Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Basic Linear: Partially Defined, Numerical Distribution, Identify the Templates game.

The game scenario presents a situation where Sukanya receives messages from three associates while on vacation. The second sentence provides numerical information about the messages, indicating that each friend leaves either one or two messages. Thus, initially, Sukanya must receive a total of three, four, five, or six messages. Because only one message is sent each day, and there are between three and six messages, the game is only Partially Defined. Partially Defined linear games are very rare, and they force you to consider all the numerical variations as you begin each question.

Rule #1. This rule indicates that L does not send the first message. Because there are only three associates who can send Sukanya a message, we can deduce that either H or J must send the first message. We can represent this as a dual-option on the diagram:

PT55-Oct2008_LGE-G2_srd1.png

The vertical bar between the third and fourth messages indicates that at least three messages must be received.

Rule #2. This rule requires that the first and last messages be from the same person. Because each associate must send a message (and thus at least three messages must be sent), if the first and last messages are from the same person then there must be more than three messages. Thus, we can deduce that at least four messages are sent:

PT55-Oct2008_LGE-G2_srd2.png

The vertical bar has moved over a space to indicate that there must be a minimum of four messages.

Of course, if the first and last messages are from the same person, and only four, five, or six messages are sent, this leads to only three basic templates:

Template #1—Four messages total:

PT55-Oct2008_LGE-G2_srd3.png

In this template, one person sends two messages, and the other two people send one message each (2-1-1).

Template #2—Five messages total:

PT55-Oct2008_LGE-G2_srd4.png

In this template, two people send two messages, and the other person sends one message each (2-2-1).

Template #3—Six messages total:

PT55-Oct2008_LGE-G2_srd5.png

In this template, each person sends two messages (2-2-2).

We will explore the template possibilities in more detail after we examine the last two rules.

Rule #3. This rule creates an HJ block, with the unusual stipulation that the block occurs exactly once within the received messages:

PT55-Oct2008_LGE-G2_srd6.png

Rule #4. This rule indicates that J sends exactly one message among the first three messages. Thus, for example, if J were to send the first message, J could not send the second or third message. Without considering the templates, at this point the diagram would appear as:

PT55-Oct2008_LGE-G2_srd7.png

However, the last two rules add enough additional information that exploring each of the three base templates is advisable.

Template #1A—Four messages total, H sends the first and last message.

From the second rule, if H sends the first message, H must also send the last message. To meet the third rule, J must send the second message, leaving L to send the third message:

PT55-Oct2008_LGE-G2_srd8.png

Template #1B—Four messages total, J sends the first and last message.

From the second rule, if J sends the first message, J must also send the last message. To meet the third rule, H must send the third message, leaving L to send the second message:

PT55-Oct2008_LGE-G2_srd9.png

Because only J or H can send the first message, the above two solutions are the only two solutions when four messages have been left.

Template #2A—Five messages total, H sends the first and last message.

When five messages are sent, the messages must be distributed in a 2-2-1 unfixed arrangement among the three associates. From the second rule, if H sends the first message, H must also send the last message. Thus, H cannot send any more messages.

To meet the third rule, J must send the second message. J cannot then send the third message due to the restriction in the last rule, and, because H has already sent two messages, H cannot send another message. Thus, only L is available to send the third message. The fourth message must be sent by either J or L:

PT55-Oct2008_LGE-G2_srd10.png

Template #2B—Five messages total, J sends the first and last message.

When five messages are sent, the messages must be distributed in a 2-2-1 unfixed arrangement among the three associates. From the second rule, if J sends the first message, J must also send the last message. Thus, J cannot send any more messages.

To meet the third rule, H must send the fourth message. From the last rule, the second and third messages cannot be sent by J, and so they must be sent by H or L, with either H sending one of the two and L sending the other, or with L sending both. At a minimum, L must send one of the two messages:

PT55-Oct2008_LGE-G2_srd11.png

Template #3A—Six messages total, H sends the first and last message.

When six messages are sent, the messages must be distributed in a 2-2-2 arrangement among the three associates. From the second rule, if H sends the first message, H must also send the last message. Thus, H cannot send any more messages.

To meet the third rule, J must send the second message. From the last rule, the third message cannot be sent by J, and, because H has already sent two messages, H cannot send another message. Thus, only L is available to send the third message.

The fourth and fifth messages must be sent by J or L, with J sending one of the two messages and L sending the other, creating a dual-option:

PT55-Oct2008_LGE-G2_srd12.png

Template #3B—Six messages total, J sends the first and last message.

When six messages are sent, the messages must be distributed in a 2-2-2 arrangement among the three associates. From the second rule, if J sends the first message, J must also send the last message. Thus, J cannot send any more messages.

To meet the third rule, H must send the fifth message. To comply with the numerical restrictions, the second, third, and fourth messages must be sent by H, L, and L, in some order:

PT55-Oct2008_LGE-G2_srd13.png

When combined, the six templates form the final setup to the game:

PT55-Oct2008_LGE-G2_srd14.png

[CEF]

Hi,

Can you please explain what kind of set up would be best on this game? Also, I struggled with #8, #10, & #11 but I think they may have been for lack of understanding the game. If you could please explain I would appreciate it!

[Nikki Siclunov]

Hi CEF,

This is an unusual game, because it is not immediately clear how many messages we're dealing with. So, first you need to figure this out. Each of 3 associates leaves at least one, but no more than two, messages. The minimum number of messages is therefore 3, and the maximum - 6. However, since the first and the last messages are from the same person, clearly we cannot have only 3 messages left. Therefore, we must have either 4, 5, or 6.

Given that Lula leaves neither the first nor the last messages, we have only H and J to leave these two messages. This provides us with two possibilities for each of the 3 numerical distributions:

4 messages:

J _ _ J

H _ _ H

5 messages:

J _ _ _ J

H _ _ _ H

6 messages:

J _ _ _ _ J

H _ _ _ _ H

The last two rules should be helpful in filling in some of the empty spaces Try it out.

[asalusti]

What is the setup for these two games? I think I completely misinterpreted the 2nd game.

Could someone show me what the set up for these games would look like?

Thanks
-Alaina

[Emily Haney-Caron]

Hi Alaina,

Thanks for posting! For this game, we know we have between 4 and 6 slots, so I might do something like this:

____ ____ ____ _ _ _ _ _ _ ____

We know there will be slots 1-3 and then a last slot; we may or may not have two additional slots in between those. You'd want to indicate in your diagram that the first and last slots will be the same, and will not be L. We also know there will be only one HJ block, so I'd add that to the diagram. You also want to make sure to include that J can only be in one of the first three slots. That's all there is to work with here; the rest of the game is really just tackling the individual questions. Not a lot of inferences!

[srcline@noctrl.edu]

Hello David,

I understand the set up and the numerical distribution , but I am lost on the inferences and none of the questions are making sense to me. Can you please explain some of these?

Thankyou
Sarah

[Nikki Siclunov]

Hi Sarah,

The optimal way to approach this game is by using the three numerical distributions to create a total of six templates: three templates where J is first and last, and three where H is first and last. See my explanation above. Each template is quite limited by the simple fact that there are only three variables we can work with, none of which can repeat more than twice. The only "inference" is that there cannot be only 3 messages, and also that in a 4-message setup, L can only leave one message.

Please review my explanation above and let me know if it helps you get started.

Thanks,