Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping Game: Defined-Moving, Unbalanced, Overloaded, Fixed Numerical Distribution, Identify the Templates.

This is a Grouping game, with colors assigned to two distinct groups, the jacket and the overalls. One of the initial steps in any grouping game is to attempt to establish, if possible, the number of elements in each group. In this instance, it takes a combination of the game scenario and rules to determine how many elements are in each group.

The second sentence of the game scenario establishes that one of the two items will always be entirely one color, and then the first two rules establish the number of colors in each costume piece if the piece is plaid. Each rule creates a separate fixed numerical distribution:

• Distribution #1
  
  This distribution comes from the first rule, which states that “If the jacket is plaid, then there must be exactly three colors in it.”
  
  The rules states that if the jacket is plaid, then it must have exactly three colors. From the second sentence in the game scenario, then, the overalls must be exactly one color. This creates a 3-1 fixed distribution:

Dec 06_M12 game #1_cr_game#1_setup_diagram_1.png

• Distribution #2
  
  This distribution comes from the second rule, which states that “If the overalls are plaid, then there must be exactly two colors in them.”
  
  The rules states that if the overalls are plaid, then they must have exactly two colors. From the second sentence in the game scenario, then, the jacket must be exactly one color. This creates a 1-2 fixed distribution:

Dec 06_M12 game #1_cr_game#1_setup_diagram_2.png

Because all the possibilities are established when each costume piece is plaid (or, alternately, when each costume piece is entirely one color), these are the only two possible distributions in the game.

With all the numerical possibilities established, you must now focus on the contents of each grouping. In this case, the contents are the colors, and the last three rules of the game address the colors of each piece of the costume. Accordingly, we will examine each of the last three rules against the two numerical possibilities.

• Rule #3
  
  This rule states that “The jacket and overalls must have exactly one color in common.” From a representational standpoint, an easy way to diagram this rule would be:
  
  Dec 06_M12 game #1_cr_game#1_setup_diagram_3.png
  However, this diagram, while useful, is not the best possible representation. A better approach would be to use internal diagramming and represent this rule directly on the diagram of each numerical possibility:
  
  Dec 06_M12 game #1_cr_game#1_setup_diagram_4.png
  In examining rules #4 and #5, we will keep in mind the operating effects of this rule.
  
  Rule #4
  
  This rule specifies that the colors of the jacket can only be green, red, and violet. In the 1-2 distribution, there is not an immediate impact from this rule (there will be an impact when this rule is combined with the rules #3 and #5). In the 3-1 distribution, however, since the jacket is a total of three colors, all three slots are filled:
  Dec 06_M12 game #1_cr_game#1_setup_diagram_5.png
  Note that at this point we have not yet considered the effects of the last rule, but even so we are not going to fill in the remaining slots with options that will ultimately be impossible. For example, in the 3-1, from rule #3 and rule #4, one could assume that the overalls must be either green, red, or violet. While this is true from those two rules, rule #5 will alter those possibilities to only red or violet, and so we will refrain from diagramming further at this time in order avoid confusion.
  
  Rule #5
  
  This rule specifies that the colors of the overalls can only be red, violet, and yellow. A comparison of the color sets of the jacket and overalls shows that they share only two colors: red and violet. Thus, in the 3-1 distribution, because the jacket has already been determined to be green, red, and violet, the effect of rule #3 is that the overalls can only be red or violet:
  
  Dec 06_M12 game #1_cr_game#1_setup_diagram_6.png
  In the 1-2 distribution, the jacket must be red or violet (it cannot be green because it would not have a color in common with the overalls). Thus, one color selection for the overalls must be red or violet. The color selection for the overalls can be any color (other than the color the jacket and overalls have in common, of course):
  
  Dec 06_M12 game #1_cr_game#1_setup_diagram_7.png
  This representation, while helpful, is not entirely satisfactory because it creates the possibility that an interpretation mistake could be made (for example, accidentally selecting the same color for both color selections for the overalls while visually scanning the arrangement). Since the jacket can only be two colors, a better approach is to create a template for each color option:
  
  Dec 06_M12 game #1_cr_game#1_setup_diagram_8.png
  Overall, there are three templates containing six solutions: two solutions in the 3-1 scenario, and four solutions in the 1-2 scenario (two solutions when red is the common color, two solutions when violet is the common color).
  
  In reviewing the color contents of six templates, the controlling effect of the color sets and rule #3 becomes apparent: red and violet are featured prominently, whereas green and yellow do not appear as much. In fact, green and yellow cannot appear in the same costume together, an inference that is tested in question #4.

Combining all of the information above leads to the following optimal setup for the game:

Dec 06_M12 game #1_cr_game#1_setup_diagram_9.png

At this point we are ready to attack the game with confidence since the setup elegantly captures all six solutions to the game.