Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Basic Linear: Balanced game.

The game scenario establishes that six slots at a benefit concert are filled by six different bands, leading to this basic setup:

PT70 -Game_#1_setup_diagram 1.png

The six slots are numbered, creating a Basic Linear diagram. Because there are six separate variables, and one for each of the six positions, this is a Balanced game.

One immediate consideration is the visual similarity between the variables U, V, and W. LSAC typically doesn’t put all three of these letters together as variables, so in your diagram you must be sure to write neatly in order to avoid any confusion while under time pressure. U and V in particular could be mistaken for each other if not written clearly.

With the basic structure in place, let us now turn to the rules.

The first rule creates a simple sequence:

PT70 -Game_#1_setup_diagram 2.png

This sequence creates two Not Laws on the diagram:

PT70 -Game_#1_setup_diagram 3.png

The second rule creates a slightly more advanced sequence:

PT70 -Game_#1_setup_diagram 4.png

This sequence adds four more Not Laws to the diagram:

PT70 -Game_#1_setup_diagram 5.png

Additionally, because both the first and second rules contain Z, they can be linked together to create a super-sequence:

PT70 -Game_#1_setup_diagram 6.png

This combination of the two rules creates two additional Not Laws. First, V cannot perform in slot five (because both Z and X must perform later than V), and second, X cannot perform in slot three (because V, Z, and W must all perform earlier than X). Let’s add both Not Laws to the diagram:

PT70 -Game_#1_setup_diagram 7.png

At this point in the setup, you should note two areas of restriction: X has been pushed back in the schedule and cannot perform in any of the first three slots, and the last slot has been reduced to having just three of the six bands as possible options. As new rules are added, continue to keep an eye on these two areas, and any new areas of restriction that appear.

The third and fourth rules are very similar, and in many respects understanding the effects of these two rules is the key to the game. The fourth rule limits Y to one of the first three slots, and the third rule limits U to one of the last three slots. The two rules can be represented as:

PT70 -Game_#1_setup_diagram 8.png

Each rule immediately creates three Not Laws, which can be added to the diagram:

PT70 -Game_#1_setup_diagram 9.png

Note that the Not Law limitations on slots one and six have created a triple-option on slot one and a dual-option on slot six.

The combination of the last two rules also produces two additional inferences:

First, because Y must be among the first three variables, and X cannot be among the first three variables, we can infer that Y X, and add that to the super-sequence:

PT70 -Game_#1_setup_diagram 10.png

Second, because Y must be among the first three variables, and U must be among the last three variables, we can infer that Y U, and add that to the super-sequence as well:

PT70 -Game_#1_setup_diagram 11.png

The above sequence addresses all six variables (and thus there are no randoms in the game). However, the sequence by itself does not fully reflect the scope of the last two rules, and thus the slot designators 1-3 and 4-6 have been placed on Y and U, respectively.

The interaction of the sequence with the last two rules also creates some powerful insights into the game. As mentioned previously, X must fill one of the last three slots. The third rule also establishes that U must fill one of the last three slots. Thus, the last three slots are filled by U, X, and exactly one other variable (not necessarily in that order). Let’s consider that information by examining just the six slots in the game, without all of the prior Not Law information:

PT70 -Game_#1_setup_diagram 12.png

Before considering the third variable that must fill the last slot above, let’s consider the grouping of the first three slots. From the fourth rule, Y must be among the first three slots. And let’s also take a look at V, which from the sequence is the most “front-running” variable. V must fill an earlier slot than Z and X, and since U performs among the last three, U must perform later than V as well. Thus, V can never be among the last three bands, meaning that V must fill one of the first three slots:

PT70 -Game_#1_setup_diagram 13.png

Of the remaining two variables—W and Z—one must then perform among the first three, and one must perform among the last three:

PT70 -Game_#1_setup_diagram 14.png

This is an incredibly powerful way to view the game, and it also simplifies the game by breaking it into two small, equal-sized pieces.

In addition, the slot limitations above add two new Not Laws to the diagram:

1. V cannot perform in slot four because, as noted above, V must fill one the first three slots.

2. X cannot perform in slot four. Either Z or W must perform in one of the last three slots, and from the sequence both Z and W perform earlier than X. Thus, X can never perform in slot four. Note that, because X cannot perform in any of the first three slots and now cannot perform fourth, X must always perform in slot five or slot six. This is an important restriction that plays a role in question #7.


Adding these last two Not Laws results in the final diagram for the game:

PT70-Oct2010_LGE-G1_srd.png