Setup and Rule Diagrams

[Dave Killoran]

• Setup and Rule Diagram Explanation

This is a Basic Linear: Balanced, Identify the Possibilities game.

The initial scenario for the game is relatively simple:

D98_Game_#4_setup_diagram 1.png

The first rule establishes that no car can be the same color as the car next to it:

D98_Game_#4_setup_diagram 2.png

The C designation in the not-block above stands for “color.” The CC not-block is a shorthand notation that indicates that no two cars of similar color can be adjacent, as required by the first rule. This representation saves the time of writing out PP, GG, and OO not-blocks. Note that in a game with only six spaces, and only three colors, this is a significantly more inhibiting rule than might at first be suspected.

The second rule is relatively easy to represent:

D98_Game_#4_setup_diagram 3.png

The third and fourth rules are represented as Not Laws under the diagram. Of course, since there are only three colors, removing one color from the options leaves only two color possibilities for that car:

D98_Game_#4_setup_diagram 4.png

The rule that states that car 1 cannot be orange leads to the important Not Law that car 2 cannot be green. Let’s examine why this is the case, using the two scenarios for car 1:

• 1. Car 1 is green. If car 1 is green, then car 2 cannot be green.
  
  2. Car 1 is purple. When car 1 is purple, then car 5 or 6 is also purple, and we can deduce that car 4 is orange. When car 4 is orange, then car 3 cannot be orange (from the rule that no two adjacent cars can be of the same color) and car 3 cannot be purple (because the two purple cars are either 1-5 or 1-6). Consequently, car 3 must be green. And, when car 3 is green, car 2 cannot be green.

Thus, regardless of whether Car 1 is purple or green, Car 2 can never be green:

D98_Game_#4_setup_diagram 5.png

An analysis of the diagram above reveals that the placement of the two P cars is restricted. Combining that with the first rule, the best approach is to attack the game by Identifying the Possibilities:

Three possibilities with P in 6:

D98_Game_#4_setup_diagram 6.png

Three possibilities with P in 5:

D98_Game_#4_setup_diagram 7.png

Thus, there are only six solutions to the game. With these solutions in hand, the questions are easy to attack.

[annabelle.swift]

Hi, when I first did this game, I did not get the inference that Car 2 can't be green if Car 1 can't be orange (I did eventually figure it out in Question 23). How did you know to explore the two scenarios for Car 1 (G or P) in order to infer that Car 2 can't be green?

After reading the setup solution, I also thought we could similarly infer that Car 3 can't be orange if Car 4 can't be green, like so:

LG Dec 1998 pt 1.JPG

If Car 4 is orange, Car 3 can't be orange since consecutive cars can't be the same color.

LG Dec 1998 pt 2.JPG

If Car 4 is purple, Car 5 can't be purple, so Car 6 must be purple. Car 1 must be green since the two purples have been used up. If Car 1 is green, Car 2 must be orange since consecutive cars can't be the same color and the purples are gone. Car 3 can't be orange if Car 2 is orange.


Is this a valid inference? If so, why was it not included in the setup solution? Was it just not relevant to the questions that followed? Thank you in advance and hope this wall of text makes sense!

[Adam Tyson]

That's a great inference, annabelle.swift, and totally valid! It's clear once all the scenarios are diagrammed that car 3 is never orange, so while we could have included that in our original setup, doing so just might have been a little bit of overkill. This game could have been approached with an eye on car 4 first, rather than car 1, and the same inferences and solutions would have emerged, leading ultimately to the same inferences. There is more than one way to attack this game, which is usually true of any game!

[bbjigglercakes]

• Setup and Rule Diagram Explanation

This is a Basic Linear: Balanced, Identify the Possibilities game.

The initial scenario for the game is relatively simple:

D98_Game_#4_setup_diagram 1.png

The first rule establishes that no car can be the same color as the car next to it:

D98_Game_#4_setup_diagram 2.png

The C designation in the not-block above stands for “color.” The CC not-block is a shorthand notation that indicates that no two cars of similar color can be adjacent, as required by the first rule. This representation saves the time of writing out PP, GG, and OO not-blocks. Note that in a game with only six spaces, and only three colors, this is a significantly more inhibiting rule than might at first be suspected.

The second rule is relatively easy to represent:

D98_Game_#4_setup_diagram 3.png

The third and fourth rules are represented as Not Laws under the diagram. Of course, since there are only three colors, removing one color from the options leaves only two color possibilities for that car:

D98_Game_#4_setup_diagram 4.png

The rule that states that car 1 cannot be orange leads to the important Not Law that car 2 cannot be green. Let’s examine why this is the case, using the two scenarios for car 1:

• 1. Car 1 is green. If car 1 is green, then car 2 cannot be green.
  
  2. Car 1 is purple. When car 1 is purple, then car 5 or 6 is also purple, and we can deduce that car 4 is orange. When car 4 is orange, then car 3 cannot be orange (from the rule that no two adjacent cars can be of the same color) and car 3 cannot be purple (because the two purple cars are either 1-5 or 1-6). Consequently, car 3 must be green. And, when car 3 is green, car 2 cannot be green.

Thus, regardless of whether Car 1 is purple or green, Car 2 can never be green:

D98_Game_#4_setup_diagram 5.png

An analysis of the diagram above reveals that the placement of the two P cars is restricted. Combining that with the first rule, the best approach is to attack the game by Identifying the Possibilities:

Three possibilities with P in 6:

D98_Game_#4_setup_diagram 6.png

Three possibilities with P in 5:

D98_Game_#4_setup_diagram 7.png

Thus, there are only six solutions to the game. With these solutions in hand, the questions are easy to attack.

My question is how does one have time to do 6 hypotheticals within 8 mins and 45 seconds... Untimed i could do it but timed at best i could answer 3/5 questions....
Hypotheticals in this game to me is the only way to attack the game...

Also the two hypotheticals that DONT work (not shown) would also take time to discard as being options

[Robert Carroll]

bbjigglercakes,

I think the fact that you got 3/5 timed is very encouraging, because the purpose of doing Possibilities upfront is that you'll take much less time on the questions. The fact that you were able to do 3 of the questions, combined with the fact that the questions basically involve no work and thus no extra time when all the Possibilities are written out, means that you were close to perfect on this anyway. Ultimately, 6 full Possibilities will solve all the questions, so it's fine to take all the time to do them. It's just about doing them quickly enough, which you seem to have been close to anyway.

Robert Carroll

[aleumass]

Hi! When I approached this game, I started with trying to find scenarios around rule 2 that car 5 or 6 must be purple. It was difficult to place things with this method. When I saw your explantion to start with Green or Purple as 1 since Orange can't be 1 made inferences a lot easier which led you to possibilities with P in 5 and P in 6. How can I come to that conclusion easier to start with what can go in 1 and start possibilities from there? Because trying to figure out the orders with P in 5/6 wasted a lot of time.

Thank you!

[Luke Haqq]

Hi aleumass!

It makes sense to start with slot 1 in filling in possibilities because the third rule narrows this slot down to only two possibilities, R or G. This is also true of slot 4 (which must be O or R), so that also could have been a place to start. Whether using slot 1 or slot 4, this would allow you to have two definite outlines, each using one of the two options. So in general, if you have a slot in a game that is filled with A/B, it's potentially worth unpacking the possibilities with A in that position and again with B. This might prove fruitful and enable you to figure out all the possibilities for a given game, or it might not yield many or any inferences. Even if it doesn't yield any inferences, this can still be useful time familiarizing yourself with a given game.

In addition, in this game it turns out that slots 1 and 4 restrict one another. This is because of the first rule that no two cars of the same color are next to each other. This makes the inference evident that slot 2 can't be G. This aspect of slots 1 and 4 restricting each other seems to be a reason why testing P in slots 5 and 6 might be less useful; when it's in slot 5, this combines with the fourth rule about car 4 and thus restricts the possibilities. But this is less the case when P is in slot 6, which doesn't directly connect to other rules. So in general, when diagramming out the possibilities, it's best to start with the most restricted variables--such as those in which there are only one of two possibilities, or specific variables that are mentioned in multiple rules.