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This is a Balanced Linear, Defined Game.
This is a basic linear game featuring six groups placed in six spaces, with exactly one group per space (a Defined, Balanced game). Games of this nature—ordered with the exact number of variables for the given spaces—are considered the easiest type of Logic Game, and this is a perfect way to start a Logic Games section. The only drawback is that there are just five questions in this game; it would be preferable to see a greater number of questions attached to such a simple game scenario.
Considering just the game scenario and the rules, you should make the following basic setup for this game:
Note that the third rule, which involves G, is represented by Not Laws on groups 2, 4, and 6, and this representation indicates that G can only be in groups 1, 3, or 5.
Although the diagram above captures the basic meaning of each rule, it does not capture the inferences created by the first and second rules (such as Not Laws, etc.). In fact, you have an interesting choice at this juncture of the game: you can either show all the Not Laws that result from the two blocks or you can show templates based on the placement of the blocks. Either approach will work, although the templates approach tends to be faster. Regardless, let’s show how each diagramming approach would unfold.
Approach 1: Diagram the Not Laws
Using the diagram above as a base, we can add Not Laws drawn from each of the first two rules.
Rule #1. This rule creates a large, flexible split-block involving P and M. Because there must be at least two groups between P and M, this block takes up a minimum of four spaces (leaving three options if the block is compressed as tightly as possible: groups 1-4, 2-5, or 3-6). Consequently, we can deduce that M can never appear in groups 1, 2, or 3, and we can deduce that P can never appear in groups 4, 5, or 6. Adding these Not Laws to the diagram, we arrive at:
Rule #2. This rule creates a fixed split block involving F and V. Because V is always two groups behind F, V can never appear in groups 1 or 2; because F is always two groups ahead of V, F can never appear in groups 5 or 6. Adding these Not Laws to the prior diagram, we arrive at:
What becomes immediately apparent from these Not Laws is that groups 2 and 6 are the most restricted, and each has only three options:
At this point in the game, you have diagrammed and considered all of the rules, so you should head towards the questions. And, since the meaning of the third rule is completely captured by the Not Laws in the diagram, you will simply need to focus on the first two rules (as an aside, of the first two rules of the game, the first rule is more problematic because it contains a degree of uncertainty—how many other groups are between P and M—that you must track throughout the game).
Approach 2: Diagram the Templates
The alternative approach is to diagram the game based on templates created by the blocks. Your first choice is which block to use as the basis for the templates. In this case, the choice should be easy: use the FV block created by the second rule. This block is the better choice because it is fixed, with exactly one group between F and V. Although the PM block is larger and takes up more space, it is an inferior choice because it is flexible, and the number of groups between P and M is not fixed; this flexibility creates more options, and ultimately, more templates.
Using the FV block, we can place the block in four positions: groups 1-3, 2-4, 3-5, and 4-6. The following diagram shows each scenario:
Of course, the other rules can also be applied to derive more information about each template. Let’s start with the third rule since it is more concrete than the first rule (thereafter, we will consider the first rule).
Rule #3. Since G must always be placed in group 1, 3, or 5, in Template #1 G must be placed in group 5 (groups 1 and 3 are already occupied by F and V). Likewise, in Template #3 G must be placed in group 1 (groups 3 and 5 are already occupied by F and V). Applying these two inferences, we can add G to Templates #1 and #3:
Rule #1. Because the block created by this rule is so large, it has somewhat limited placement options around F and V, especially in the Templates #1 and #3, which are more restricted now that G has been placed in each. Let’s examine the effect of the first rule on each template:
Template #1: Because P and M must be separated by at least two groups, in this template P must be placed in group 2 and M must be placed in group 6. The only remaining group is group 4, which must be filled by J, the random. Thus, this template has only one solution:
Template #2: The PM block has several options within Template #2. It can be placed in groups 1-5, 1-6, or 3-6. Consequently, this template will not fill in as completely as Template #1. Aside from the general position of P (group 1 or 3) and M (group 5 or 6), we can deduce that group 6 will be filled by J or M (from the initial Not Laws, group 6 cannot be filled by F, G, or P, and, in this template, group 6 cannot be filled by V, leaving only J or M). Adding all the information together, this template is still only partially complete:
Note that if G marches in group 1 or 3, that will create a chain reaction forcing P into the remainder of group 1 or 3. If P is forced into group 3, then M must be in group 6 (and J must be in group 5).
Template #3: As in Template #1, the placement of the PM block is limited in this template. Because P and M must be separated by at least two groups, in this template P must be placed in group 2 and M must be placed in group 6. The only remaining group is group 4, which must be filled by J, the random. Thus, this template has only one solution:
Template #4: At first glance it may appear that not much can be done with this template. However, the size of the PM block again leads to a useful inference. The PM block can only be placed in groups 1-5 or 2-5. Consequently, we can infer that M must always march in group 5 in Template #4, that P can only march in group 1 or 2, and that either G or J must march in group 5:
Compiling all four templates, we arrive at the following setup:
After applying all the rules, we have two very complete and powerful templates, and two other templates that contain a fair amount of information. We are now ready to attack the questions, and we will use the templates as they are more efficient that the Not Law setup.
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