Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

This is a Basic Linear: Balanced game.

The game scenario establishes that a realtor shows five houses, one in each of five neighborhoods. Our only job is to determine the order in which the houses are shown:

PT72_Game_#2_setup_diagram 1.png

This creates a Basic Linear diagram, and because there are five variables for five positions, this is a Balanced game. With the basic structure in place, let us now turn to the rules.

The first rule establishes that R must be either first or second:

PT72_Game_#2_setup_diagram 2.png

This Split Dual-Option creates three Not Laws:

PT72_Game_#2_setup_diagram 3.png

The second rule establishes another Split Dual-Option: T must be either first or fifth:

PT72_Game_#2_setup_diagram 4.png

As with the previous rule, Not Laws can again be drawn:

PT72_Game_#2_setup_diagram 5.png

An alternative approach to this Split Dual-Option would be to use arrows:

PT72_Game_#2_setup_diagram 6.png

Whichever approach you take, avoid representing multiple Split Dual-Options in a way that could lead to erroneous inferences. For instance, your diagram should not look like this:

PT72_Game_#2_setup_diagram 7.png

A Dual Option in the first position is unwarranted, because there are other variables—besides R and T—that could potentially be placed there. Always represent your rules in a way that avoids misinterpretation, even if that requires a slight deviation from the norm.

The third rule establishes a Dual Option for Q or V in the third position:

PT72_Game_#2_setup_diagram 8.png

The final rule establishes a QS Not-Block:

PT72_Game_#2_setup_diagram 9.png

Thus, we arrive at the final setup for this game:

PT72_Game_#2_setup_diagram 10.png

There are more inferences in this game, but they require undertaking a Templates-based approach. Such an approach would be tremendously helpful in this game for several reasons:

• 1. The number of variables is incredibly small: the fewer the variables, the fewer solutions.
  
  2. The game is Balanced, with a 1-1 distribution between variables and available spaces.
  
  3. The rules are incredibly restrictive. Notice, for instance, the implication of the Q/V Dual-Option: if Q were third, and S and Q cannot be next to each other, then S must be either first or fifth, trading places with T. On the other hand, if V were third, exactly one of Q or S must be positioned on either side of V to avoid violating the last rule of the game. And, since R must be either first or second, the first two variables must be R and either Q or S, not necessarily in that order.

This line of reasoning produces the following templates:

• 1. If Q is third, then S and T will trade places at positions 1 and 5. Consequently, R must be second, and V—fourth:

PT72_Game_#2_setup_diagram 11.png

• 2. If V is third, it is best to split the solutions in two different directions, based on whether R is first or second. In either solution, we must ensure that S and Q are never adjacent to each other:

PT72_Game_#2_setup_diagram 12.png

PT72_Game_#2_setup_diagram 13.png

While a Templates approach typically requires a significant investment of time, here the benefits outweigh the costs by a wide margin. This is because there are only three templates, and each of them is restricted enough to render the creation of local diagrams in Questions #7, #9 and #11 unnecessary. Additionally, the templates reveal several key Not Laws, which would have been exceptionally difficult to infer without the templates: Q cannot be fifth, and V cannot be either second or fifth. These inferences help answer Question #10.

The game epitomizes the central proposition that is inherent in a Templates approach: you will spend a bit more time in the setup, but this time will be regained in the lightning-fast execution of the questions.

[LustingFor!L]

Was it wrong to set this up as a template game? I initially came up with the following,

1. R in 1 and Q in 3 --> Was able to see that it was not possible, because nowhere to put "S" without violating a rule.
2. R in 1 and V in 3
3. T in 1 and Q in 3
4. T in 1 and V in 3 --> Was able to see that there was no where to put "Q" and "S" without violating a rule.

Then I realized you could have R 2 and T 5. Ultimately, I can see its not really a template game, but it helped me make inferences and see what rules were driving the game. I was able to get all of the questions except #12, too time consuming.

What would have been a better approach/less time consuming setup?

[David Boyle]

Was it wrong to set this up as a template game? I initially came up with the following,

1. R in 1 and Q in 3 --> Was able to see that it was not possible, because nowhere to put "S" without violating a rule.
2. R in 1 and V in 3
3. T in 1 and Q in 3
4. T in 1 and V in 3 --> Was able to see that there was no where to put "Q" and "S" without violating a rule.

Then I realized you could have R 2 and T 5. Ultimately, I can see its not really a template game, but it helped me make inferences and see what rules were driving the game. I was able to get all of the questions except #12, too time consuming.

What would have been a better approach/less time consuming setup?

Hello,

There may be various approaches, but you may in fact be able to make templates, or even approach identifying all the possibilities.
First off, when we see that R can be first, in rule 1, and T can be first, in rule 2, that should set off some alarm bells of a good kind. ...If we put R in 1, then T can't be there, so it must be in 5. That alone narrows down things, but from rule 3, you also know that 3 must have Q or V. But if it had Q, then where would S go? since under rule 4, Q and S can't be adjacent. So, V must be 3rd, and Q or S must be 2nd or 4th.
Let's try a second scenario, where T gets to be first, and R has to be second. Can V go third? If it did, Q and S would be forced into 4 and 5, which violates rule 4. So, Q is third, V in fourth (separating Q and S), and S in fifth.
All that's left is putting R in second and T in fifth. If we do those and put Q in third, S can't be fourth, so it's back in 1, and V must be 4th.
Finally, if you put R in second and T in fifth, but now we put V in third, then Q and S can alternate between 1 and 4.

What you did above has some validity, and it even hits some of the points I hit; but you leap from either R or T alone, straight to what Q or V is doing. First map how R and T interact with each other, as I did, then you can go on to the Q/V issue. That really clears things up, as we can see.

David