Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is an Advanced Linear: Unbalanced: Overloaded, Numerical Distribution game.

The game scenario discusses two variable sets: the music CDs and the ratings. Because the ratings have an inherent order, they should be chosen as the base, and the CDs should be stacked on top of the ratings.

Although the ratings have an inherent order, deciding whether to order them 1-2-3-4 or 4-3-2-1 can be difficult. For some students, the 1-2-3-4 order is the natural way to draw the diagram, as that aligns with the normal way linear games are presented. For other students, the 4-3-2-1 ordering makes sense, because the 4 star rating is the highest rating, and thus should be first. In our diagram, we will use the 1-2-3-4 ordering as that is the ordering that most students naturally tend to use (but should you choose the 4-3-2-1 arrangement, you can solve the game equally well).

Finally, because there are six CDs that fall into four ratings, this game is Unbalanced: Overloaded. In the discussion of the first rule we will explore the Numerical Distribution in this game in more detail. The initial setup for the game thus appears as:

PT54-Jun2008_LGE-G2_srd1.png

Rule #1. This rule addresses the numerical possibilities within the game. From the game scenario, we know that each CD receives exactly one rating, but that leaves many different possible Numerical Distributions. The first part of this rule establishes that each rating must be assigned to at least one CD, which by itself limits the number of CD-to-rating distributions to two (3-1-1-1 and 2-2-1-1, both unfixed). However, because the rule also limits the number of CDs to receive the same rating to two, the 3-1-1-1 distribution is eliminated, and thus the CDs must be spread across the ratings in an unfixed 2-2-1-1 arrangement.

Rule #2. This rule creates a horizontal block:

PT54-Jun2008_LGE-G2_srd2.png

This block yields two Not Laws: H cannot receive a one-star rating, and N cannot receive a four-star rating.

With the information from the first two rules, the setup now appears as:

PT54-Jun2008_LGE-G2_srd3.png

Rule #3. This rule establishes a vertical block:

PT54-Jun2008_LGE-G2_srd4.png

This block must be one of the “2s” in the 2-2-1-1 distribution that controls this game.

Rule #4. This is an unusual rule. If at most one CD is rated more highly than Q, then Q can never receive a rating of one star or two stars (if Q did receive one or two stars, then at least two CDs total would receive three and four stars, thus violating this rule). Therefore, Q receives either three or four stars.

If Q receives three stars, then in order to conform to this rule only one CD can receive a four-star rating (and thus the I/H/R block established in the third rule could not receive four stars). This inference can be diagrammed as Q3 41.

If Q receives four stars, then another CD can receive a four-star rating (but another CD does not have to receive a four-star rating). These two facts lead to an important inference: I can never receive a four-star rating. Let’s recap the pathway to this inference:

Q can only receive a three-star or four-star rating. If Q receives a three-star rating, then I cannot receive a four-star rating because then two CDs would receive a higher rating than Q (I and either H or R). If Q receives a four-star rating, then I cannot also receive a four-star rating because I requires either H or R to receive the same rating, and if Q, I, and H/R receive a four-star rating there is a violation of the first rule. Thus, I can never receive a four-star rating.

With this final inference in place, and the recognition that S is a random, we can create the complete setup for this game:

PT54-Jun2008_LGE-G2_srd5.png

[srcline@noctrl.edu]

Hello,

A bit confused on the set up for this game. I had a vertical setup:

onestar: _

twostar: _

threestart: _

fourstar: _

I diagrammed the rules as
H 1 more > N

HIF in block
or RIF in block
1 CD > Q

I dont think I have a grasp on the rules.

I Figured that Q could not be on the fourth star .

Thankyou
Sarah

[Emily Haney-Caron]

Hi Sarah,

I'd do a horizontal setup here, because we'll need to have two positions available for each rating, and horizontal makes that a bit easier to represent.

So you want 1, 2, 3, and 4 going across, each with 2 spaces above it, with the knowledge that possibly only one space will be filled.

The second rule, then, is an NH block.

The third rule tells us we have one of the two following blocks:

H R
I I

The 4th rule actually tells us Q cannot be first or second (because then more than 1 other would receive more stars than Q). Also, it tells us that if Q is in 3, then only 1 CD can be in 4 (since only 1 can receive more stars than Q).

With that setup, try to approach the questions again and see what you think.

[adlindsey]

I'm confused and could be wrong but isn't the "N" should be an "R" in

The third rule tells us we either have a block of:
H N
I OR I

H R
I OR I

I don't see an N in the 3rd rule.

[Dave Killoran]

Good catch! I think Emily made a simple typo there while transitioning from discussing the second rule (which contains N) to the third rule (which doesn't). I fixed it above so now it's correct.

Thanks!

[ray57]

Hello,

The last rule of this game states that "at most one CD received more stars than Quasi did"--does this mean that exactly one movie received more stars than Quasi? Or if a movie did receive more stars than Quasi then it could only be one movie?
As you can see the interpretation of this rule can make or break the game because if exactly one movie received more stars than Quasi then Q would have to have 3 stars. Thanks for your help.

Ray

[Rachael Wilkenfeld]

Hi Ray,

The rule means that if any CD received more stars than Q, there could only be 1 such CD. This means that Q is either in 4 (which or without another CD), or Q is in 3, with only 1 CD in 4.

Hope that helps clear things up,
Rachael

[emma4832]

I had trouble determining whether to set the CDs or the stars as the base here. I chose the CDs and had a really hard time with the game. Could someone provide some insight into why/how to decide to use the stars as the base here?

Thank you!

[Jeremy Press]

Hi Emma,

The stars are the best base here because this is a linear game with linear-style rules, and the stars are the "inherently ordered" variable set (meaning, 1 star is always less than 2 stars is always less than 3 stars, etc.). In linear games, the inherently ordered variable set always makes the ideal base.

If you were wondering how you should've picked up on the linear nature of this game, and the linear-style rules, check out especially these rules:

1. "Headstrong received exactly one more star than Nice did."
2. "At most one CD received more stars than Quasi did."

These rules are linear-style rules, in which the relationships between two variables are compared on a spectrum that runs from least to most. Given these rules, a linear setup will be ideal, and the "inherently ordered" variable set will automatically be chosen as the base.

I hope this helps!

Jeremy