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 Dave Killoran
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Setup and Rule Diagram Explanation

This is a Grouping: Defined-Fixed, Unbalanced: Underfunded, Numerical Distribution game.

The game scenario presents a situation where four people move three pieces of furniture, two people per piece. As there are only four people, and six positions to fill, some of the people must move more than one piece of furniture. This creates two possible unfixed numerical distributions of furniture to people: 3-1-1-1 and 2-2-1-1. Thus, either one person moves all three pieces of furniture and the other three move exactly one piece each, or two people move two pieces each and the other two move one piece each.


Rule #1. This rule creates a conditional relationship featuring a double-arrow:


Rule #2. This rule creates a simple conditional relationship:


Rule #3. This rule creates a negative grouping relationship, best shown with a vertical block as this provides the best visual impact:


As there is no order to the people moving the furniture, it does not matter if G is on top of J in this block, or if J is on top.

One of the effects of this rule is to impact the numerical distributions. In the 3-1-1-1 distribution, because G and J cannot move the same piece of furniture, neither G nor J can be the person that moves all three pieces. Thus, in this distribution, G and J must move one piece of furniture, and H or M must move three pieces of furniture:


In the 2-2-1-1 distribution, G and J cannot both move two pieces of furniture, otherwise there would be overlap and they would both have to move the same piece of furniture (this deduction is made via the Overlap Principle). Thus, at least one of G or J must move a single piece of furniture:


This information leads to the final setup for the game:

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The rule that stipulates "Grace helps move the sofa if, but only if, Heather helps move the recliner" (Rule #1) is confusing me.

I'm not sure why this translates into a double-arrow... Why isn't it simply a single directional conditional arrow like the second rule? Also, will all "if, and only if" sentencing structures predicate the double arrow?

Thanks, Joe
 Steve Stein
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Hey Joe,

Good question--the phrase "if but only if" is confusing for many students. In fact, I just recently wrote a blog post about this very issue (see ... ut-only-if).

As it turns out, "if but only if" is equivalent to "if and only if."
Consider, for example, the rule "A will go if and only if B goes"
This rule could also be phrased "A will go if but only if B goes"

Regardless which phrasing is chosen, we yield two basic rules:
A will go if B goes; A will go only if B goes.

Including diagrams, it becomes clear that this leads to the double-not arrow:

A will go if B goes (B :arrow: A)

A will go only if B goes (A :arrow: B)

Together the two rules yield the double arrow relationship: A :dbl: B

I hope that's helpful--please let me know whether this clears up this challenging conditional issue--thanks!

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That clears it up. Thanks!
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Hi PowerScore,
The explanation for this game makes reference to the Overlap Principle. I am taking the on-line full course and looking for help in locating the discussion of the OP in the lecture materials.

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Thank you Dave!
Seriously - you guys and gals rock. Thanks so much for the (always) quick responses!

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Any way someone could post a full setup and diagram?
 Jonathan Evans
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Hi, Eveitch,

This game is pretty open-ended (i.e. not very deduction heavy), but a baseline setup could look like the following:


I hope this helps!

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