Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping: Defined-Fixed, Unbalanced: Overloaded game.

This game is quite difficult, in part because of the extra juggler available to fill the two groups, and in part because of the assignment of positions within each group.

The initial diagram for the game appears as follows:

J97_Game_#4_setup_diagram 1.png

The first rule establishes that G and H, if they are assigned to team, must be assigned to the front position. This means that G and H cannot be assigned to the middle or rear position on either team, inferences which can be shown as side Not Laws:

J97_Game_#4_setup_diagram 2.png

Note that this also means that G and H cannot be assigned to the same team, which will be shown in the final setup with a not-block.

The second rule establishes that K, when it is assigned to a team, must be assigned to the middle position. This results in K Not Laws on the front and rear positions:

J97_Game_#4_setup_diagram 3.png

The third rule indicates that L, if assigned, must be assigned to team 1. Thus, L cannot be assigned to team 2, and this is shown with a Not Law under the team 2 column:

J97_Game_#4_setup_diagram 4.png

Note that the rear position on team 2 is now so limited that only N, P, or Q could be assigned to it. This can be shown with a triple-option:

J97_Game_#4_setup_diagram 5.png

The fourth and fifth rules establish three global not-blocks:

J97_Game_#4_setup_diagram 6.png

The triple-option, the prior Not Laws, and these not-blocks can also be used to draw several interesting inferences:

• 1. P cannot be assigned to the front or middle position on team 2 because then there would be no juggler for the rear position on team 2 (the not-blocks would eliminate N and Q from being on team 2).
  
  2. If N is assigned to the front or middle position on team 2, then Q must be assigned to the rear position on team 2 (the PN not-block eliminates P from consideration).
  
  3. If Q is assigned to the front or middle position on team 2, then N must be assigned to the rear position on team 2 (the PQ not-block eliminates P from consideration).

Regrettably, these inferences do not play a significant role in the game.

The sixth rule presents an annoying conditional relationship that references exact team positions for H and Q:

J97_Game_#4_setup_diagram 7.png

Because H must be in the front position for team 1 or 2 if assigned, the contrapositive is also useful:

J97_Game_#4_setup_diagram 8.png

Thus, if Q is assigned to any position other than the middle position of team 1, and H is assigned, H must be assigned to the front position on team 1.

The prior information can be assembled to create the final setup for the game:

J97_Game_#4_setup_diagram 9.png

Note also that if any one juggler is unassigned, then the remaining six jugglers must all be assigned.

[parytownson]

I have a question about the contrapositive of Rule 6.

How did you deduce that if Q is NOT in the middle position of team 1, then H must therefore be in front position of team 1? Because it seems as though that is a mistaken necessary condition since the following hypothetical could exist:

Team 1

Front: L, Middle: N, Rear: P

Team 2

Front: G, Middle: K, Rear: P

As seen above, H is not present in this hypothetical. It is the missing juggler. With that being the case then, wouldn't this prove that "if Q is not in middle position of team 1, then H is on front position of team 1" is an incorrect contrapositive of Rule 6? As such, shouldn't the necessary condition instead be: "If Q is not in the middle position of team 1, then H is not on team 2?"

[parytownson]

I have a question about the contrapositive of Rule 6.

How did you deduce that if Q is NOT in the middle position of team 1, then H must therefore be in front position of team 1? Because it seems as though that is a mistaken necessary condition since the following hypothetical could exist:

Team 1

Front: L, Middle: N, Rear: P

Team 2

Front: G, Middle: K, Rear: P

As seen above, H is not present in this hypothetical. It is the missing juggler. With that being the case then, wouldn't this prove that "if Q is not in middle position of team 1, then H is on front position of team 1" is an incorrect contrapositive of Rule 6? As such, shouldn't the necessary condition instead be: "If Q is not in the middle position of team 1, then H is not on team 2?"

Typo:

I meant to say for Team 1 the positions are the follow:

Front: L, Middle: N, Rear: Q

[Robert Carroll]

pary,

You're correct! The diagram of the contrapositive doesn't make this clear, but the explanation does note that H is front of team 1 as long as H is assigned:

Thus, if Q is assigned to any position other than the middle position of team 1, and H is assigned, H must be assigned to the front position on team 1.

I made the relevant phrase bold to emphasize it.

Robert Carroll