Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

This is a Grouping Game: Partially Defined.

The scenario in this game lists six technicians, each one of which is repairing between one and three types of machines. Thus, each technician is a separate “group,” and one of the tasks of the game is to attempt to determine how many machines each technician repairs.

The rules contain a great deal of numerical information, which is not surprising given the uncertainty in the scenario about group sizes. We know from the scenario that each technician repairs at least one type of machine, so let’s look at each of the six rules and see what other information we can gather about group sizes:


Rule #1:

• This rule establishes that exactly four technicians (including X) must repair R.

Rule #2:

• This rule establishes that Y repairs at least two types of machines, T and V.

Rule #3:

• This rule starts off a run of four rules that all address the size of the groups in the game. In this case, S and Y cannot repair the same type of machine. Because each technician must repair at least one type of machine, this rule effectively eliminates S and Y from each repairing all three types of machines. When combined with the second rule, then, we can deduce that Y repairs exactly two types of machine, and those machines are T and V. Of course, if Y repairs T and V, then S cannot repair T and V, and thus S must repair exactly one machine, R.

The first three rules thus establish the following setup:

Dec 05_M12_game#3_L6_explanations_game#2_setup_diagram_1.png

Note that the circled numbers above S and Y indicate that those technicians are fixed at those group sizes. Of course, this is only a partial setup based on the first three rules. Let’s continue on by examining the remaining rules:

Rule #4:

• This rule establishes that Z repairs more types of machines than Y. Because we have already established that Y repairs exactly two types of machines, we can now infer that Z repairs three types of machines, and since there are only three types of machines in total, Z must repair R, T, and V:

Dec 05_M12_game#3_L6_explanations_game#2_setup_diagram_2.png

Rule #5:

• This rule is similar to the third rule, but in this instance W and S do not repair any of the same types of machines. We can thus infer that W does not repair R, and that W repairs either one or two machines (either T or V or both):

Dec 05_M12_game#3_L6_explanations_game#2_setup_diagram_3.png

Rule #6:

• The final rule establishes that U repairs exactly two types of machines. And, due to the constraints created by the first rule, we can determine that U must repair R, and that the second machine U repairs is then T or V. The only remaining point of indeterminacy in the game is X. We know X must repair R, but it is also possible (but not necessary) that X repairs one or two additional machines. This uncertainty is identified with a 1/2/3 notation above X:

Dec 05_M12_game#3_L6_explanations_game#2_setup_diagram_4.png

This last setup is our final setup, and note how much information the rules have provided. Although the game scenario did not establish individual group sizes, the rules provided enough information so that every group is defined except for W and X, and only the machines in U, W, and X are not perfectly determined. Thus, the game is Partially Defined, and we know that we must keep a close watch on U, W, and X in the game as these are the only points of uncertainty.

[Vexans]

Hello, can you explain how we can determine quickly that the people are the groups in this game. It's a bit tricky when people are far more often chosen as the variable rather than the group. When attempting it I designated the types of machines as the group and consequently missed the entire game.

[James Finch]

Hi Vexans,

The key to setting this game up correctly, using the technicians as the groups rather than what is being repaired, is looking at all the rules and thinking through them before beginning the diagram. The first rule suggests that we should use the RTV variable set as groups, but the rest are either as or more easily diagrammed using the technicians as a base, which then in turn makes it much easier to see the inferences than using RTV as a base.

My best advice for this and any other game is to take a moment and think about the rules before deciding on a diagram--always use a diagram that makes the most sense and is easiest to transfer the written rules into visual shorthand. And if you're having difficulty setting a diagram up, try another set-up, as the extra time setting up a second but better diagram will likely pay off by making it easier to correctly answer the questions.

Hope this helps!

[TargTru99^]

Greetings,

I have been watching the video for Lesson 6 of the live online course. At around 43:29, the third rule listed in the game is diagrammed in the lesson as S Y. I am simply not convinced that this is the correct symbolization of this rule. The third rule only states that any machine repaired by Y is not repaired by S. I cannot understand why it would be logically true that any machine repaired by Stacey cannot be repaired by Yolanda, nor can I understand why Yolanda cannot repair the radio as a result. This seems to me like a Mistaken Reversal being made. Please help me out.

[Stephanie Oswalt]

Hi Targ,

Thanks for the question! I have moved your post to the thread discussing the topic. Please review the above full explanation and related discussion, and let us know if this helps, or if you still have additional questions! Thanks!

[Dave Killoran]

A question from a student:

Hi Dave. I am taking your advanced logic games course. I am in Lesson 2 - Part 1 and I strongly disagree that the technicians should be the base. If you put R, T, and V as the base and use the rules, you can actually complete almost the entire game before going to the questions. The only variables outstanding will be U, which must either repair T or V; and W which could repair T, V, or both. What am I missing?

Also, X could repair T or V or neither. This is referring to December 2005 Game 3. I don't have the game itself so I can't go to the questions and work them using the two separate bases.

Hi,

While any variable set can technically be used as the base, there are often certain reasons why one base would be superior to another. The Technicians base in this game has a number of strengths, so I'd say that labelling it something similar to "strongly disagree" is not a position I'd agree with Let's look at some of the strengths of the Technicians' base first:

1. Creates a very strong numerical distribution where four of the six technicians are fully determined, and of the remaining two one is partially defined. This occurs in part because each technician must be between 1 and 3 machines, which creates a nice way to control numbers and see what's possible.
2. There ends up being a relatively small amount of uncertainty in the setup, which is perhaps the best suggestion that this is a viable and strong approach.
3. If you review the questions, I believe most are easier answered with the Technicians base.


The above are fairly compelling to me, and so I would never disagree with someone who chose to set the game up this way. It will work, and quite successfully!

That said, I also believe that each person will have personal preferences that makes things easier for them. Maybe that's the case here for you, and if so, my opinion is more or less irrelevant. I always say that if something feels right and it works, go with it

For me, using the Machines as the base doesn't give me the initial clarity of numerical control. Yes, the first rule is super helpful and fixes Radios at 4, but afterward you have to work harder imho to track things like Yolanda is fixed at 2, and to me it's harder to "block her off" at 2 than it is in the Technicians base. Essentially, the Machines leaves more things floating around for you to deduce and note whereas the Technicians base doesn't. Remember, though: these bases both will show the same information ultimately (since they are about the same game), and ultimately it's a matter of which is better to attack the questions with me (which is true of any game). For me, that's the Technician's base solidly.

Thanks!

[Adam Tyson]

Thanks for the question, TargTru99^! Consider that rule first as given - anything that Y repairs, S cannot repair. You might diagram that this way:

Y S

What's the contrapositive? Reverse and negate, and you get:

S Y

That type of rule, with a positive Sufficient Condition and a negative Necessary Condition, is what generates the double not arrow that we used in the lesson:

Y S

It's not a Mistaken Reversal at all, but a contrapositive! S repairs Radios because S cannot repair TVs or VCRs. Since S repairs Radios, Y cannot repair them, per the rule as diagrammed above. If Y had Radios, they both would have them, and S would be repairing something that Y repairs!

Remember to consider the contrapositive of any conditional rule by reversing the order and negating the terms. When the Sufficient Condition is positive and the Necessary Condition is negative, you can use the double not arrow as a shortcut to mean "these two can never be together" or "these two can have nothing in common."

[ArizonaRobin]

A question from a student:

Hi Dave. I am taking your advanced logic games course. I am in Lesson 2 - Part 1 and I strongly disagree that the technicians should be the base. If you put R, T, and V as the base and use the rules, you can actually complete almost the entire game before going to the questions. The only variables outstanding will be U, which must either repair T or V; and W which could repair T, V, or both. What am I missing?

Also, X could repair T or V or neither. This is referring to December 2005 Game 3. I don't have the game itself so I can't go to the questions and work them using the two separate bases.

Hi,

While any variable set can technically be used as the base, there are often certain reasons why one base would be superior to another. The Technicians base in this game has a number of strengths, so I'd say that labelling it something similar to "strongly disagree" is not a position I'd agree with Let's look at some of the strengths of the Technicians' base first:

1. Creates a very strong numerical distribution where four of the six technicians are fully determined, and of the remaining two one is partially defined. This occurs in part because each technician must be between 1 and 3 machines, which creates a nice way to control numbers and see what's possible.
2. There ends up being a relatively small amount of uncertainty in the setup, which is perhaps the best suggestion that this is a viable and strong approach.
3. If you review the questions, I believe most are easier answered with the Technicians base.


The above are fairly compelling to me, and so I would never disagree with someone who chose to set the game up this way. It will work, and quite successfully!

That said, I also believe that each person will have personal preferences that makes things easier for them. Maybe that's the case here for you, and if so, my opinion is more or less irrelevant. I always say that if something feels right and it works, go with it

For me, using the Machines as the base doesn't give me the initial clarity of numerical control. Yes, the first rule is super helpful and fixes Radios at 4, but afterward you have to work harder imho to track things like Yolanda is fixed at 2, and to me it's harder to "block her off" at 2 than it is in the Technicians base. Essentially, the Machines leaves more things floating around for you to deduce and note whereas the Technicians base doesn't. Remember, though: these bases both will show the same information ultimately (since they are about the same game), and ultimately it's a matter of which is better to attack the questions with me (which is true of any game). For me, that's the Technician's base solidly.

Thanks!

Dave,

Thank you for taking the time to answer this question for me. I went back to the lesson and set it up both my way (RST base) and your way (technicians base) to compare them. I don't have access to the questions so I can't judge on that part but my goal was to see how accessible the information would be to me once I set it up both ways.

For me, blocking Yolanda at 2 was a simple not law and the fact that Stacy can't repair any of Yolanda's machines made that part easy. I also found it easy to just put Zane into all 3 since Yolanda was repairing 2 machines. Over the course of the rules, it was pretty simple to deduce the 4 technicians who would work on R. I found these rules equally simple to maintain on both setups.

The big difference for me was after all the rules are displayed on the diagram. There is uncertainty for 3 variables, U, W, and X and both W and X are undefined on the amount of machines they will repair. On my setup I was able to put W and X over to the side and put U in the diagram since U has to repair one of T or V. On your setup I was able to show all of this on the diagram itself.

After going through all this, it became apparent to me that using the technicians base as you suggested is the superior way to diagram. That is because everything is shown on the diagram. All those extra seconds that I would spend on each question addressing the uncertainties with a machines base would add up over the course of the game. With the technicians base, I was able to show the uncertainties so I could stop thinking about it when I got to the questions.

This experience was somewhat mind-blowing to me. Your suggestion for the technicians to be the base was so foreign to the way that I would normally do things that it seemed really wrong, especially when I had another way to do it. However, I struggle with timing in the logic games section so I am going to train myself to think about the base selection in the way that you're suggesting. Thank you for taking the time to work with me on this. I think it will make a huge difference to me on test day!

[Mmjd12]

Hi,

I understood the second rule to mean that Y could repair both Televisions and VCRs. In other words, Y could repair T or V or both, but doesn't necessarily have to do both. How can you tell by the language used in the rule that that's what was meant?

[Dave Killoran]

Hi,

I understood the second rule to mean that Y could repair both Televisions and VCRs. In other words, Y could repair T or V or both, but doesn't necessarily have to do both. How can you tell by the language used in the rule that that's what was meant?

Hi M,

In this case it is because there are no exceptions made to the "both," meaning there's no "or" or "either" or anything synonymous to those ideas there. It's just a flat statement that Yolanda repairs both: "Yolanda repairs both televisions and VCRs" (italics added).

Thanks!