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#39270
Setup and Rule Diagram Explanation
This is a Grouping: Partially Defined, Identify the Templates game.
The game scenario establishes that at least two photographers must be assigned to each of two graduation ceremonies—Silva and Thorne. There are six photographers available, none of whom can be assigned to both ceremonies. The scenario also indicates that not all photographers need to be assigned. To keep track of those photographers who are not assigned to either ceremony, it is advisable to create a third (“Unassigned”) group in our setup as shown below:
Because only the minimum number of photographers in each group is established in advance, the game is Partially Defined. Given the high level of uncertainty inherent in this setup, it would likely counterproductive to examine the Numerical Distributions that govern the assignment of photographers to ceremonies.
The first rule states that F and H must be assigned to one of the ceremonies:
This rule also generates two Not Laws applicable to the “Unassigned” group:
The second rule stipulates that if L and M are both assigned, it must be to different ceremonies. In other words, L and M cannot be assigned together:
Note that this rule cannot be represented using Dual Options (L/M) in either ceremony, because it is entirely possible that one—or both—of L and M is not assigned to either ceremony.
The third rule establishes the following conditional relationship:
Since L is common to the second and third rules, it is worth combining them as follows:
The last rule contains a negative sufficient condition:
This rule deserves a closer look. First, the sufficient condition would be met when K is assigned to Silva, but also when K remains unassigned: in either situation, H and M will be forced into Thorne. Furthermore, recall that F and H must always be assigned together to one of the ceremonies. So, whenever K is assigned to Silva or else remains unassigned, all three of H, M, and F must be assigned to Thorne:
The implications of the last rule are significant, because (1) the sufficient condition is readily satisfied whenever K is not assigned to Thorne, and (2) the necessary condition requires three of the six available variables to be assigned to Thorne. The threshold for meeting the sufficient condition is low, whereas the burden carried by the necessary condition is high. When such low-threshold, high-burden conditional relationships appear in a game, they are often a driving force.
To help us understand how the last rule affects the game, let’s explore the options that result when K is assigned to each of the three available groups:
If K is assigned to Silva, we need to ensure that F, H and M are all assigned to Thorne in accordance with the first and the fourth rules:
With only two variables remaining (G and L), we need to ensure that at least one of them is assigned to Silva. However, by the contrapositive of the conditional chain resulting from the combination of the second and the third rules, if M is assigned to Thorne, then G cannot be assigned to Silva (and G must be assigned to thorne or Unassigned). Therefore, L must be assigned to Silva:
On to Template 2, where K is assigned to Thorne:
The fourth rule has no bearing on the assignment of K to Thorne (to conclude otherwise would be a Mistaken Negation of that rule). Notice, however, that F and H must still be assigned together to either Silva or Thorne. We could, then, examine the placement of that block in each of the two groups:
The remaining rules do not allow us to make additional inferences in Template 2A, so it is best to leave that solution as is. Template 2B, however, is heavily restricted by the second and the third rules. In Template 2B, at least two of the remaining three variables—G, L, and M—must be assigned to Silva. Since M and L are never assigned to the same group (second rule), it follows that G must be assigned to Silva. This triggers the third rule, according to which L must be assigned to Thorne whenever G is assigned to Silva. The remaining variable—M—must be assigned to Silva in order to ensure that each group is assigned at least two variables:
Last, let’s examine the third Template, where K is unassigned:
If K is unassigned, we need to ensure that F, H and M are all assigned to Thorne in accordance with the first and the fourth rules:
To ensure that each group has at least two variables in it, the remaining two variables—G and L—must both be assigned to Silva:
This is clearly impossible, because if G is assigned to Silva, then L must be assigned to Thorne (third rule). Consequently, we can infer that K cannot be unassigned, i.e. K must always be assigned to one of the two ceremonies.
The final diagram for the game should look like this:
This is a Grouping: Partially Defined, Identify the Templates game.
The game scenario establishes that at least two photographers must be assigned to each of two graduation ceremonies—Silva and Thorne. There are six photographers available, none of whom can be assigned to both ceremonies. The scenario also indicates that not all photographers need to be assigned. To keep track of those photographers who are not assigned to either ceremony, it is advisable to create a third (“Unassigned”) group in our setup as shown below:
Because only the minimum number of photographers in each group is established in advance, the game is Partially Defined. Given the high level of uncertainty inherent in this setup, it would likely counterproductive to examine the Numerical Distributions that govern the assignment of photographers to ceremonies.
The first rule states that F and H must be assigned to one of the ceremonies:
This rule also generates two Not Laws applicable to the “Unassigned” group:
The second rule stipulates that if L and M are both assigned, it must be to different ceremonies. In other words, L and M cannot be assigned together:
Note that this rule cannot be represented using Dual Options (L/M) in either ceremony, because it is entirely possible that one—or both—of L and M is not assigned to either ceremony.
The third rule establishes the following conditional relationship:
Since L is common to the second and third rules, it is worth combining them as follows:
The last rule contains a negative sufficient condition:
This rule deserves a closer look. First, the sufficient condition would be met when K is assigned to Silva, but also when K remains unassigned: in either situation, H and M will be forced into Thorne. Furthermore, recall that F and H must always be assigned together to one of the ceremonies. So, whenever K is assigned to Silva or else remains unassigned, all three of H, M, and F must be assigned to Thorne:
The implications of the last rule are significant, because (1) the sufficient condition is readily satisfied whenever K is not assigned to Thorne, and (2) the necessary condition requires three of the six available variables to be assigned to Thorne. The threshold for meeting the sufficient condition is low, whereas the burden carried by the necessary condition is high. When such low-threshold, high-burden conditional relationships appear in a game, they are often a driving force.
To help us understand how the last rule affects the game, let’s explore the options that result when K is assigned to each of the three available groups:
If K is assigned to Silva, we need to ensure that F, H and M are all assigned to Thorne in accordance with the first and the fourth rules:
With only two variables remaining (G and L), we need to ensure that at least one of them is assigned to Silva. However, by the contrapositive of the conditional chain resulting from the combination of the second and the third rules, if M is assigned to Thorne, then G cannot be assigned to Silva (and G must be assigned to thorne or Unassigned). Therefore, L must be assigned to Silva:
On to Template 2, where K is assigned to Thorne:
The fourth rule has no bearing on the assignment of K to Thorne (to conclude otherwise would be a Mistaken Negation of that rule). Notice, however, that F and H must still be assigned together to either Silva or Thorne. We could, then, examine the placement of that block in each of the two groups:
The remaining rules do not allow us to make additional inferences in Template 2A, so it is best to leave that solution as is. Template 2B, however, is heavily restricted by the second and the third rules. In Template 2B, at least two of the remaining three variables—G, L, and M—must be assigned to Silva. Since M and L are never assigned to the same group (second rule), it follows that G must be assigned to Silva. This triggers the third rule, according to which L must be assigned to Thorne whenever G is assigned to Silva. The remaining variable—M—must be assigned to Silva in order to ensure that each group is assigned at least two variables:
Last, let’s examine the third Template, where K is unassigned:
If K is unassigned, we need to ensure that F, H and M are all assigned to Thorne in accordance with the first and the fourth rules:
To ensure that each group has at least two variables in it, the remaining two variables—G and L—must both be assigned to Silva:
This is clearly impossible, because if G is assigned to Silva, then L must be assigned to Thorne (third rule). Consequently, we can infer that K cannot be unassigned, i.e. K must always be assigned to one of the two ceremonies.
The final diagram for the game should look like this:
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