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#41655
Setup and Rule Diagram Explanation
This is a Basic Linear: Unbalanced, Underfunded, Identify the Templates game.
The game scenario establishes that five television programs must be shown over the course of three hours, beginning at 1 PM. At first glance, this appears to be an Unbalanced, Overloaded game, as the number of variables (5) exceeds the number of hourly slots (3). However, notice that four of the programs are half-hour long—R, S, T, and W—and one is an hour-long—G. So, we are probably better off using “half-hour” slots as our base, so that we have a total of six slots for the five programs.
Technically, then, the game is Unbalanced, Underfunded. To make it Balanced, we need to represent the hour-long program (G) using a GG Block, because that program must take up two adjacent “half-hour” time slots in our base:
Remember, however, that the GG Block counts as one program, not two! So, if the GG block occupied the first two spaces, for instance, then the program beginning at 2:00 would be the second program shown, not the third. This is an important “kink” that you must keep in mind as you attack the questions. Some students hoped to avoid this complexity by using the order in which the programs are shown as the base, regardless of the hourly slots:
While initially attractive in its simplicity, such a setup is a terrible idea. For one thing, it does not allow us to represent the first or the second rules internally, making it exceptionally difficult to keep track of whether each slot in our setup refers to the hour or the half-hour. This would depend on where G is relative to the rest of the variables: if G is the first program shown, then the second slot would be referring to 2:00, because G is an hour-long program. But if any of the other programs were shown first, then the second slot would be referring to 1:30, because they are all half-hour long. Confusing? You bet! This is a mess we would rather avoid at all costs. This is why it is critical to examine the rules before settling on a particular setup: the setup must allow you to represent the rules clearly and accurately, so as to avoid unnecessary confusion down the road.
With our original structure in place, let us now turn to the rules.
The first rule establishes that G must start on the hour rather than the half hour:
Representing this rule can be challenging. Many students mistakenly decided to create G Not-Laws at the 1:30, 2:30, and 3:30 time slots. Although G cannot start on the half hour, it will clearly run into the “half hour” slot following its starting time, because G lasts a full hour. So, we cannot create any Not-Laws to reflect the application of this rule, unless we use G1 G2 subscripts, which only adds complexity to an already complicated setup. Indeed, the only way to competently represent what must be true is to create three Templates based on the three possible placements of the GG Block:
What may strike you as unusual is our decision to undertake a Templates-based approach early on, without fully evaluating if the game is sufficiently restricted. Admittedly, the decision was partially prompted by our realization that there is no other way to clearly represent the first rule. Furthermore, even if the remaining rules are not sufficiently restrictive (though they are), the rule generates only three templates.
The second rule tells us that T starts on the half hour rather than the hour. Since T is a half-hour long program, this rule generates three T Not Laws:
The second rule also generates Split-Dual options for T in each of our three templates, because each template contains only two possible time slots that can accommodate T:
The third rule establishes the following sequence:
R 
S
This sequence generates two Not Laws, an R Not Law, and an S Not Law:
The fourth rule establishes the following conditional block:
Examine the meaning of this rule carefully. The rule does not generate a simple WT Block, because if W is not shown earlier than T, then the rule is not activated and a WT Block does not need to be formed. The block is only necessary if W is shown earlier than T! So, for instance, if W were shown at 1:00, then W would be shown earlier than all other programs, including T, triggering the Block formation. Likewise, if T were shown at 3:30, then all other programs would be shown earlier than T, including W, requiring W to be shown immediately before T, at 3:00.
Now let’s examine this rule in the context of our templates. Each template allows T to be placed in one of two possible positions. Observant test-takers will notice that exactly one of these two positions, in each template, triggers the sufficient condition in the last rule! If T is shown at 3:30—as may be the case in Templates 1 and 2—then W would need to be shown earlier than T, triggering the WT Block. In Template 3, if T were shown at 2:30, then W would again be shown earlier than T, and immediately before it.
It should be apparent by now that the precise location of T within each template can affect the placement of W. This, in turn, would restrict the placement of the remaining two variables, R and S, which—as we know—are bound by a sequencing rule. The optimal way to proceed, therefore, is to split each Template into two, based on the scheduled showing of T:
As mentioned earlier, W would be shown earlier than T by default in Templates 1B, 2B, and 3B. Consequently, W must be shown immediately before T in all three of these templates. The remaining variables—R and S—must be sequenced in accordance with the third rule. Templates 1B, 2B, and 3B are therefore completely determined:
Interestingly, the last rule virtually no effect on the placement of W in Templates 1A, 2A, and 3A. This is because if W is shown earlier than T in any of these templates, W would have to be shown immediately before T, regardless of the last rule. Alternatively, if W is shown after T, then the last rule does not apply and W can be shown anywhere after T. Either way, we can completely disregard the last rule when analyzing any of these three templates. The only rule we need to consider is the R S sequence, which generates two Dual-options in each template:
With these six templates in place, we arrive at the final setup for this game:
This game epitomizes the central proposition that is inherent in a Templates approach: you will spend more time in the setup, but this time will be regained in the lightning-fast execution of the questions. Here, the approach is advantageous for several reasons: First, each template is severely restricted, which obviates the need to create local diagrams for any of the local questions. Second, the first rule would be difficult to represent without the use of templates, making such an approach virtually essential to the avoidance of misinterpretation. Third, the templates already take into account the implications of the conditional rule, so you no longer run the risk of misapplying it as you attack the questions. Finally, consider the fact that this game contains seven questions, rather than, say, five. The higher the number of questions, the more benefit you can expect to derive from your initial investment of time.
This is a Basic Linear: Unbalanced, Underfunded, Identify the Templates game.
The game scenario establishes that five television programs must be shown over the course of three hours, beginning at 1 PM. At first glance, this appears to be an Unbalanced, Overloaded game, as the number of variables (5) exceeds the number of hourly slots (3). However, notice that four of the programs are half-hour long—R, S, T, and W—and one is an hour-long—G. So, we are probably better off using “half-hour” slots as our base, so that we have a total of six slots for the five programs.
Technically, then, the game is Unbalanced, Underfunded. To make it Balanced, we need to represent the hour-long program (G) using a GG Block, because that program must take up two adjacent “half-hour” time slots in our base:
Remember, however, that the GG Block counts as one program, not two! So, if the GG block occupied the first two spaces, for instance, then the program beginning at 2:00 would be the second program shown, not the third. This is an important “kink” that you must keep in mind as you attack the questions. Some students hoped to avoid this complexity by using the order in which the programs are shown as the base, regardless of the hourly slots:
While initially attractive in its simplicity, such a setup is a terrible idea. For one thing, it does not allow us to represent the first or the second rules internally, making it exceptionally difficult to keep track of whether each slot in our setup refers to the hour or the half-hour. This would depend on where G is relative to the rest of the variables: if G is the first program shown, then the second slot would be referring to 2:00, because G is an hour-long program. But if any of the other programs were shown first, then the second slot would be referring to 1:30, because they are all half-hour long. Confusing? You bet! This is a mess we would rather avoid at all costs. This is why it is critical to examine the rules before settling on a particular setup: the setup must allow you to represent the rules clearly and accurately, so as to avoid unnecessary confusion down the road.
With our original structure in place, let us now turn to the rules.
The first rule establishes that G must start on the hour rather than the half hour:
Representing this rule can be challenging. Many students mistakenly decided to create G Not-Laws at the 1:30, 2:30, and 3:30 time slots. Although G cannot start on the half hour, it will clearly run into the “half hour” slot following its starting time, because G lasts a full hour. So, we cannot create any Not-Laws to reflect the application of this rule, unless we use G1 G2 subscripts, which only adds complexity to an already complicated setup. Indeed, the only way to competently represent what must be true is to create three Templates based on the three possible placements of the GG Block:
What may strike you as unusual is our decision to undertake a Templates-based approach early on, without fully evaluating if the game is sufficiently restricted. Admittedly, the decision was partially prompted by our realization that there is no other way to clearly represent the first rule. Furthermore, even if the remaining rules are not sufficiently restrictive (though they are), the rule generates only three templates.
The second rule tells us that T starts on the half hour rather than the hour. Since T is a half-hour long program, this rule generates three T Not Laws:
The second rule also generates Split-Dual options for T in each of our three templates, because each template contains only two possible time slots that can accommodate T:
The third rule establishes the following sequence:
R SThis sequence generates two Not Laws, an R Not Law, and an S Not Law:
The fourth rule establishes the following conditional block:
Examine the meaning of this rule carefully. The rule does not generate a simple WT Block, because if W is not shown earlier than T, then the rule is not activated and a WT Block does not need to be formed. The block is only necessary if W is shown earlier than T! So, for instance, if W were shown at 1:00, then W would be shown earlier than all other programs, including T, triggering the Block formation. Likewise, if T were shown at 3:30, then all other programs would be shown earlier than T, including W, requiring W to be shown immediately before T, at 3:00.
Now let’s examine this rule in the context of our templates. Each template allows T to be placed in one of two possible positions. Observant test-takers will notice that exactly one of these two positions, in each template, triggers the sufficient condition in the last rule! If T is shown at 3:30—as may be the case in Templates 1 and 2—then W would need to be shown earlier than T, triggering the WT Block. In Template 3, if T were shown at 2:30, then W would again be shown earlier than T, and immediately before it.
It should be apparent by now that the precise location of T within each template can affect the placement of W. This, in turn, would restrict the placement of the remaining two variables, R and S, which—as we know—are bound by a sequencing rule. The optimal way to proceed, therefore, is to split each Template into two, based on the scheduled showing of T:
As mentioned earlier, W would be shown earlier than T by default in Templates 1B, 2B, and 3B. Consequently, W must be shown immediately before T in all three of these templates. The remaining variables—R and S—must be sequenced in accordance with the third rule. Templates 1B, 2B, and 3B are therefore completely determined:
Interestingly, the last rule virtually no effect on the placement of W in Templates 1A, 2A, and 3A. This is because if W is shown earlier than T in any of these templates, W would have to be shown immediately before T, regardless of the last rule. Alternatively, if W is shown after T, then the last rule does not apply and W can be shown anywhere after T. Either way, we can completely disregard the last rule when analyzing any of these three templates. The only rule we need to consider is the R S sequence, which generates two Dual-options in each template:
With these six templates in place, we arrive at the final setup for this game:
This game epitomizes the central proposition that is inherent in a Templates approach: you will spend more time in the setup, but this time will be regained in the lightning-fast execution of the questions. Here, the approach is advantageous for several reasons: First, each template is severely restricted, which obviates the need to create local diagrams for any of the local questions. Second, the first rule would be difficult to represent without the use of templates, making such an approach virtually essential to the avoidance of misinterpretation. Third, the templates already take into account the implications of the conditional rule, so you no longer run the risk of misapplying it as you attack the questions. Finally, consider the fact that this game contains seven questions, rather than, say, five. The higher the number of questions, the more benefit you can expect to derive from your initial investment of time.
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