- PowerScore Staff
- Posts: 8223
- Joined: Feb 02, 2011
This is a Grouping: Defined-Moving, Unbalanced: Overloaded, Numerical Distributions, Identify the Templates game.
This game requires that we select five colored threads out of available six, and assign each of the five threads to one of three rugs. The rugs can be represented as three functionally identical groups (R), as none of the rules apply to specific rugs. To keep track of the unassigned colored thread, we need to create an “Unused” group (U), as shown below:
Since exactly five colored threads are used to weave the rugs, but it is yet unknown how many colored threads are used in each rug, the game is Defined-Moving. In addition, since there are six thread colors, but only five colors are used to weave the rugs, the game is Unbalanced: Overloaded.
The scenario indicates that each rug is either solid or multicolored, suggesting that the minimum number of thread colors per rug is one. Although the maximum is not initially pre-determined, there are only two ways to distribute five variables into three groups:
- 3 - 1 - 1
2 - 2 - 1
The first rule indicates that if W is used, two other colors must be used as well:
We cannot use W in the 2-2-1 distribution, because that distribution does not allow for a rug with three colors:
We could use W in the 3-1-1 distribution, but not in either of the solid rugs:
Note: in the 3-1-1 template, a tricky point is that W could be the unused thread color.
The second rule establishes a conditional relationship between O and P:
This rule has several important implications:
- 1. Since exactly five thread colors must be used, P must always be a color used in one of the rugs. This is because if P were not used, by the contrapositive O could not be used either, which leaves only four colors available for the three rugs.
2. Since O requires the use of P, O cannot be used in any of the solid rugs.
3. The 2-2-1 distribution precludes the use of W, previously discussed. Consequently, O and P must be used in one of the rugs with two colors.
4. Since W and O cannot be used in any of the solid rugs, but at least one of them must always be used, we can conclude that either W or O—or both—are used in a multicolored rug.
The last three rules are all negative Grouping rules, and each eliminates a pair of variables from appearing in the same rug:
These rules have an important implication to the 2-2-1 distribution. To prevent F and T from appearing in the same rug, we need to ensure that Y appears in one of the two multicolored rugs (via the Hurdle the Uncertainty principle):
Since P and T appear in multiple rules, their placement in the 3-1-1 distribution should be carefully tracked.
The final diagram for the game should look like this:
The open-ended nature of the 3-1-1 distribution may be troubling for some test-takers who crave a higher level of certainty in their original setup. One possible approach would be to examine the entire range of possible solutions based on which thread color is not used. However, this would require the creation of five additional templates, as any one of the thread colors (except for P) could end up in the “not used” group. This is a situation where the cost of template-creation probably outweighs the benefits.