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This is a Grouping: Partially Defined, Numerical Distribution, Identify the Templates game.
This game is extremely challenging, and it requires you to fully understand the implications of some relatively standard rules.
In the game scenario, an artisan is creating three stained glass windows from a selection of five colors. Each color is used at least once, and each window contains at least two colors. This leads to an initial setup as follows:
However, a major point of concern within this scenario is the use of “at least” in relation to both the use of the colors and the number of colors within each window. The use of “at least” opens the door to many more combinations than would exist if the word “exactly” had been used in each instance. For example, without considering the rules, initially each of the three windows could be made from all five colors. This use of “at least” is the first sign that this game may be harder than average.
The first rule establishes that exactly one window will contain both green and purple glass:
Carefully note the wording in this rule however; it does not state that one of the windows will be only green and purple glass, just that exactly one of the windows contains both green and purple glass, which means that other colors could be present. Again, this allows for more options than would otherwise be the case.
The second rule is fairly straightforward, and it establishes that exactly two of the windows contain rose glass:
This rule means that our minimum color set is now: G O P R R Y.
The third rule is actually two rules in one, and each rule is a negative grouping rule. Because we have a vertical component to our Grouping diagram, we will show these two rules as vertical not-blocks:
One of the implications of this rule is that none of Y, G, and O can be used three times. Because each color must be used in at least one window, if, for example, O was used three times (once in each window), then Y could not be used in any window. Thus, Y, G, and O are each used in one or two windows, and this numerical fact plays an important role in the game.
The final rule has a powerful effect on the game. This rule contains a negative sufficient condition, and most students initially diagram it as follows:
Of course, the contrapositive of this rule is:
When both representations are considered together, the rule means that when one of O or P is not used in a window, the other color must be used. Because every window must then contain O or P (or possibly both), this can be represented directly on the diagram as an O/P option for each window:
Note that while P could be used in all three windows, O can only be used in at most two windows (see the discussion of the third rule above), so one window will always contain P and not O.
This is an incredibly powerful rule that must be accounted for in the making of each window, and by itself it answers question #8. Consider also what this rule means for Y. From the third rule Y cannot be used in a window that contains O, and thus whenever Y is used in a window, P must also be used:
While this inference is not directly tested in the questions (how unfortunate!), it does play a role in answering questions such as #12.
At this juncture, our setup appears as follows:
At this point, most students use this as the final setup and move on to the questions. And, with the information already derived from our cursory examination of the rules, they will answer questions #7 and #8 with relative ease. Plus, the inferences derived during our discussion will carry them to overall success on the game.
But, there is a further step that can be taken, and it is one that rests on recognizing that the specific window numbers are not that critical in this game. Take a moment to glance at the rules and the seven questions. You may notice that the composition of the windows is not attached to specific window numbers: none of the rules references a specific window number, and only question #7 references specific window numbers. Thus, we can also attack this came by considering the composition of each window irrespective of the window numbers. While this approach is not necessary for game success, it is worthwhile to consider from a strategic standpoint, and as an instructive lesson for future games that might be similar.
When the game is considered without referencing window numbers, the initial scenario would appear as follows:
But, some of the rules—such as the first rule—establish further restrictions on the color combinations, such as the following:
In addition, there is also the fact that Y is a very powerful variable in this game. Y must be used at least once, but it cannot be used in all three windows due to the third rule. Thus, Y must be used once or twice (and it correspondingly eliminates certain colors from being used and also requires P to be used). From this standpoint, the use of Y suggests a templating approach, one when Y is used once, and one when Y is used twice. Let’s examine both:
Template #1: Y used in exactly one window
When Y is used in exactly one window, then the GP block from the first rule must be used in a different window (remember, from the third rule G and Y cannot be used in the same window). The “third” window then retains the choice of O or P. The use of each color must still be tracked, and the second rule involving R must also be tracked. This template thus provides a base, but it still allows for multiple options.
Template #2: Y used in exactly two windows
This template provides a bit more detailed information. Two of the windows contain Y, and thus they also contain P. The other window contains the GP block from the first rule, as well as O. The use of each color must still be tracked, and the second rule involving R must also be tracked.
Is having the templates necessary to solving this game or even to solving it quickly? No, the rules and the resulting inferences will allow for success (thus, in the questions, we will generally use the rules to solve each problem and only occasionally reference the templates). But, again, this approach is slightly more efficient and gives you a better overall picture of the possible directions the game can take. Therefore, templates are useful to consider, especially because a higher percentage of recent games have hinged more on group composition rather than on particular group names or group numbers, and this feature at times benefits from a template approach.
Overall, this is a very challenging game, and it qualifies as one of the hardest games to ever appear on the modern LSAT. Almost every student struggles to complete this game within the time constraint, so do not be discouraged if you had trouble with this game. Instead, study each rule and its implications so that the next time the test makers attempt to use these ideas to create difficulty you will be prepared.