Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping: Defined-Moving, Unbalanced: Overloaded, Numerical Distribution game.

This is a very challenging game. Initially, the game looks like a standard Overloaded Grouping game:

D00_Game_#3_setup_diagram 1.png

Because each selection has two characteristics—type of stone (ruby, sapphire, topaz) and a specific name (F, G, etc)—there are two spaces for each of the six selections.

The first rule reserves at least two of the six selections for topazes:

D00_Game_#3_setup_diagram 2.png

Note that the rule is somewhat open-ended as it specifies that at least two of the topazes are selected, so the above diagram only represents the minimum that must occur.

The second rule is conditional:

• 2S 1R

Of course, if exactly two sapphires are selected, and exactly one ruby is selected, then the remaining three stones must be topazes:

• 2S 1R 3T

Thus, if exactly two sapphires are selected the six stone types are fully determined. More on this rule in a moment.

The third rule contains two negative grouping relationships:

• W H
  
  W Z

W and H are different types, so tracking this rule is a bit more challenging. W and Z are both topazes, and thus the maximum number of topazes that can be selected is three: X, Y, W/Z. In turn, this affects the second rule, which results in the topazes being selected. If the second rule is enacted, then the three topazes must include X and Y:

• 2S 1R 3T (X, Y, W/Z)

Note that because at least two topazes must be selected from the first rule, and W and Z cannot both be selected, we can infer that X or Y or both must always be selected.

The fourth rule is a simple conditional rule:

• M W

Of course, W appears in both the third and fourth rules, and combining those two rules leads to the following two inferences:

• M H
  
  M Z

Given that the first two rules address the number of each stone type in the game, a quick review of the numerical facts is worthwhile:

• Minimum 2 T (from the first rule)
  Maximum 3 T (W and Z won’t go together, making 4 impossible)
  Maximum 3 R (there are only 3 Rs)
  Maximum 3 S (there are only 3 Ss)
  2S 1R 3T (thus a 2-2-2 distribution is impossible)

Using these restrictions, the following Numerical Distributions can be identified:

D00_Game_#3_setup_diagram 3.png

The variety of distributions is one reason this game is difficult, but, fortunately, the distributions can be used to answer both question #14 and question #16.

Adding all of the information together produces the final setup:

D00_Game_#3_setup_diagram 4.png

D00_Game_#3_setup_diagram 3.png

[ivan.l99]

I admittedly drew five templates initially consisting of Wx, Wy, Xy, Xz, and Yz. This seems far off from what I should have done. With the set up, these questions were more manageable. I would not have thought to draw out those six distributions had I not referred to this forum. Any good way to avoid this in the future?

[Jeff Wren]

Hi ivan,

The short answer is that any time that a game has anything other than a simple 1 to 1 balanced numerical distribution (for example 7 runners placed into 7 positions with no ties), you should figure out the numerical distribution or distributions. For those balanced 1 to 1 games, the numerical distribution is so strait-forward/obvious that there's really nothing to "figure out."

To identify games with these more complex numerical distributions, look for games with an uneven number of variables to available spaces (overloaded or underfunded with repeating variables), games with subgroups (like this one with the types of jewels), games with rules about numbers (at least two topazes, more red balls than white balls must be selected, etc.).

As for templates, while it's true that templates often overlap with numerical distributions (often creating one or two templates for each distribution can be very helpful for some games depending on the number of distributions and the other rules/restrictions), they don't always overlap.

In this game, the large number of distributions (6 is on the high side, 2 or 3 is more common) combined with the different possibilities within those distributions is too many to be useful. Templates are generally ideal for games that have 2-3 major branches to them, (sometimes 4), more than that and you are risking spending too much time during the setup.

Numerical distributions and Identify the Templates are discussed in lesson 9 of The PowerScore LSAT Course and chapter 9 of "The Logic Games Bible."