Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping/Linear Combination: Defined-Fixed, Unbalanced: Underfunded game.

This is a difficult game, so do not be upset if you struggled. As mentioned previously in our books and courses, Unbalanced games are typically more difficult than Balanced games. It is incumbent upon you to always keep track of whether or not a game is Balanced, and then use that information to your advantage. For example, if this were the first game in a Logic Games section, the best strategy might be to skip this game and return to it later in the section. There would surely be easier games in the section; you could attack them first, then return to this more difficult game having already answered a number of questions correctly. For those students who took the December 1996 LSAT, it was fortunate that the most difficult game of the test appeared last in the section. Thus, most students had already successfully completed many of the easier questions in the section when they arrived at this game.

Let us take a moment to examine the Underfunded aspect of the game. There are seven products that must fill eight advertising periods (7 into 8). In order to compensate for this shortfall, exactly one of the products is advertised twice. This doubling produces a 2-1-1-1-1-1-1 numerical distribution. Regrettably, we cannot ascertain exactly which product is doubled, and this greatly contributes to the difficulty of the game.

Since the products are advertised over a four-week period, and two products are advertised each week, our setup will feature two slots per week, diagrammed in stacks:

d96_Game_#4_setup_diagram 1.png

Note that you do not want to draw out the 8 spaces on one horizontal line. Having two stacks creates a vertical component and that allows for better representation of the different types of blocks (e.g. the HJ block versus the blocks involving G).

Using this structure, most students diagram the game in a manner similar to the following:

d96_Game_#4_setup_diagram 2.png

Several of the rules are quite tricky. The first rule, which states that “J is not advertised during a given week unless H is advertised during the immediately preceding week,” can be partially represented as a block because of the “immediately preceding” qualifier. However, because of the “unless” portion of the rule, the block only occurs when J is present. Hence, the rule is diagrammed as a conditional statement with an arrow. When J is advertised, H must be advertised in the preceding week. This rule automatically means that J cannot be doubled because this would cause H to be doubled as well (and only one variable can be doubled).

However, the normal Not Laws that follow from a block do not occur in this case because of the fact that one of the products is doubled (and thus the problems created by the Underfunded aspect of the game begin). While it is true that J cannot be advertised during week 1, it is not true that H cannot be advertised during week 4. Since H could be the doubled product, H could be advertised during week 1 and week 4, for example (J would be advertised during week 2).

The second rule also produces two notable inferences. The rule states that the product that is advertised twice is advertised during week 4, but not during week 3. Thus, the only two options would be to advertise that doubled product on weeks 1 and 4 (1-4) or weeks 2 and 4 (2-4). Because the major point of uncertainty in this game is the doubled product, this information is very valuable. Since the doubled product must be advertised during weeks 1 and 4 or weeks 2 and 4, any variable that appears in week 3 cannot be the doubled product and therefore cannot appear in any week except week 3. By combining this inference with the last rule, we can infer that O cannot be advertised during weeks 1, 2, or 4. The second rule also allows us to infer that any product advertised during weeks 1, 2, or 4 cannot be advertised during week 3, since the doubling does not allow for week 3 to be used. By combining this inference with the rule involving K, we can infer that since K must be advertised during week 1 or 2 at the least, K cannot be advertised during week 3. K could still be advertised during week 4 since K could be the doubled product. Thus, a K Not Law cannot be placed on week 4.

There is still one critical inference yet to be uncovered, but because very few students discover this inference during the setup, we will continue on to the questions now, and then discuss the inference when it arises in question #20. The game diagram above is therefore only partially complete. We will fill in the rest of the diagram as we analyze the questions. We take this approach in an effort to more realistically deconstruct the way most students attack this game, and thereby provide more insightful and useful analysis. Of course it would be preferable for you to discover all inferences in a game before proceeding to the questions, but there will be times when this does not occur. How you react to that situation is just as important as your ability to make initial inferences.

[Annah]

Hello Dave,

In Game#6 on page 121, there are two conditional statements present in the question. The first states 'J is not advertised during a given week unless H is advertised during the immediately preceding week.' The third condition states 'G is not advertised during a given week unless either J or O is also advertised that week.'

In the explanation for this question the first condition has been diagrammed as J HJ stating that because of the 'unless' portion of the question the HJ horizontal block only occurs when J is present and the presence of H individually has no bearing on J.

However, for the second condition the diagram is portrayed differently. Would it not mimic the condition above and read
G G/J
OR G G/O (vertically)
- implying that similarly, as above, only when G is present does the G/J or G/O vertical block occur?

I would be grateful if you could elaborate on instances where conditional statements lead to blocks dependent on the occurrence of one variable and conditional statements where no such dependency occurs because in this specific question, the two statements are practically similar but yield different diagrams.

Thank you

[Dave Killoran]

Hi Anna,

Thanks for the question. As you note, the rules are similar, so we do want to explore that. Before looking at that, however, let's consider that in reality, every type of block representation could be presented conditionally. For example, consider a Basic Linear game where there are seven different variables each filling one of seven separate spaces. If you have a rule like "A is the variable immediately before B," we'd all show that as an AB block. But, there would be nothing wrong with showing that as A AB, or B AB. They both capture the same truth. Why don't we do that then? Because it wastes time. It takes extra time to write out "A " and "B ," and it's unnecessary, so we typically skip that. Now, as you know from our prior conversation over at http://forum.powerscore.com/lsat/viewto ... f=7&t=3754, when we have the possibility of multiples of certain variables, that changes the scenario and typically requires that the arrows be used.

Moving back to this game, we know there is the possibility of multiples in the game, but that doesn't actually create the difference here, interestingly. G can't be one of the multiples (see #21), and thus I removed that "G " for the sake of efficiency. If you did write it out in that fashion, there is no harm in having done so. In the JH rule, I would also have removed the "J " because J can't be doubled, except for one primary consideration: if you show this rule solely as a horizontal HJ block, almost everyone will immediately make Not Laws for J on week 1 and H on week 4, which would not be correct. That problem doesn't occur with the G rule, which features vertical blocks, and hence the difference. Thus, if we go to the root cause, although the two rules are similar, because one is a vertical block and the other a horizontal block, they aren't quite the same, and I accounted for that while setting up the game. Tricky nonetheless, and a good question overall!

Please let me know if that helps. Thanks!

[Annah]

I understand your explanation perfectly! Thank you so much for your reply!

[srcline@noctrl.edu]

Hello

I couldn't really get far into this game. I had a vertical setup:

W1:

W2:

W3: O

W4:

1st rule diagrammed as :
J adv. during the week H adv. imm. preceding week

3rd rule:
G adv during the week J or O adv during the week

So does G, J and O have to be together?

Also I am not understanding the second rule. Much appreciated if someone can explain the rules and set up of this game.

Thankyou
Sarah

[Clay Cooper]

Hi Sarah,

This is an interesting game. Thanks for your question.

To answer your first specific question - do G, J, and O have to go together? No - in fact, they cannot go together. In this game, exactly two products are advertised each week. That means we will have four groups of two variables each; it could never be the case that G, J, and O all go together.

I'll now answer your second point, that you had a hard time with the second rule, which says that whatever product appears twice must appear in week 4 and must not appear in week 3. It sounds like maybe you didn't catch the part of the setup in which we are told that of the seven products, all but one will appear exactly once, and the other one will appear twice. If you notice, we have seven variables, but four groups of two to fill - therefore, we need an eighth variable.

The setup and the second rule tell us: that this eighth variable will be one of our original variables, repeated; that it will appear in week 4; and that it will not appear in week 3.

Your setup looks fine except that I would add blanks beside each week to indicate that there will be two variables appearing there. For instance,

1 __ __
2 __ __
3 __ O
4 __ __

would be great.

Also, you have correctly diagrammed the two conditional rules you typed out. The third conditional rule, that G is not advertised in a week unless O or J is, would simply look like this, following the unless formula:

G is advertised in a week --> the other product that week is O or J

The fourth rule simply means that K must appear in either the first or second week; probably the most efficient way to indicate this rule at the outset would be by drawing not laws for K under weeks three and four. Then, since we know K must appear at least once, we know it must do so in week one or two.

That covers the setup and the rules. Does it make a little more sense now?

[yrivers]

I have a quick question on the rules. Rule 2: product advertised during two of the wks is advertised in wk4 but not wk3.

This gives us the inference that the repeats are:
1) Wk1 & wk4
2) Wk2 & wk4

Why not wk4 & wk4? Confused on this part since there's one slot in wk 4 that the repeat variable could fall into? Thanks!

[James Finch]

Hi Y,

The way I read the setup is that the word "pair" precludes having the same product advertised in both slots during the same week. Moreover, the second rule states that there is a product advertised during two separate weeks, not slots, and if that product filled both slots during week 4 it would still have to be advertised in either week 1 or week 2. That would mean that one of the products would not be advertised, which is contrary to our setup, as we would have one product taking up 3 slots and thus only 5 slots left for 6 variables.

Hope this clears it up!

[DesignLaw806]

Clay,

I noticed your explanation for rule 4, K is advertised in week 1 or 2 with not-laws for week 3 and 4. However, K can in fact be one of the doubled products in week 4 as explained in the initial setup post.

Hi Sarah,

This is an interesting game. Thanks for your question.

To answer your first specific question - do G, J, and O have to go together? No - in fact, they cannot go together. In this game, exactly two products are advertised each week. That means we will have four groups of two variables each; it could never be the case that G, J, and O all go together.

I'll now answer your second point, that you had a hard time with the second rule, which says that whatever product appears twice must appear in week 4 and must not appear in week 3. It sounds like maybe you didn't catch the part of the setup in which we are told that of the seven products, all but one will appear exactly once, and the other one will appear twice. If you notice, we have seven variables, but four groups of two to fill - therefore, we need an eighth variable.

The setup and the second rule tell us: that this eighth variable will be one of our original variables, repeated; that it will appear in week 4; and that it will not appear in week 3.

Your setup looks fine except that I would add blanks beside each week to indicate that there will be two variables appearing there. For instance,

1 __ __
2 __ __
3 __ O
4 __ __

would be great.

Also, you have correctly diagrammed the two conditional rules you typed out. The third conditional rule, that G is not advertised in a week unless O or J is, would simply look like this, following the unless formula:

G is advertised in a week --> the other product that week is O or J

The fourth rule simply means that K must appear in either the first or second week; probably the most efficient way to indicate this rule at the outset would be by drawing not laws for K under weeks three and four. Then, since we know K must appear at least once, we know it must do so in week one or two.

That covers the setup and the rules. Does it make a little more sense now?

[George George]

@DesignLaw806 I agree. It would be best practice to use the Slash Option for K in 1 and 2 (as the original setup explanation shows), e.g. K/ /K and not use the Not Laws for K under spots 3 and 4 (in case it repeats). Good catch!