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This is a Basic Linear: Balanced game.
The game scenario establishes that seven paintings will be arranged in a row of seven positions. The “entrance” serves as a point of orientation. If it helps, you can jot it down to the left of the first position, as shown below:
This creates a Basic Linear diagram, and because there are seven variables for seven positions, this is a Balanced game. With the basic structure in place, let us now turn to the rules.
The first rule establishes the following sequence:
- T W
The second rule establishes a split-block, indicating that R and M are separated by exactly one space:
As with the previous rule, Not Laws can again be drawn:
The third rule establishes a rotating PS block:
Alternatively, you can represent this rule with a circle, as shown below:
Because either P or S can be closer to the entrance, this rotating block does not generate any Not Laws.
The fourth rule establishes the following conditional relationship:
By the contrapositive, we also know that:
As a result, V must be either third or fourth, creating a Split Dual-Option. Of course, V cannot be positioned anywhere else, which generates five additional V Not Laws (representing these Not Laws is optional, since V is already shown to be either third or fourth):
At this point, each of the rules has been represented, and appropriate Not Laws drawn. Let’s now look at some inferences. Consider that in this game there are two blocks—a rotating block and a Split block—which together occupy five of the seven spaces on our diagram. Additionally, we have a 2-variable sequence, and severe limitations on the placement of V. There is no overlap between the variables in those elements, so one can deduce that there must be some restrictions in their movements. Let’s examine the interaction between the two blocks as it relates to V:
The R _ M block requires three consecutive spaces, whereas the PS/SP block requires two consecutive spaces. Since V is limited to one of only two spaces, the convergence of these three elements creates an important inference:
- If either P or S were fourth, then V would have to be third, and the other half of the rotating PS block would have to be fifth. However, this alignment fully occupies three adjacent positions right in the middle of our diagram (third, fourth, and fifth), leaving no room for the R _ M block. Thus, neither P nor S can be positioned fourth.
There are more inferences in this game—question #20 makes that apparent—but figuring out that inference while under time pressure is exceptionally difficult, and so we will examine that inference when we arrive at question #20.
The incredibly restrictive nature of the rules also opens up the possibility of a Templates-based approach. Due to its size, the R _ M block can only occupy five distinct positions on the diagram, each of which stands to limit the positioning of V, and also determine—to an extent—the location of the rotating PS block. Also, note that P and S are interchangeable, since there are no rules that apply specifically to one of the two variables but not to the other. Therefore, a closer examination of the two options presented by the rotating block (PS or SP) is unwarranted.
One possible way to begin creating the Templates would be to examine the location of the R _ M block, because it is the rule that restricts the most variables (2) to the fewest number of spaces (5 each). By comparison, the rotating PS block restricts 2 variables to as many as 7 spaces each, whereas the V Dual-option restricts only 1 variable to 2 possible spaces. The R _ M block, therefore, provides the highest level of restriction, and should be analyzed first:
Next, we need to position the rotating PS block, while also ensuring that the remaining two variables—T and W—are sequenced in accordance with the first rule. Templates 1, 2 and 4 make this task relatively straightforward:
In Template 3, the rotating PS block can occupy two distinctly different positions (1—2 or else 6—7). This has affects T and W, allowing for two subsidiary templates (we will draw these out here for the purposes of full coverage, but during the game you would not want to spend the extra time drawing out both sub-templates since the PS/TW relationships is relatively clear):
Finally, in Template 5 the rotating PS block is allowed a greater degree of mobility. Nevertheless, you should immediately notice that it cannot occupy the third and fourth positions, because this would interfere with the positioning of V. Therefore, the rotating PS block can only occupy positions 1—2 or else 2—3, which allows for two additional subsidiary templates. Again, we will draw these out here for the purposes of full coverage, but during the game you would not want to spend the extra time drawing out each sub-template:
The full list of seven templates is recreated below:
While a Templates approach requires a significant investment of time and may appear daunting at first, it is ultimately advantageous in this game: First, each template is severely restricted, which makes it unnecessary to create local diagrams for any of the local questions (Questions #18, #19, #21, #22, and #23). Second, the templates reveal another important inference—namely, that W cannot be third—which immediately answers Question #20. Third, the optimal approach is to show just the five main templates (and to not show all the sub-templates). Last, 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 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. Ultimately, this approach is optional here, and you can successfully solve the questions without it. Consequently, our solution to each question will reference the relevant templates, but we will also discuss how to solve the questions without the use of templates.