Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

This is an Advanced Linear: Balanced, Identify the Templates game.

The game scenario establishes that each of five students—G, H, J, K, and L—works exactly two nonoverlapping shifts at a gallery over the course of five consecutive days, Monday through Friday. The base should be the days of the week, due to the inherent order of this variable set. 

Since each student must work exactly two of the ten shifts, we can create a Balanced setup by simply duplicating each of the five variables, as shown above.

Now let’s examine each rule.

The first rule indicates that no student works both shifts on any day. Visually, this rule is best represented with a vertical Not-Block. However, instead of writing five Not-Blocks (one for each student), we can simply use S (for student):

The second rule states that on two consecutive days, L works the second shift. This creates the following Block:

Note the L Side Not-Law on your main diagram:

The third rule is similar to the second rule, requiring G to work the first shift on two nonconsecutive days. Visually, this creates two vertical G Blocks that cannot be adjacent, which can be represented using a G Not-Block:

Note the G Side Not-Law on your main diagram:

The fourth rule stipulates that K works on Tuesday and Friday. Since we don’t know whether she works the first or the second shift on either day, we can overcome this uncertainty by placing K Not-Laws on Monday, Wednesday, and Thursday:

The fifth rule creates a vertical HJ Block. Since we don’t know the exact shifts that H and J work, the block is rotating:

With K working on both Tuesday and Friday, the HJ Block can only be placed on Monday, Wednesday, or Thursday. However, this does not warrant placing H and J Not-Laws on Tuesday and Friday. Why? Because each student must work on exactly two shifts; while H and J could work together on two separate days, they don’t need to: they only need to work on the same day as each other once.

The last rule prohibits G and L from working on the same day as each other, creating a vertical GL Not-Block:

At this point, all the rules have been diagrammed, and relevant Not Laws drawn. Your main setup should look like this:

There are more inferences in this game, but they would be hard to come by. For instance, did you notice that G and K must work, respectively, the first and second shifts on Friday? Or that J cannot work either of the two shifts on Tuesday?

Here’s a better strategy: instead of trying to infer additional Not-Laws, recognize the opportunity of a Templates-based approach! Of particular importance is the interaction between the second, third, and sixth rules, all of which concern G and L: First, the two second shifts assigned to L must be consecutive, whereas the two first shifts assigned to G cannot be. Moreover, G and L can never overlap, which limits the possible placements of the LL-block even further.

At first glance, there appear to be four possible ways to allocate the LL-block:

• 1. LL-block goes on Monday-Tuesday:
  
  To ensure that G and L never work on the same day as each other, G’s two nonconsecutive first shifts must fall on Wednesday and Friday. Thanks to the fourth rule, K must work on Tuesday (with L) and on Friday (with G). Last, remember you need a full day for the HJ-Block, which can only be placed on Thursday. We have 8 of 10 spaces filled, with only the first shift on Monday and the second shift on Wednesday open, and only H and J to fill them.
  
  
  2. LL-block goes on Tuesday-Wednesday:
  
  In this Template, G must work on Monday and Friday. (Note that G cannot work on Thursday, because we need a full day for the HJ-Block, and K must work on Friday). As in Template 1, K must work on Tuesday (with L) and on Friday (with G). The HJ-Block is again on Thursday. We again have 8 of 10 spaces filled, with only the second shift on Monday and the first shift on Wednesday still open, and only H and J to fill them.
  
  
  3. LL-block goes on Wednesday-Thursday:
  
  Here, the two nonconsecutive days on which G must work the first shift are Tuesday and Friday. (Note that G cannot work on Monday for the same reason it couldn’t work on Thursday in Template 2: we need a full day reserved for the HJ-Block, and with K already taking a Tuesday spot, placing G on Monday would mean that there is something taking a spot from HJ every single day.) K must work on Tuesday (with G) and on Friday (also with G). The only day on which H and J can work together is Monday.
  
  
  4. LL-block goes on Thursday-Friday:
  
  In this solution, the only two nonconsecutive days on which G can work are Monday and Wednesday. However, we run into a problem: The HJ block needs an empty day, and this leaves none. G works on Monday and Wednesday, K works on Tuesday and Friday, and L works on Thursday and Friday. So this fourth option is out!

The game epitomizes the central proposition that is inherent in a Templates approach: you will spend a bit more time in the setup, but this time will be regained in the lightning-fast execution of the questions. The templates also reveal several powerful inferences that would have been virtually impossible to make otherwise:

• 1. G and K work the first and second shifts on Friday.
  2. Either K or G works the first shift on Tuesday.
  3. Either K or L works the second shift on Tuesday.
  4. Either H or J works the first shift on Thursday.
  5. J never works on Tuesday or Friday.

Although this step is optional, you can now return to your main diagram and input the newly-discovered inferences before you proceed to the questions:

[Pragmatism]

So, I OJ Simpsoned the world out of rules around L and G. The way I interpreted them was as follows, “On two consecutive days, L works the 2nd shift.” Interpretation: If there is a student that works on 2 consecutive days, then that person ought to be L and be placed on the second shift. I did the same for G, thus made this game SIGNIFICANTLY harder than it ought to be, because getting that rule clarified this game became so much easier.

Question, how did you get that reading of the rule? Also, what would the rule have to say (not explicitly in conditional terms) for it to be interpreted the way I read that rule?

Thank you

[Francis O'Rourke]

Hi, Pragmatism!

The straightforward answer is that the phrase "on two consecutive days" tells you that something must happen on two days, one immediately after the other, but the particular days do not matter.

That is a bit abstract. So instead of thinking about how we can get that interpretation directly, it is easier to understand what this rule leaves out. I want you to imagine a slightly different and more common type of rule.

Say that you have a rule that states "X must work on day 1." This is an extremely common rule. It gives us a direct inference that X must be placed on Day 1, so you would likely fill in Day 1 with X. You would not create a conditional statement here, as that would just complicate the matter.

Now think about this rule: "on day 1 or day 2 Y must work." You have likely already seen this sort of rule numerous times as well. It creates a split option for us, where Y must work on one of those two days. You would likely not create a conditional statement here.

The difference between these rules is that the second rule gives us less certainty. The structure of the two statements is also reversed, but that does not change the meaning of the rule.

You probably see where I am going with this. Imagine that we have a rule with even less certainty. For example "Z must work on some day of the week." This is very minimal information. It merely tells us that we need to include Z at some point on our diagram. Once again, there is no conditional statement. The rule tells us that something is definitely happening.

The meaning of this last rule would not be changed if we reversed the order of the sentence. "Z must work on some day" would mean the same thing as "on some day, Z must work."

I suspect that the issue you had with this rule had to do with either the sentence's nonstandard syntax, or the small amount of certainty that it provided.

If the rule had stated "On Monday and Tuesday, L must work the second shift." I suspect that you would have simply filled in L for Monday and Tuesday second shift, instead of creating a conditional statement. It is important to recognize that this rule leaves out some common information, such as a specific day and conditional indicators.

Additionally, the rule looks somewhat more like a conditional statement with the "on two days," phrase appearing before the "L works the second shift" phrase. The order of information does not change the meaning of the rule.

I hope that this helps, but let me know if you have any further questions!

As for your final question, in order to make this a conditional rule as you interpreted it, I would expect wording to be extremely close to what you provided: "If there is a student that works the second shift on 2 consecutive days, then that person will be L."

[Pragmatism]

I LOVED how you clarified that for me so vividly. That makes a lot more sense now. Also, you were right about the syntax is what threw me off with that rule, but having that clarified this game and future games as such will become significantly easier then how I translated that rule. Honestly, if LSAT had made that game with that rule being a conditional statement, that game would have been a nightmare. I hope LSAT is not reading this and getting any ideas for future games.

[ThePabs_]

Doing four templates seemed very difficult at the time but makes so much sense now that it is laid out. How long should it be taking on average for a question like this to do the full setup?

[Jeremy Press]

Hi ThePabs_,

It's always a little dicey to be too precise about how long an aspect of any individual game "should" take to complete. This is because different test takers have different goals for overall section performance, and different strengths on individual types of games (some test takers will find linear games like this one and the first game on this section to be easier, while others will find grouping games like the second and fourth games on this section to be easier).

However, assuming you want to complete the entire logic games section (in this case all 23 questions), and without adjusting too much for contextual factors like which games you find easiest, I'd assess that you'd want to complete the templates in this scenario within about 6 minutes or less. I'm allotting about 2 minutes for: (a) reading the scenario; (b) diagramming the basic game board, variables, rules, and simple inferences; and (c) recognizing the possibility of using templates. I'm then allotting about 1 minute for diagramming each templated possibility (and recognizing that the fourth apparent possibility, with the LL block in the second shift on Thursday and Friday, doesn't work). If you were able to complete the templates within 6 minutes, it's entirely realistic to finish all of the questions on the game within about 3 minutes, putting you just slightly above the target time of 8:45 per game that allows you to complete all 4 games in 35 minutes. Understand that you can have a little leeway with that timing (for either the scenario/templating, or the questions) if you were able to complete the other games in the section in less than the 8:45 target time per game.

I hope this helps!

Jeremy

[davidl113]

Hi,

I've been a bit hesitant to the use of templates because I fear that I might waste a lot of time creating them, making me rush and consequently making incorrect inferences. However, I'm convinced that they're KEY to games like this one. What are some indicators I should be looking for to know that a template approach is the best approach? I'd really appreciate some advice on this!

[Claire Horan]

Hi Davidl113,

This is a great question that I hope everyone conTEMPLATES.



I crack myself up. But on to the answer.

First, practice with templates will increase your speed at making them and your understanding of when they will be helpful. While studying, you may want to err on the side of making templates so that you learn when not to use them. In other words, you can't learn from your mistakes if you don't make any. Practice will help you develop your intuition about when they are helpful and when not.

That said, I've found that a good rule of thumb is to make templates when the effects of a rule or its interactions with other rules are difficult to figure out without them. For example, let's say I have a rule that says A finishes the race before B. I can visualize the effects of this rule quite easily by writing the rule as A-B and by using not-laws once I have more information.

In contrast, if a rule says A finishes before B only if A finishes first. In that case, I will have trouble using this rule with other rules without a template. I will make template for when A finishes first, and one where A does not finish first. Note that these are the only two possibilities when using templates it's very important that they add up to include every possible solution.

I hope my rule of thumb helps, but remember: practice, practice, practice and they will become more comfortable, quick, and useful for you!