Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is an Advanced Linear Game.

While there are three variable sets presented in the game scenario, the days of the week should be chosen as the base. Remember, when days of the week appear in a game, they should almost always be used as the base of the game.

With the days of the week as the base, two stacks can be created—one for the parking lots and one for the prices (again, leave ample vertical space between the two stacks for individual Not Laws):

O04_Game_#4_setup_diagram 1.png

Unlike the third game on this test, the setup for this game will not conclude with most of the spaces occupied. However, the rules still yield a fair amount of information, and several of the rules in particular require more than a surface analysis.

Rule #1. This simple rule results in a “15” being placed on Thursday in the Cost stack.

Rule #2. To capture the literal meaning of this rule, diagram it as follows:

$X > $Z

If X costs more than Z, then we can conclude that X cannot cost $10 and that Z cannot cost $15. Hence, X must cost $12 or $15, and Z must cost $10 or $12:

X 12/15


Z 10/12

The connection between X and Z is such that if X is $12, then Z must be $10, and if Z is $12, then X must be $15:

X12 Z10


Z12 X15

At this point, since the first rule has already established that the Thursday lot costs $15, we can infer that Anastasia cannot park in lot Z on Thursday, and that she must park in lot X or Y. That inference can be shown with a dual-option on Thursday:

O04_Game_#4_setup_diagram 2.png

Rule #3. This rule establishes a connection between the cost of the Wednesday and Friday lots, namely that the Wednesday lot must cost more than the Friday lot. The literal meaning of this rule can be diagrammed as:

$W > $F

Functionally, we can conclude that the Wednesday lot cannot cost $10 and that the Friday lot cannot cost $15. Hence, the Wednesday lot must cost $12 or $15, and the Friday lot must cost $10 or $12:

Wednesday 12/15


Friday 10/12

These inferences can be shown with by dual-options in the Cost row on Wednesday and Friday:

O04_Game_#4_setup_diagram 3.png

The connection between Wednesday and Friday is such that if the Wednesday lot is $12, then the Friday lot must be $10, and if the Friday lot is $12, then the Wednesday lot must be $15:


Wednesday12 Friday10


Friday12 Wednesday15


Rule #4. This rule establishes a numerical distribution for the lots. Like the prior two rules there is a simple diagram for this rule, but the relationships that follow from the rule prove more useful than the diagram of the rule itself. The rule can be diagrammed as:

#Z > #X

Numerically, then, there must be more than one day on which Anastasia parks in lot Z. And, because the game scenario establishes that Anastasia parks in each lot at least once, we can infer that she cannot park in Z more than three times (parking in X and Y one time each leaves a maximum of three days to park in Z). Thus, Anastasia must park in lot Z either two or three times, leading to the following two distributions:

• If Anastasia parks in lot Z three times, then she must park in lot X once and in lot Y once, creating the following fixed 3-1-1 distribution (Anastasia cannot park in lot X twice because then there would not be a day for her to park in lot Y):
  
  Z Z Z X Y
  
  If Anastasia parks in lot Z two times, then she can only park in lot X once, and the remaining two times she must park in lot Y, creating the following fixed 2-1-2 distribution:
  
  Z Z X Y Y

The two scenarios above are the only two possible distributions of the parking lots to the five days. Note that regardless of the scenario, Anastasia can only park in lot X a single time.

Because the fourth rule establishes fixed numerical distribution but does not allow us to place any of the variables within the diagram, the linear diagram created during the discussion of rule #3 is the final diagram for the game (excluding the diagramming of each of the rules, of course). In reviewing this diagram, you should note that Monday and Tuesday are completely open, and thus, when attacking the questions you should not initially look to Monday or Tuesday to solve a question. Wednesday, Thursday, and Friday are much better starting points for your analysis. And, although the linear diagram of the five days has a number of open spaces, the second, third, and fourth rules provide a wealth of information with which to attack the questions.

[Snomen]

Nowhere in the Rules and description does it say that the numbers are kinda fixed…, So how are we supposed to know that if X is 15 Y cannot be 15? It just says that they cost for the day.. Can anyone explain this ?
Thank you!

[Adam Tyson]

The scenario DOES fix the costs per lot, snomen! It reads "One of the lots costs $10 for the day, another costs $12, and the other costs $15." So we aren't just saying that on some days she pays 10 and on others she pays 12, etc. This part of the scenario ties each individual lot to a fixed cost. Don't forget that the scenario itself also includes rules and restrictions!