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#41251
Setup and Rule Diagram Explanation
This is a Grouping: Defined-Moving, Balanced, Numerical Distribution, Identify the Templates game.
The game scenario establishes that seven sales representatives work in three zones:
Each representative must work in a zone, but the game scenario does not establish exactly how many representatives work in each zone. Thus, the rules will have to provide information on that aspect of the game.
The first rule creates a dual-option on zone 1:
Because the rule also stipulates that P and T cannot work in the zone together, a vertical not-block also exists:
Note that, although this PT not-block is limited to Zone 1 by the rule, due to the other rules this not-block applies to all zones (same for the not-block created in the next rule)
The second rule is identical in nature to the first rule, but it addresses T and U in Zone 2:
Note that, because T is common to both rules, only three basic solutions exist for satisfying the first two rules. More on this fact later in the setup.
The third rule creates a PQ block:
The fourth rule creates an SU block:
Note that the third rule can be connected to the first rule via P, and that the fourth rule can be connected to the second rule via U:
The fifth rule is numerical, and establishes that Zone 3 has more sales representatives than Zone 2. Because Zone 2 must have at least one representative (from the second rule), we can infer that Zone 3 must have at least two representatives. Because Zone 1 must have at least one representative (from the first rule), the minimum Numerical Distribution for the three zones is:
Thus, four representatives are already assigned, leaving just three representatives available for placement among the three zones. What, then, are the possible Numerical Distributions for this game based just on the numbers? To solve this problem, focus on the relationship between Zone 2 and Zone 3. First show all of the options for Zone 3 when Zone 2 has one representative, then show all of the options for Zone 3 when Zone 2 has two representatives, and so on. Remember, there are three representatives left to add to the minimums, so each solution must add up to seven.
When Zone 2 has one representative, Zone 3 can have two to five representatives (at least initially, before considering the rules):
When Zone 2 has two representatives, Zone 3 can have three or four representatives:
If Zone 2 attempted to have three representatives, then Zone 3 would have to have four representatives, which will not work since that would assign all seven representatives to Zones 2 and 3, leaving no representative for Zone 1.
Note that the 1-1-5 distribution above is impossible due to the actions of the first four rules. Because P is in a block, the only way to meet the first rule is for T to be the only representative in Zone 1. But, because S is in a block, the only way to meet the second rule is for T to be the only representative in Zone 2. Clearly, those two conditions are incompatible, and thus the 1-1-5 distribution cannot occur. Thus, these are the only five possible fixed distributions in the game:
With the Numerical Distributions in place, and the powerful limitations created by the first four rules, the best approach at this juncture is to show templates based on the possible placements of P, T, and U in the first two zones, while at the same time using the distribution information to make further inferences. These are the three base templates we will use:
Of course, these templates just satisfy the requirements of the first two rules (while adding in the blocks from the third and fourth rules). Now, let’s further analyze each template in greater detail.
This is a Grouping: Defined-Moving, Balanced, Numerical Distribution, Identify the Templates game.
The game scenario establishes that seven sales representatives work in three zones:
Each representative must work in a zone, but the game scenario does not establish exactly how many representatives work in each zone. Thus, the rules will have to provide information on that aspect of the game.
The first rule creates a dual-option on zone 1:
Because the rule also stipulates that P and T cannot work in the zone together, a vertical not-block also exists:
Note that, although this PT not-block is limited to Zone 1 by the rule, due to the other rules this not-block applies to all zones (same for the not-block created in the next rule)
The second rule is identical in nature to the first rule, but it addresses T and U in Zone 2:
Note that, because T is common to both rules, only three basic solutions exist for satisfying the first two rules. More on this fact later in the setup.
The third rule creates a PQ block:
The fourth rule creates an SU block:
Note that the third rule can be connected to the first rule via P, and that the fourth rule can be connected to the second rule via U:
The fifth rule is numerical, and establishes that Zone 3 has more sales representatives than Zone 2. Because Zone 2 must have at least one representative (from the second rule), we can infer that Zone 3 must have at least two representatives. Because Zone 1 must have at least one representative (from the first rule), the minimum Numerical Distribution for the three zones is:
Thus, four representatives are already assigned, leaving just three representatives available for placement among the three zones. What, then, are the possible Numerical Distributions for this game based just on the numbers? To solve this problem, focus on the relationship between Zone 2 and Zone 3. First show all of the options for Zone 3 when Zone 2 has one representative, then show all of the options for Zone 3 when Zone 2 has two representatives, and so on. Remember, there are three representatives left to add to the minimums, so each solution must add up to seven.
When Zone 2 has one representative, Zone 3 can have two to five representatives (at least initially, before considering the rules):
When Zone 2 has two representatives, Zone 3 can have three or four representatives:
If Zone 2 attempted to have three representatives, then Zone 3 would have to have four representatives, which will not work since that would assign all seven representatives to Zones 2 and 3, leaving no representative for Zone 1.
Note that the 1-1-5 distribution above is impossible due to the actions of the first four rules. Because P is in a block, the only way to meet the first rule is for T to be the only representative in Zone 1. But, because S is in a block, the only way to meet the second rule is for T to be the only representative in Zone 2. Clearly, those two conditions are incompatible, and thus the 1-1-5 distribution cannot occur. Thus, these are the only five possible fixed distributions in the game:
With the Numerical Distributions in place, and the powerful limitations created by the first four rules, the best approach at this juncture is to show templates based on the possible placements of P, T, and U in the first two zones, while at the same time using the distribution information to make further inferences. These are the three base templates we will use:
- Template #1: T in Zone 1, SU in Zone 2
- Template #2: PQ in Zone 1, T in Zone 2
- Template #3: PQ in Zone 1, SU in Zone 2
Of course, these templates just satisfy the requirements of the first two rules (while adding in the blocks from the third and fourth rules). Now, let’s further analyze each template in greater detail.
- Template #1: T in Zone 1, SU in Zone 2
- This template may initially appear to have no further restrictions on the placement of the remaining variables (K, M, P, and Q), but consider the options for the PQ block. When Zone 2 has two representatives, then only two fixed distributions exist: 2-2-3 and 1-2-4. Thus, the PQ block cannot be placed into either Zone 1 (as that would make three representatives) or Zone 2 (as that would make four representatives). Thus, in this template, the PQ block must be placed into Zone 3.
Further, of the two remaining representatives—K and M—at least one must be placed into Zone 3 in order to meet the requirements of the fifth rule. The remainder of K and M can then work in Zone 1 or Zone 3. The combination of this information leads to the final template:
- Template #2: PQ in Zone 1, T in Zone 2
This template contains the greatest number of possibilities. The only restriction in this template is that the SU block cannot be placed into Zone 2 as that would result in three representatives in that zone, a violation of the numerical rules of the game. Thus, the SU block must work in Zone 1 or Zone 3. Additionally, both of K and M cannot work in Zone 2 due to the numerical limitations.
- Template #3: PQ in Zone 1, SU in Zone 2
This template is the most restricted of the three templates. Because there are two representatives already working in Zone 2, there must be at least three representatives working in Zone 3. But, since Zone 1 already has P and Q, only K, M, and T are available to work in Zone 3, and thus we can infer that all three must work in Zone 3:
- Thus, only one solution exists in this template.
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