Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

This is a Grouping/Linear Combination, Numerical Distribution game.

This game contains two variable sets: the six petri dishes numbered 1-6, and the three shelves labeled bottom, middle, and top. Although the petri dishes may appear to have an inherent order due to the 1-6 numbering, these dishes can be placed in any arrangement, and thus do not have a true order. The shelves do have an inherent order based on verticality, and thus the initial setup for the game should appear as follows:

With the basic structure of the game established, one immediate question is the assignment of petri dishes to the three shelves. The game scenario provides no information about how the dishes are assigned, and there is not even a minimum of one dish per shelf established. At this point, then, anything is possible, so you should expect the rules to provide some numerical information.

The first rule directly addresses the assignment of petri dishes, and establishes a maximum of three petri dishes on a shelf. This rule then allows for the following unfixed numerical distributions of the six petri dishes to the three shelves:

The remaining rules do not have further impact on the number of possible numerical distributions, and the three distributions above are the only distributions possible in the game. However, additional information can be determined about some of the distributions based on the rules. More on that later in this discussion.

The second rule creates a sequence between dish 2 and dish 6:

Accordingly, dish 2 can never be stored on the bottom shelf, and dish 6 can never be stored on the top shelf. These two Not Laws can be shown directly next to each shelf on the main diagram:

The third rule creates a rotating vertical block composed of dish 5 and dish 6, which can be diagrammed in different ways according to your preferences:

Because the block rotates, no Not Laws can be drawn. Note also that a 5 Not Law cannot be drawn on the middle shelf (as a result of the dish 6 Not Law on the top shelf), because dish 5 could be stored on the middle shelf and dish 6 could be stored on the bottom shelf. The one thing that can be deduced from this block is that either dish 5 or dish 6 is always stored on the middle shelf. We’ll show this fact as a dual-option on the final diagram.

The fourth rule creates a not-block. This not-block is shown horizontally because dish 1 and 4 cannot be on the same shelf:

Although this block could rotate, because the order on each shelf is not tracked in this game, the rotating aspect of the block is not relevant.

The prior diagrams accurately represent each of the rules and the resulting inferences. However, as mentioned previously, these rules do impact some of the distributions identified previously. Let’s discuss those implications before presenting a final diagram for the game.

• The 3-3-0 distribution
  In the 3-3-0 distribution, only two of the shelves are in use. Because dishes 5 and 6 are in a vertical block, the two shelves in use must be consecutive, meaning that the middle shelf is always used in this distribution (this inference answers question #9).
  
  Further, because dishes 1 and 4 are on different shelves, they are separated in the distribution with each taking a space in each group of 3, as are dishes 5 and 6:

• This means that dishes 2 and 3 must be on different shelves as well in this distribution:

• The 3-2-1 distribution
  This distribution is somewhat restricted because the variable pairs (1 and 4, 5 and 6) have limitations, but the number of solutions within this distribution is still fairly significant because the pairs can rotate. The questions test this distribution several times, so we’ll examine this distribution in more detail when it arises in the questions.
  
  The 2-2-2 distribution
  This distribution has a large number of possible solutions.
  
  All of this information combined leads to the final setup for the game:

Keep in mind that because the 3-3-0 distribution can occur, there does not have to be a dish stored on every shelf. With the setup in hand, let’s move to the questions.