Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is an Unbalanced: Underfunded Basic Linear game.

J94_Game_#2_setup_diagram 1.png

A careful reading of the rules reveals that there are several numerical options available for the assignment of firefighters to days, and thus what appears to be a simple linear game is actually somewhat complex. Because numbers always play an important role in Logic Games, let us take a moment to review how you could have recognized that this game was not a simple 1-to-1 relationship:

1. The game scenario never specifies that each firefighter must work.

• Many students make the mistake of assuming that because five firefighters are listed, that all five must work. However, if that is the case, you will always see confirmation of that fact in either the game scenario or rules. This game contains no such confirmation, and so it is not the case that every firefighter must work. Do not make the mistake of assuming that the first rule states that every firefighter works. Instead, the first rule indicates that exactly one firefighter is assigned each day (and not that every firefighter is assigned).
  
  Remember, the statements that are omitted in the rules are just as important as the statements that are made.

2. The second rule reveals that some firefighters are able to work more than one day per week.

• The second rule directly addresses numbers, and so this is the most apparent tip-off that something is going on with the firefighter assignments in this game. While this rule limits firefighters to no more than two days worked, applying this rule in combination with the point made in item #1 above leads us to three unfixed numerical distributions for the game:
  
  2-2-1: Two firefighters work two days each, one firefighter works one day.
  Two firefighters do not work.
  
  2-1-1-1: One firefighter works two days, three firefighters each work one day.
  One firefighter does not work.
  
  1-1-1-1-1: Each firefighter works one day. When this distribution is in play, the game acts like a standard Basic Linear game.

With the understanding that not all of the firefighters must work, the rules are no longer simple blocks and sequences. Instead, for the rules to be in effect, you must know that all of the firefighters mentioned in each rule are working. Otherwise, the rule is not in force.

The third rule is shown as a CC not-block, which is more efficient than showing each of the five firefighters in individually lettered not-blocks.

The fourth rule is now conditional, as F and J must both work in order for the F J sequence to apply. Because the rule only enacts if both F and J work, no Not Laws can be drawn.

The fifth rule is also conditional, and only occurs when H works. Of course, via the contrapositive, if G does not work, then H cannot work, and the other three firefighters would be in the 2-2-1 distribution. Note that the conditional nature of this rule means that G could be the first firefighter to work, a scenario that can occur if H does not work. Thus, no G Not Law appears under Monday.

[morios]

This LG question is addressed in the LG bible, but I'm not sure how to arrive at the conclusion that each firefighter doesn't need to work.

The setup, summarized:
There are five firefighters
None work on Sat/Sun, and on Mon-Fri, exactly one of the firefighters works each day.

However it is noted in the book that not every firefighter is required to work. My question is - isn't that addressed in "exactly one of the firefighters works each day" making this a 1-1 numerical distribution?

[Nikki Siclunov]

Since the game does not contain a "clean-up" rule such as "There are five firefighters, each of whom works on at least one day during the week," it is entirely possible that only one of the firefighters works on every day during the 5-day workweek. The statement "exactly one of the firefighters works each day" does not mean that a different firefighter works each day.

Hope this helps!

[Dave Killoran]

Hey Morios,

Nikki is exactly right here. What they did in this game is rather insidious because they give the appearance of creating a classic 1-to-1 relationship without actually doing so. There are two sides to establishing a 1-to-1 relationship: first that there is exactly one firefighter per day, and second that each firefighter must work just one time. While they clearly stated the first part, they never state the second part, and thus we don't have the 1-to-1 scenario we are all so used to seeing.

As a useful exercise, compare the language used in the firefighters game to language used in other, true 1-to-1 games and look for the difference in how the relationships are stated. You'll quickly see how real 1-to1 games always state both sides.

Thanks!

[LSAT99.9]

Hi,
For this question the rule "Fuentes never works later in the week than Jackson" is diagrammed as " F - J" , but why can't F work on the same day as Jackson? Because I thought that just because the rule says somebody can't work after another person this doesn't eliminate the possibility of working on the SAME day as somebody.

Thank you!!

[Adam Tyson]

Check those rules again - rule 1 tells us that "exactly one of the firefighters works each day". No ties allowed!

[jessamynlockard]

Hi,
So I know you say that the last rule is a conditional rule, but I don't understand why that means the conditional means " if G does not work, then H cannot work".

Based on the diagram: H "HG" block, my contrapositive looks like: NOT "HG block" NOT H. which I suppose isn't a very useful inference.

[Shannon Parker]

Hi,
So I know you say that the last rule is a conditional rule, but I don't understand why that means the conditional means " if G does not work, then H cannot work".

Based on the diagram: H "HG" block, my contrapositive looks like: NOT "HG block" NOT H. which I suppose isn't a very useful inference.

Your diagram is correct. If there is not an HG block there cannot be an H. You cannot have an HG block without a G, therefore if there is not a G there can be no H.

Hope this helps.
Shannon

[amagari]

What is the contrapositive of the rule involving F and J? I believe I did it correctly but it doesn't seem to make any sense. And how would it influence the game?

With Question 11, the contrapositive played a role and I want to make sure I don't miss anything in my understanding of the F and J rule.

[Adam Tyson]

That contrapositive would be spelled out as "If F is not before J, then F and J do not both work," amagari! In diagram form, something like this:

F J F or J

That would give you two scenarios, perhaps: 1. If J ever works Monday, then F cannot work that week; 2. if F ever works Friday, then J cannot work that week.

You're right about question 11! If G is out, H is out, and our workers must be F, I, and J. Two of them work twice, one works once, and the F/J rule means that those two cannot both work twice (because at least one F would be after at least one J).

Is that how you made it work? Let us know!