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Setup and Rule Diagram Explanation
This is a Grouping: Defined-Fixed, Unbalanced: Underfunded game.
This game marks an increase in difficulty in this section, primarily because this game is not as clean as the prior two.
The game begins by introducing two separate variable sets:
- Six Meetings over two semesters: Fall: 1, 2, 3; and Spring: 1, 2, 3
Five Cities: H, M, O, T, V
The six meetings might initially appear to have a sense of order, but as it turns out, the Fall/Spring Groups are key, and the Linear aspect is not relevant. Thus, those two semester groups can be selected as the base, and the diagram can be represented vertically or horizontally. Here's the setup appearing vertically:
- ___ ___
Let's take a moment to analyze the effects of that last point. Since every city must appear at least once, as soon as you know the the three cities in either semester (and thus that semester is filled), you automatically know two of the cities in the other semester. So, for example, let's say you knew that H, M, and O hosted in the fall; because T and V must also host, they must automatically be two of the cities in the spring. This is quite powerful when making hypotheticals in this game, and helps solve questions like #15 and #17 (where knowing 2 of the 3 then leads to a limited number of solutions).
With the base diagram in place, let's analyze the rules individually. Afterwards, we will dive into some of the interactions that lead to inferences:
- Rule #1: This is a conditional rule that establishes that when H hosts in the fall, then M must also host in the fall:
H fall M fall
The immediate thought for many students upon seeing a conditional rule in a game with two groups is to take the contrapositive, which would be:
M fall H fall
Of course, if this is a game with two groups, can't we apply two-value system thinking and change "not fall" into "spring" in the second diagram? Regrettably, no. This is not a two-value system game because you can have repeating variables, and thus, as my colleagues have jointly put it: "It's not a true two-value system, because there are three options for any given variable: (1) meeting in Fall (but NOT in Spring), (2) meeting in Spring (but NOT in Fall), and (3) meeting in both." And thus, "M being in the Spring does not prove H is in the Spring, because M could be in the Spring AND in the Fall."
Understanding that this is not a true two-value game is critical to not being fooled into making false inferences!
Rule #2: This rule is similar in form to the first rule, and can be diagrammed as:
V spring T spring
The contrapositive is:
T spring V spring
Rule #3: This rule reserves a space in each semester for M or V:
- _____ _____
- _____ _____
The rules are fairly straightforward to represent (aside from the tricky non-two-value system), and at this point in the game you can either proceed directly to the questions, or you can consider templates. Templates could be built around the variations on satisfying M and V rule, or on the variations of what occurs with placing the H and V from the first two rules. The former approach is fairly time-consuming and rather open, and thus doesn't strike us as the best use of time. The second approach is more useful since some of the templates fill in almost completely, and is a reasonable choice for those who take it. In this case, we chose not to use the Template approach, but it is defensible; it comes down to the approach you prefer.
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