mistake. Having two separate diagrams—one keeping track of the order in which the books are discussed, and another one splitting the books into two separate groups—would not take into account the relationship between the sequencing and the grouping elements in the game. It would be cumbersome, for instance, to represent the first rule if these elements are being tracked separately. Last, but not least, make sure to always read through the rules before deciding on a particular setup, especially if you believe that there are multiple ways of organizing the information presented in the scenario.
With that in mind, your initial setup should look like this:
Now let’s examine each rule.
The first rule applies only to the top stack, and deserves a closer look. The rule establishes that no two books that are summarized are discussed in consecutive weeks. In other words, if we have two consecutively discussed books, they cannot both be summarized. To represent this rule visually, a simple SS Not-Block applicable to the top stack is all we need:
Additionally, note that the top stack entails a two-value system (books are either summarized or not summarized). So, if any book is summarized, we would immediately know that any books immediately adjacent to it are not summarized:
The second rule is the trickiest rule in this game, if not on the entire test: it involves both a negative sufficient condition and a double necessary condition. As such, it deserves a closer look (we’ll show the “summarized” reference with a subscript “S”). The rule establishes that there are two necessary conditions that must be satisfied if N is not summarized: R is summarized, and T is summarized. Thus, the proper diagram for this statement is:
It is worth noting that the rule does not prevent all three of N, R, and T from being summarized! For instance, if N is summarized, we know nothing about whether R or T is summarized: to conclude otherwise would be a Mistaken Negation. However, if either one of the necessary conditions is not met, then the sufficient condition cannot be met either. In other words, if either R or T is not summarized, then N must be summarized:
Note that in the contrapositive, the “and” in the necessary condition is changed to “or.” According to the contrapositive, if either member of the pair is not summarized, then the other member of that pair must be summarized. In other words:
You should immediately notice that these rules can be combined to create the following super-sequence:
As with any sequencing game, representing all the Not Laws that result from this chain would be a mistake (there are 24 of them). Instead, let’s examine what must be true, given the incredibly restrictive nature of the rules. Since each of the six books being discussed takes part in the super-sequence sequence, we can easily infer that O must be discussed fourth: three books must be discussed earlier than O, and two books must be discussed later than O. Thus, O divides the linear setup in two separate sections:
Now that we have a solid grasp of the sequence in which the six books are discussed, let’s return to the rules governing the order in which the books are summarized. A few key deductions can be made:
Furthermore, notice that R and K are discussed consecutively, and therefore cannot both be summarized:
This inference ties with the second rule to create the following chain relationship:
From the contrapositive of this relationship, we can infer that whenever K is summarized, R is not summarized, forcing N to be summarized:
Rest assured that most test-takers failed to spot either of these inferences, which was OK—they played a minor role in attacking the questions. Nevertheless, your ability to make them shows a level of engagement with the rules that will doubtless be helpful later on.
The final diagram for the game should look like this:
This is an Advanced Linear: Balanced game.
The game scenario establishes that six books will be discussed over the course of six consecutive weeks, one book per week. The base should be the weeks, due to the inherent order of this variable set. Additionally, we need to determine which of the six books are summarized (at least one must be). Whether a book is summarized or not constitutes a second variable set, one that should be stacked above the base. This strategy makes sense, because we need to keep track of two distinctly different attributes for each week: which book is being discussed, and whether or not that book is summarized. Additionally, the order in which the books are being discussed affects whether or not they are summarized (see, e.g. first rule). This sort of relationship between the two variable sets is unique to Advanced Linear games, and is best represented using a stacked setup.
Some test-takers would understandably interpret the summarizing aspect of the game as adding a grouping dimension to it, and therefore create a two-value grouping base. This is a mistake. Having two separate diagrams—one keeping track of the order in which the books are discussed, and another one splitting the books into two separate groups—would not take into account the relationship between the sequencing and the grouping elements in the game. It would be cumbersome, for instance, to represent the first rule if these elements are being tracked separately. Last, but not least, make sure to always read through the rules before deciding on a particular setup, especially if you believe that there are multiple ways of organizing the information presented in the scenario.
With that in mind, your initial setup should look like this:
Now let’s examine each rule.
The first rule applies only to the top stack, and deserves a closer look. The rule establishes that no two books that are summarized are discussed in consecutive weeks. In other words, if we have two consecutively discussed books, they cannot both be summarized. To represent this rule visually, a simple SS Not-Block applicable to the top stack is all we need:
Additionally, note that the top stack entails a two-value system (books are either summarized or not summarized). So, if any book is summarized, we would immediately know that any books immediately adjacent to it are not summarized:
The second rule is the trickiest rule in this game, if not on the entire test: it involves both a negative sufficient condition and a double necessary condition. As such, it deserves a closer look (we’ll show the “summarized” reference with a subscript “S”). The rule establishes that there are two necessary conditions that must be satisfied if N is not summarized: R is summarized, and T is summarized. Thus, the proper diagram for this statement is:
It is worth noting that the rule does not prevent all three of N, R, and T from being summarized! For instance, if N is summarized, we know nothing about whether R or T is summarized: to conclude otherwise would be a Mistaken Negation. However, if either one of the necessary conditions is not met, then the sufficient condition cannot be met either. In other words, if either R or T is not summarized, then N must be summarized:
Note that in the contrapositive, the “and” in the necessary condition is changed to “or.” According to the contrapositive, if either member of the pair is not summarized, then the other member of that pair must be summarized. In other words:
- Either N or R must be summarized (and both can be summarized as well)
Either N or T must be summarized (and both can be summarized as well)
You should immediately notice that these rules can be combined to create the following super-sequence:
As with any sequencing game, representing all the Not Laws that result from this chain would be a mistake (there are 24 of them). Instead, let’s examine what must be true, given the incredibly restrictive nature of the rules. Since each of the six books being discussed takes part in the super-sequence sequence, we can easily infer that O must be discussed fourth: three books must be discussed earlier than O, and two books must be discussed later than O. Thus, O divides the linear setup in two separate sections:
Now that we have a solid grasp of the sequence in which the six books are discussed, let’s return to the rules governing the order in which the books are summarized. A few key deductions can be made:
- As we know from our discussion of the second rule, either N or T must be summarized. However, since no two consecutive books can be summarized (first rule), if F is discussed second, then F cannot be summarized. This is because if F were discussed second and was summarized, we would need to ensure that neither the first (N) nor the third (T) books are also summarized—which is impossible. Thus:
Furthermore, notice that R and K are discussed consecutively, and therefore cannot both be summarized:
This inference ties with the second rule to create the following chain relationship:
From the contrapositive of this relationship, we can infer that whenever K is summarized, R is not summarized, forcing N to be summarized:
Rest assured that most test-takers failed to spot either of these inferences, which was OK—they played a minor role in attacking the questions. Nevertheless, your ability to make them shows a level of engagement with the rules that will doubtless be helpful later on.
The final diagram for the game should look like this:
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Fail