Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping: Partially Defined, Numerical Distribution game.

The game scenario establishes that five students will review three plays. However, in a twist that increases the difficulty of the game, each student reviews one or more of the three plays. This turns the game into a Partially Defined scenario, and, prior to considering the rules, each student can theoretically review each of the three plays, and each play can be reviewed by all five students. However, the rules reduce the number of possibilities significantly.

From a diagramming standpoint, either the plays or the students can be chosen as the base. This type of situation has come up before (a good example is October 2001, Game #2), and it can be difficult to decide which should be the base (although, fortunately, the game can be done successfully with either variable set as the base). One way to help to determine the base is to glance at the rules—given the uncertainty of the numbers in each variable set, if the rules tend to define one group more than the other, that group will typically be the better base. In this case, the rules seem to focus on the students, and thus the students will be the base for the game, leading to the following initial scenario:

PT42-Dec2003_LGE-G4_srd1.png

The first rule directly addresses the numbers, stating that both K and L review fewer plays than M:

PT42-Dec2003_LGE-G4_srd2.png

Thus, M must review either 2 or 3 plays, and K and L must review 1 or 2 plays.

The second rule creates a negative grouping relationship between J and L, and J and M:

PT42-Dec2003_LGE-G4_srd3.png

The first rule establishes that M reviewed 2 or 3 plays. This rule indicates that J and M cannot have any reviewed plays in common. Thus, M cannot review all 3 plays (this would automatically create a violation of this rule), and therefore M must review exactly 2 plays (and J must review exactly 1 play). In addition, from the first rule, then, K and L must each review exactly 1 play:

PT42-Dec2003_LGE-G4_srd4.png

The third rule assigns T to both K and O:

PT42-Dec2003_LGE-G4_srd5.png

With T assigned to K, K is closed to reviewing additional plays.

The fourth rule creates two students with identical reviewing rosters. Because of the numerical distribution, those two students have to be either two of J, K, and L, or both M and O.

Two final notes:

O can review 1, 2, or 3 plays.

J and M must jointly review all 3 plays. Thus, if J reviews a play, M reviews the other 2 plays. If J does not review a play, then M must review that play. If M reviews a play, then J does not review that play.

With this information in hand, it is time to move on to the questions.

[deana]

Hi,

I set it up with STU as the base and JKLMOto represent the five students. I represented the rules like this:

K<M
L<M
These two rules represent the the first rule, kramer and lopez each review fewer plays than Megregian

L/J
M/J
these are for the second rule

And I just made a note that two review the same plays exactly.

Then I wrote K and O over the T since it said they review that play.

Outside of that I know there are relationships and inferences I am missing but can't find them. Especially during a timed section.

Thanks so much I know this is a huge question!

[Jon Denning]

Hey Deana - thanks for the question. For game 4, I think part of your difficulty likely arose from picking the plays as the base. When I set this game up and considered which base to use, I began to think about which base--plays or people--would give me more defined groups numerically. And what I found is that the rules actually tell you a lot more about the size of the groups that would be formed if the people are the base. Consider the first two: If M reviews more than K or L, and M does not review any that J reviews, then M must review exactly 2 (can't be all 3 because then it would share with J), and J K and L must review exactly 1. That's huge! You now know the exact number for 4 of the 5 people. Using them as the base then tells you a lot about group size, and that's a really critical piece when working with grouping games. Definitely something to keep in mind as you continue preparing.

The final two rules fill in T for K and O, and tell you that exactly two people must match. Now simply move through the questions making sure you follow the rules (M and J cannot share, for instance), and considering the restrictions we just imposed on group size. Go back and redo this game with people as the base and see if it becomes easier.


Okay well I hope that helps. If you are still struggling with individual questions feel free to let us know and we'll help you out with those too. Thanks!

Jon

[moshei24]

Hi!

The last rule states, "Exactly two of the students review exactly the same play or plays as each other."

Does that rule just mean that no matter what the end result is, two of the students need to review the same play(s)?

What would be a good way to diagram that?

Thanks!

[Dave Killoran]

Yes, it means there are exactly two students with identical reviewing rosters. Because of the numerical distribution, those two students have to be either two of J, K, and L, or both M and O.

Diagramming this rule isn't necessarily dictated by one path, so whatever way helps you best remember the rule is fine. That might mean writing "2 same" or "2 identical," or creating a block representation to indicate the sameness. The rest of the setup is so clearly defined that this is the only rule you really need to "off diagram," so to speak, so as long you know what it means, you should be fine.

Thanks!

[moshei24]

Thank you.

But J and L can't go together, so it would have to be either J and K or K and L from those 3. M can't be with any of them matched up. And can't O be 1 or 2 or 3? Meaning that it could also be J and O, K and O, L and O, or M and O?

Thank you!

[Dave Killoran]

Sorry, I should slow down when I write these answers In my haste I accidentally left out K and O on that original list.

Basically, there are four pairs that could satisfy the rule: JK, KL, KO, and MO.

JO and LO won't work because they would be based around reviewing only T, and since K already reviews only T, you'd actually have three reviewers reviewing identical play rosters (JOK and LOK).

Thanks!

[moshei24]

Oh, right.

My mistake. Thanks!

[rpark8214]

Hi,
I am a bit confused with the last rule: "Exactly two of the students review exactly the same or plays as each other."

For example, if Sunset was reviewed by MO, and Tamerlane was reviewed by KMO, would that not be exactly two students reviewing the same plays as each other? Or is the rule telling me that Sunset and Tamerlane must be reviewed only by the same two people. Thanks!

[AthenaDalton]

Hi rpark,

What this rule means is that exactly two students will review the same plays as each other -- so the focus should be on the students, not necessarily the plays. Another way of thinking of it is that the editorial board of this student newspaper can only give the exact same assignment to two students.

For example, this is one of the possible solutions to question 21 of this game:

J reviews: S
K reviews: T
L reviews: U
M reviews: T, U
O reviews: T, U

As you can see, students M and O review exactly the same two plays as each other. It's ok that these two plays (T and U) are also reviewed by K and L. We're only concerned with whether two students review exactly the same plays as each other, not whether certain plays get reviewed by two or more students.

Hope this helps!

Athena Dalton