Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Mapping—Supplied Diagram, Identify the Possibilities game.

This game might seem somewhat intimidating, but in reality it is just an exercise in connecting the dots. While at first there may appear to be a large number of solutions, two factors combine to limit the number of solutions to six total: the limited number of segments, and the limited ways those segments can connect the chalets. Let’s examine both factors in more detail.

There are only five total segments (or paths) that must be shown. Because two of the five line segments are already given (J-N and K-L), only three more remain to be drawn. With only three segments left, there are a limited number of possible combinations of those segments.

The three segments are also limited in the way they can be drawn because of the rule that disallows the paths to cross, and the rule that maximizes the number of connections to a chalet to two. Those two rules further reduce the number of possible solutions, and should suggest to you that one way to solve this game would be to Identify the Possibilities.

Let’s look at the initial setup to the game:

The first thing that should be apparent to you is that M is limited in the chalets it can connect to. Because of the actions of the first and fourth rules, M cannot connect to K, L, or O. Thus, M must be directly connected to either J or N, but it cannot be connected to both or that would be a violation of the third rule. At the same time, from the second rule M must connect to another chalet, so M must be connected to J, or M must be connected to N, but not both. This limitation is one of the keys to the game.

With the that limitation in mind, let’s examine the various solutions to the game:

Scenario Set 1: J and M connected

When J and M are connected, three segments are used, and two remain to be placed:

In this configuration, M is now “closed” to further connections, as is J (J is connected to two other chalets, and thus from the third rule J can not connect to any more chalets). Thus, only K, L, N, and O are available to connect the remaining two segments. From here, only four possible connections exist: K-N, L-O, N-O, and K-O. Any viable solution will use exactly two of these connections, so examine those possibilities and make sure they meet the rules.

1. K-N connections

When K and N are connected, N and K are then closed off to further connection (from the third rule), and so the only remaining possible connection is L-O:

2. L-O connections

When L and O are connected, L is then closed off to further connection (from the third rule), and so the only remaining possible connection is K-N (which we just covered above) or N-O:

3. N-O connections

When N and O are connected, N is then closed off to further connection (from the third rule), and so the only remaining possible connection is L-O (which we just covered above) or K-O:

4. K-O connections

When K and O are connected, K is then closed off to further connection (from the third rule), and so the only remaining possible connection is N-O, which was in the prior discussion.
Let’s now examine the next set of possibilities, the ones that occur when M and N are connected.

Scenario Set 2: M and N connected

When M and N are connected, three segments are used, and two remain to be placed:

In this configuration, M is now “closed” to further connections, as is N (N is connected to two other chalets, and thus from the third rule N can not connect to any more chalets). Thus, only J, K, L, and O are available to connect the remaining two segments. From here, only four possible connections exist: J-K, J-O, L-O, and K-O. Any viable solution will use exactly two of these connections, so examine those possibilities and make sure they meet the rules.

1. J-K connections

When J and K are connected, J and K are then closed off to further connection (from the third rule), and so the only remaining possible connection is L-O:

2. J-O connections

When J and O are connected, J is then closed off to further connection (from the third rule), and so the only remaining possible connection is K-O or L-O:

J-O and K-O

J-O and L-O

3. L-O connections

When L and O are connected, L is then closed off to further connection (from the third rule), and so the only remaining possible connection is J-O, which we just covered above. K-O is not a viable connection because then all six chalets would not be connected in a single, continuous path as specified in the scenario.

4. K-O connections

When K and O are connected, K is then closed off to further connection (from the third rule), and so the only remaining possible connection is J-O, which was covered in an earlier discussion.

So, there are only six possible solutions to the game:

[Dave Killoran]

The Forum doesn't have the tools to post replicas of each solution, but we'll post an image of the diagram at some point. Happily, there are only 6complete solutions to this game. The proper setup, then, is to show all 6 prior to starting the game.

Four of the solutions are relatively straightforward, just diagram the six chalets and draw lines between them according to the rules. The two hardest solutions feature a line from the bottom left chalet all the way across to the top right chalet (a big diagonal)—get those two, and the game becomes a lot easier.

Thanks!

EDIT: diagrams are now posted at the top of the thread!

[twendell]

Can this set up be provided?

[Jeremy Press]

Hi twendell,

Sure! See the pictures posted below for each option. Note that, with the exception of template #1 and #6, they're all derived from local questions. #2 corresponds to question 19. #3 corresponds to question 20. #4 corresponds to question 21. And #5 corresponds to question 24.

I hope this helps!

Jeremy

[twendell]

It does! I was missing those last two with the longer connections! Thanks

[saygracealways]

Hi PowerScore,

Thank you for posting the template possibilities on the forum. I understand all six of the template possibilities, but I am struggling to understand why my attached template is not possible?

It seems to abide by all the global rules (5 segments in total, each chalet is directly connected to another segment, no cross over segments, no more than 2 chalets per segment).

Could you please let me know what I'm missing here? I think this is why I can't seem to understand question #23? It seems to me that M can be connected to two other chalets if M-N, M-J, J-N, K-L and L-O. Am I making the same error here?

Last but not least, how do can I judge if I should Identify all the Possibilities before jumping into solving the questions? Personally, I thought that drawing them all out would take too much time, and so I drew out individual templates depending on the local questions, especially because I couldn't figure out that there are only 6 workable solutions to the game.

Thank you so much and hope to hear from you soon!

[juandresmc]

Hi PowerScore,

Thank you for posting the template possibilities on the forum. I understand all six of the template possibilities, but I am struggling to understand why my attached template is not possible?

WhatsApp Image 2020-05-05 at 10.42.31 AM.jpeg

It seems to abide by all the global rules (5 segments in total, each chalet is directly connected to another segment, no cross over segments, no more than 2 chalets per segment).

Could you please let me know what I'm missing here? I think this is why I can't seem to understand question #23? It seems to me that M can be connected to two other chalets if M-N, M-J, J-N, K-L and L-O. Am I making the same error here?

Last but not least, how do can I judge if I should Identify all the Possibilities before jumping into solving the questions? Personally, I thought that drawing them all out would take too much time, and so I drew out individual templates depending on the local questions, especially because I couldn't figure out that there are only 6 workable solutions to the game.

Thank you so much and hope to hear from you soon!

Hi saygracealways,

While I was doing this game I ran into the same problem you mentioned in your post. After reviewing the rules many times, I finally understood that the path must connect ALL 6 chalets; thus, your diagram is incorrect because it fails to connect all of them (it breaks them into “independent” groups).

Furthermore, taking this into account, that’s why (C) is the correct answer choice in question #23. If you connect M to both J and N, you would not be able to connect all 6 chalets (either you leave O or K-L segment out).

Hope this helps!

Regards,

Andrés

[Adam Tyson]

Just responding to say that Andrés has you covered here, saygracealways. Nice work, Andrés! The rules do require that all 6 chalets be connected to each other: "a single continuous path that connects all of the chalets."

While Identify the Possibilities does turn out to work here pretty nicely, I didn't see that myself at first. I handled it as more of a template game - Either M is connected to J, in which case J is all done and the next segment of the path has to go from N to either K or O, or else M is connected to N, in which case N is all done and the next segment of the path has to go from J to either K or O. L is sitting off in the corner and may be all done or else may connect to O. I chose to stop there and move to the questions, figuring that they would mostly be local and I would do the additional diagrams as called for. Fast and easy and we are off to the races!

That said, don;t be afraid to try a Possibilities approach if you don't see another way, AND if you get the sense that the number of possibilities is fairly limited. How many ways can we do this single, continuous path? I might not have said "6" right away, but I would have thought "not many, probably" and maybe gone that extra step. Experiment until you get a feel for when it's the right way to go.

[sakfi]

Hello,
How do we ensure that we have all the templates? Or is it possible to work out all the questions without necessarily having all the templates?
Thank you!

[Adam Tyson]

That's the trick with doing templates, sakfi: you have to recognize what it is that's driving them in the first place, which has to be something that has a few limited options. Then you build your templates around those limitations, so there's no way you can miss one.

Dave's approach at the top of this thread is technically not a Templates approach, but follows the more exhaustive Identify the Possibilities approach, in which you take every option to every possible outcome. Here, each time a connection is made, the remaining possible connections gets narrowed, and so you just need to follow up and chart them all. This is a rare approach, because most games have far too many possible solutions.

I described a less complete templates approach that is based on just two possible "starting positions": either M connects to N, or else it connects to J. There can't be any other option for M, so you can feel confident that you haven't missed something. You could also take them a little further and create 4 templates: when M connects to J, then N connects to either K or O, and when M connects to N, then J connects to either K or O. You might try connecting N to L, but you'll quickly discover that it won't work, because O will end up cut off from everything else. So that's another way you can be sure you have found all the templates: you try something else and it doesn't work!

Yes, you could do this game with no templates, but you should expect it take a very long time and involve a lot of work on the questions and a lot of time testing answer choices, and those are not good options. When in doubt, do a little more work up front to be better prepared once you get to the questions.