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Setup and Rule Diagrams

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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
D00_Game_#3_setup_diagram 1.png (6.97 KiB) Viewed 217 times

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
D00_Game_#3_setup_diagram 2.png (7.19 KiB) Viewed 217 times

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 :arrow: 1R

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

    ..... ..... ..... ..... ..... 2S :arrow: 1R :arrow: 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 :dblline: H

    ..... ..... ..... ..... ..... W :dblline: 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 :arrow: 1R :arrow: 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 :arrow: W

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

    ..... ..... ..... ..... ..... M :dblline: H

    ..... ..... ..... ..... ..... ..... M :dblline: 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 :arrow: 1R :arrow: 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
D00_Game_#3_setup_diagram 3.png (6.64 KiB) Viewed 217 times

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 4.png (17.57 KiB) Viewed 216 times

D00_Game_#3_setup_diagram 3.png
D00_Game_#3_setup_diagram 3.png (6.64 KiB) Viewed 217 times