Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

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

The game scenario establishes that seven departments of a law firm are distributed over three floors of an office building. With seven departments for three floors, you should immediately watch for rules and conditions that help establish numerical guidelines for each floor.

While the game scenario establishes a maximum of four departments per floor, there is no clear stipulation that there must be a department on each floor. However, the rules work in a manner that establishes that there is always at least one department per floor.

The initial scenario appears as follows:

PT59_Game_#1_setup_diagram 1.png

The first rule establishes a horizontal PT block:

PT59_Game_#1_setup_diagram 2.png

The second rule creates a vertical block:

PT59_Game_#1_setup_diagram 3.png

The third rule stipulates that L must occupy a floor by itself:

PT59_Game_#1_setup_diagram 4.png

F and S are thus randoms.

These rules, while individually simple, interact in powerful ways. The most powerful interaction is between the blocks created in the second and third rules. Because the HI block is stretched across two floors, and L is alone on its floor, these blocks cannot overlap. Thus, there are only two basic scenarios that can occur:

PT59_Game_#1_setup_diagram 5.png

Thus, just from the interaction of these two rules, each of the floors must have at least one department assigned to it. Of course, the placement of L then affects all of the remaining departments, including the PT block, F, and S. Of particular note, the PT block is limited to two positions in each scenario, leaving just F and S uncertain (although limited too, of course, as they cannot be assigned to the same floor as L). More sub-templates could be made, but the rules seem clear enough that making those extra templates seems like it could be wasteful; if a scenario is needed in a question, you can simply make it at that time.

At the start of the analysis, the numerical restrictions were mentioned. Let us revisit those now that we have a better understanding of the game.

The numerical distribution in this game must account for seven departments assigned to three floors, with a minimum of one department per floor, and a maximum of four departments per floor. One of the floors must always have exactly one department (L). Thus, only two partially unfixed numerical distributions exist of departments to floors:

PT59_Game_#1_setup_diagram 6.png

With the two distributions identified, let us examine each more closely, as many of the variables can be determined within each distribution.

Distribution #1: 4 - 2 - 1

In the 4-2-1, the floor with 1 department is L, as L occupies an entire floor. Because H and I are on different floors, they then occupy one space within the floors with 2 and 4 departments. This forces the PT block onto the floor with 4 departments (as there is not enough remaining spaces on the floor with 2 departments), leaving F and S to rotate between the floors with 2 and 4 departments. Thus, the composition of each group is largely determined:

PT59_Game_#1_setup_diagram 7.png

Distribution #2: 3 - 3 - 1

In the 3-3-1, the 1 is L, as L occupies an entire floor. Because H and I are in different floors, they then occupy one space within the 3 and 3 groups. This forces the PT block onto one of the floors with 3 departments, leaving F and S to fill the other floor with 3 departments. Thus, the composition of each group is again largely determined:

PT59_Game_#1_setup_diagram 8.png

[SGD2021]

Hello,

What part of the game scenario/rules tells us that each variable can only go once? In other words, how do we know that there can't be a repeat of any of the variables? Since the scenario does mention that "no department is to be on more than one floor," would we just assume by common sense that there can't be, for example, two I departments on one floor, so that's why we immediately know there can't be repeated variables? Or is it from the distributions that we know there can't be repeated variables (so we'd have to write them out to know this)? In other words, what would be the quickest way to realize there can't be repeats of variables?

Thank you!

[Robert Carroll]

SGD,

The scenario says there are "seven departments" and then lists seven different departments. If, as in your hypothetical, there were more than one injury department, then we'd have at least eight departments - those two injury departments, and then the six remaining departments.

Robert Carroll