Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

This is a Basic Linear: Balanced game.

The game scenario establishes that each of seven professors gives a lecture at seven different times:

P S T V W Y Z7

oct12_game_2_diagram_1.png

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

The first rule establishes the following sequence:

• P W

This sequence creates a P Not Law and a W Not Law:

oct12_game_2_diagram_2.png

The second rule establishes a second sequence:

• T S

This creates two additional Not Laws, a T Not Law and an S Not Law:

oct12_game_2_diagram_3.png

The third rule establishes a third sequence:

• V Z

This adds two more Not Laws, a V Not Law and a Z Not Law:

oct12_game_2_diagram_4.png

The fourth rule eliminates S from the last four spaces, adding four more Not Laws:

oct12_game_2_diagram_5.png

The diagrammatic representation of this rule varies depending on preference, but using Not Laws allows you to see that the first and last spaces are becoming increasingly restricted. Represented off to the side of the diagram, the rule would appear as:

• S 2, 3

The fifth rule adds a Y Not Law on 7:

oct12_game_2_diagram_6.png

Rules like this one can feel minor or inconsequential, but—when combined with the Not Laws from the prior four rules—it results in a limited selection of professors for the last lecture (W or Z):

oct12_game_2_diagram_7.png

The last rule is a bit more challenging, but remember that “if, but only if” introduces a bi-conditional relationship best represented with a double-arrow. The proper conditional diagram is thus:

• P1 Y V

You should also diagram the contrapositive, which is—naturally—also a bi-conditional statement:

• P1 V Y

At this point, each of the rules has been represented, and appropriate Not Laws drawn. Let’s now look at some inferences. The first inference comes from linking the second and fourth rules, because S appears in both of these rules. Since T lectures before S, but S lectures no later than third, we can conclude that T lectures either first or second, whereas S lectures either second or third. You can use T and S Dual Options to represent these inferences, or simply notate a T S sequence in brackets spanning the first three spaces on your diagram, remembering to add the corresponding T Not Laws:

oct12_game_2_diagram_8.png

Next, consider the restrictions placed by the first (P W) and the third rules (V Z) of the game, as they relate to T and S:

• If W lectured second or third, then P would have to lecture either first or second. However, this alignment would occupy two of the first three positions, leaving no room for T and S. Thus, W cannot lecture second or third.
  
  If Z lectured second or third, then V would have to lecture either first or second. However, this alignment would occupy two of the first three positions, leaving no room for T and S. Thus, Z cannot lecture second or third.

Adding these Not Laws to the diagram produces the following setup:

oct12_game_2_diagram_9.png

Finally, let’s take a closer look at the “if, but only if” double-arrow construction, and examine the restrictions it places on the game:

• P1 Y V
  
  P1 V Y

The conditional relationship between P, Y, and V leads to a complex, but critical Not Law: if Y lectured first, Y would be lecturing before any of the other six variables, including V. But, due to the bi-directional nature of the double-arrow, this Y V sequence would require P to lecture first, which is a logical impossibility since only one professor can lecture at any given time. To avoid this paradox, Y cannot lecture first. Consequently, only P, T, or V can lecture first.

Also note that the T S sequence must occupy two of the first three positions. So, if any professor other than T lectures first (i.e. P or V), then T must be second and S—third:

• P1 T2 S3
  
  V1 T2 S3

Adding the Y Not Law to the diagram, along with the rest of the inferences we’ve made, we arrive at the final setup for the game:

oct12_game_2_diagram_10.png

Some test-takers may also notice two additional inferences resulting from the last rule, which—admittedly—are too difficult to make under time pressure and are not necessary for attacking the questions. They are listed below for academic purposes only:

• If V lectured sixth, and Y never lectures seventh, it follows that Y must lecture before V. From the last rule, we would know that P must lecture first:
  
  V6 P1
  
  If V lectured second, it follows that Y must lecture after V (because Y never lectures first, as discussed above). But then, by the contrapositive of the bi-conditional rule above, P cannot lecture first. As a result, the only professor who can lecture first is T:
  
  V2 T1

[Lsatcat]

Hi instructors,

Are templates helpful in this game's set up -- with P in the first slot in one template, and T or V in the first slot in the other template? I was able to get all of the questions correct without templates, and simply by diagramming all of the not laws and connecting the conditional rules , but I wonder whether templates would have been more efficient.

Thank you!

Cat

[Jason Schultz]

Hi lsatcat,

In this case, templates could be useful. I generally look to create templates when you have more than one rule which creates an either/or situation. In this case, you have a couple.

Because of all the not laws created by the sequencing rules, either W or Z must be last.

Because S comes after T but cannot be later than 3rd, it must be 2nd or 3rd.

Either P is first, in which case Y>V, or it isn't, in which case V>Y

Combining the two of those rules limits the game to four possibilities (S2W7, S2Z7, S3W7, S3Z7) so you can build templates from them to guide the game.

S2W7: With S at 2, T must be one. This means P cannot be one, which also means V>Y

T S (V>Y/Z, P) W

S2Z7: Same as above.

T S (V>Y, P>W) Z

With S at 3 you start to get a little more complex because you don't know who is #1, T, V, or P.
P1S3W7

P T S Y V Z W

P1S3Z7

P T S (Y>V, W) Z

Setting up the templates for the remainder is probably more time than its worth, in my estimation. But when straying outside this group I would start by figuring out where the TS group goes, and who should be last.

[DrewKing]

Hi, can you assist me and explain rule 6 of the game. I do not quite understand the explanation above regarding the double arrows along with the contrapositive. What is that little line under the letter P? Very confusing, also which section of the online course books 1-12 discusses this best? Section 2 explains double arrows, but mostly in logical reasoning. Where in the book for logic games is it located?

[Emily Haney-Caron]

Hi Drew,

The line next to the P is a 1, indicating that Powell is in the first position. The double arrow is needed because the relationship goes both directions; if P is first, then Y is before V, and if Y is before V, then P is first.

[DrewKing]

No no, not the number 1, Under the letter P in the contrapostive's, the letter P is underlined, does that represent something?

[Adam Tyson]

Drew, that line is running THROUGH the P, not under it. It's a line of negation — if P is not 1st, then V is before Y, and vice versa. Sometimes our diagramming tools in this text-based forum get a little wonky, so the line of negation may look more like an underline. Think of it as a slash through the P — we just don't have a way to diagram that perfectly here.

[biskam]

I missed making the following not laws: Y under 1 and W and Z under 2 and 3, which made answering 7 and 8 because they took so much longer to solve under process of elimination.

So I missed making the following inferences: Next, consider the restrictions placed by the first (P W) and the third rules (V Z) of the game, as they relate to T and S:

If W lectured second or third, then P would have to lecture either first or second. However, this alignment would occupy two of the first three positions, leaving no room for T and S. Thus, W cannot lecture second or third.

If Z lectured second or third, then V would have to lecture either first or second. However, this alignment would occupy two of the first three positions, leaving no room for T and S. Thus, Z cannot lecture second or third.

I did this q a second time and still missed these inferences because I never would've thought to think while diagramming, what happens if W or Z is lectured second (or third).

In an effort to become better at inferencing, can someone please tell me how I should've identified this inference? I realize now that spaces one through 3 are limited to only three variables, and putting either W/Z second/third would force four variables into 3 spots, but can someone tell me how I should've seen this earlier? Is there a natural logic that I'm not clicking into where I should've seen that I should try to plug in w/z into 2?

Am I making sense? Basically trying to see if there's a general rule/phenomenon that I can spot in the future and therefore better diagram

Thank you!

[AthenaDalton]

Hi Biskam,

Thanks for your question! While it's not essential to find every inference up front, you can increase your chances of finding more inferences by taking a systematic approach.

Whenever you see a rule that says that "A must come before B" you can note that B cannot be placed in the first space on the diagram, and A cannot be placed in the last space. These are easy inferences to make in every game.

A similar approach can work here. Once you find that two of the first three spaces must contain T and S, you can list a few "not laws" that apply to those spaces. If a rule says "A must come before B" you can note that B (and any other letters that must come later in the sequence than another letter) cannot fit into spaces 1, 2 and 3.

Again, don't sweat it if you don't find every inference up front. It's common to find additional inferences as you work through the questions on a game.

Best of luck studying!

Athena

[nowornever]

Can someone please explain the P1 contrapositive?
I understand if P1 then Y-V, however if P is not in 1, this does not necessarily mean V-Y, does it?
Y-V could still occur if P was elsewhere.

What is the contrapositive to P1 -> Y-V

Thanks for everyone's help! Obviously I'm missing something.

_______

I think I may have just figured it out after I posted this. It has to do with the "if and only if" part.
Is there somewhere in the bibles or online that I can use to drill these "if and only if" and making contrapositives? Thanks!