Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

This is an Unbalanced: Underfunded Basic Linear game.

pt50_s06_g1_1.png

This game is Unbalanced: Underfunded because there are only four airlines to fill the six stops. Thus, some airlines must be used multiple times. Note also that the game does not specify that every airline is used. Thus, some viable solutions will not contain all four airlines. This makes the game much trickier. Remember, always read the rules carefully to see if every slot is filled, and if every variable must be used. In this game every slot is filled, but not every variable must be used. That creates a greater number of possible solutions.

• * With the first rule, instead of drawing Not-blocks for each airline such as not LL, etc, we chose to create an
  S06_Game_#1_setup_diagram 1.png
  to represent that the same “Airline” cannot appear consecutively.
  
  * The second and third rules can be diagrammed internally, meaning directly on the main diagram. Simply draw a double arrow between stops 1 and 6, and another arrow between stops 2 and 4.
  
  * The fourth rule is drawn as two separate Not-blocks, but the operational effect of neither L nor M appearing before N is that if N appears in stops 2-6, then O must precede it. N can appear in stop 1 because neither L nor M would precede it.
  
  * The last rule creates a dual-option on stop 5, and the interlaced effects of all of the rules create a limited scenario in each part of the dual-option. Thus, the last rule creates two basic templates for the game. Let us examine each.
  
  If N is chosen for stop 5, then from the first rule N cannot be chosen for stops 4 and 6. If N cannot be chosen for stop 6, then from the second rule N cannot be chosen for stop 1. Similarly, if N cannot be chosen for stop 4, then from the third rule N cannot be chosen for stop 2. From the fourth rule we can infer that L and M cannot be chosen for stop 4, and thus O must be chosen for stop 4. When O is chosen for stop 4, from the third rule we can infer that O must be chosen for stop 2. If O is chosen for stops 2 and 4, then from the first rule O cannot be chosen for stops 1 and 3. Only L or M is then available for stop 1 (and from the second rule, stop 6 as well):
  
  N chosen for stop 5:

pt50_s06_g1_2.png

If O is chosen for stop 5, then from the first rule O cannot be chosen for stop 4 or stop 6. If O cannot be chosen for stop 4, then from the third rule O cannot be chosen for stop 2. If O cannot be chosen for stop 6, then from the second rule O cannot be chosen for stop 1. Of course, from the fourth rule, if O cannot be chosen for stops 1 or 2, then N cannot be chosen for stops 2 or 3 (and then from the third rule N cannot be chosen for stop 4):

• O chosen for stop 5:

pt50_s06_g1_3.png

Taking a moment to examine the restriction on each stop, only L or M is available to fill stops 2 and 4.

• O chosen for stop 5:

pt50_s06_g1_4.png

Thus, the final rule—in combination with the other rules—triggers a powerful chain reaction where two very strong templates emerge for the game. With these two templates in hand, we are now ready to efficiently attack the game.

[DSquared]

My question concerns the Complete Games Sections, Sept. 2006 Logic Games (the first game, including questions #1-5) in the PowerScore LSAT Logic Games Bible Workbook.

Rule #4 says, "Whenever she chooses either L or M at a stop, she does not choose N at the next stop." Rule #5 says, "At stop 5, she chooses N or O."

Beginning with question #2 that states, "If the traveler chooses N at stop 5, which one of the following could be an accurate list of the airlines she chooses at stops 1, 2, and 3, respectively?"

The explanation for the answer choice that is listed as "correct", which is on page 237, says that "...Only L or M is then available for stop 1 (and from the second rule, stop 6 as well): when N is chosen for stop 5."

I am confused by the possibility of “L” OR “M” being possible answers for “stop 6” in this scenario because of the contrapositive of Rule #4. Obviously, I MUST not be doing something right, but I cannot figure out why the contrapositive of Rule #4 is not taken, and then applied. Maybe I am taking the contrapositive of Rule #4 incorrectly? Please help!
I diagrammed Rule #4 as: L or 'not’N
M

When I took the contrapositive of this rule, I diagrammed it as: N ’not’L and
‘not’M

('not' X means the Letter with a slash through it, bc I couldn't figure out how to make that symbol).

If this is the correct contrapositive, then with stop 5 being ‘N’ in question #2, then stop 6 CANNOT be ‘L’ or ‘M’ as the explanation on page 237 says that it is (which then makes stop 1 also unable to be ‘L’ or ‘M’), and by Rule #1, then neither stop 6 nor stop 1 can be ANYTHING BUT ‘O’.

If I have diagrammed & am using the contrapositive of Rule #4 correctly, then this screws up the answers given as “correct” for questions #3 and #5 as well.

Please tell me what I am doing wrong!
Thanks!

[Dave Killoran]

Hi Dsquared,

Let me see if I can help out there. Let's first start with the fourth rule itself and review it again.

The place where you are running into problems in your diagram (and thus subsequently in the question) is that your representation of the rule does not account for the linearity in the game. Let's look again at the rule language, and I'll highlight the key word that I think you may be overlooking:

• Whenever she chooses either L or M at a stop, she does not choose N at the next stop.

So, this rule is about consecutive stops where L or M are placed "earlier" (lower number) and then N can't be at the very next stop after (higher number). So, for example, LN in that order is impossible, but NL is actually ok.

This linear aspect is why when we represented the rule we turned it into two not-blocks of not LN and not MN.

Your representation of the rule doesn't account for the linear ordering, and as drawn just means that L or M can never be next to N in any order. The actual rule is only half as powerful as that, and that's why all those answers started to look incorrect to you. Try looking at those questions again with the linearity considered, and I'll bet they will look correct this time around.

Please let me know if that helps. Thanks!

[Dave Killoran]

No problem--glad I could help!

Ok, going back into your representation, the problem is that the original rule as you represented it is not accurate. Thus, when you took the contrapositive, it too was inaccurate (I'll note that you took the proper contrapositive of the rule, but when the original wrongly reflects the game, the contrapositive will also wrongly reflect the game too).

So, to resolve this problem, we have to go back to your original representation of the rule. What you had was:

L
or N
M

This is more of a grouping representation because inherently, there's nothing linear here. But, I think what you did was you mentally added the linear component, as in, "this means the two aren't next to each other." As I mentioned before, this is a problem because the rule isn't universal about not being next to each other; the rule is about L/M not being immediately before N. To solve this problem, then, you have to represent the rule differently.

In our rule diagram you can see we used not-blocks to handle the linearity. If you want to make this a conditional diagram, we still need to show the linearity somehow. This means representing space numbers, for example:

L1 N2

The contrapositive then makes sense:

N2 L1

The problem here is that there are multiple slots for L, so you have to diagram for L2, L3, etc. And you have to do the same for M. So, you'd have a lot of diagrams to draw--too many, in fact! That's why we simplified this rule to its core meaning, which is that the variables in question can't be consecutive:

L N

M N

This captures the linear element and it captures the "sliding" aspect that exists because the variables can move across a spectrum of possible placements.

Please let me know if that helps. Thanks!

[yrivers]

Hi,

I have a very general question: "she chooses the same airline at stop 1 as she does at stop 6."

How is this a double arrow as opposed to 6 --> 1? The specific language, as I read it, states that 1 reflects 6 but not necessarily vice versa. I would've expected something like, "She chooses the same airline at stop 1 as she does at stop 6, and the same airline at stop 6 as she does at stop 1."

Thanks,
Yaesul

[Dave Killoran]

Hi Yaesul,

Thanks for the question!

This is one of those interesting rules where although the initial rule is phrased as a single-direction conditional (namely 6 1), in the game it turns into a double-arrow because of the emphasis on the stops being the "same airline." Let's talk about it!

In this rule, the initial diagram is 6 1. But let's think about the implications of what happens when certain choices are made for these spaces. For example, let's use L, and let's use it in isolation of the other rules (in other words, we won't think about the other rules for the moment). If is the chosen airline for stop 6, well then from the rule we know that L is the chosen airline at stop 1. Ok, that makes sense so far.

Now reverse it. Let's say L is the chosen airline at stop 1. At first that doesn't seem to imply that L must be chosen at 6, but stop for a moment and think about what's occurring at stop 6. If stop 6 is M, for example, then from the rule we know stop 1 would have to be M, but that can't happen because we have L there already. That same pattern of analysis happens for every other variable, excluding L. So, even though the rule might not seem to initially imply that it reverses, because it says "chooses the same airline," it ultimately means that once you know the airline at one of the two stops (1 or 6), you know it at the other, and hence it becomes a double-arrow.

Please let me know if that helps. Thanks!

[yrivers]

That's hugely helpful. Thank you, Dave! Appreciate it!

[ssnasir]

Hi there,

I mess up on conditionals in logic games at time and I think I did it again here. I took the fourth rule as If M or L NO N at the next stop. My contrapositive for that was if there is N No M and L which I know is wrong, but I can't understand why.

It would be a great help if you could please let me know. Thank you!

[KelseyWoods]

Hi ssnasir!

You actually correctly diagrammed the conditional statement, it's just not the clearest way to understand the relationship.

Let's break down the statement:

"Whenever she chooses either L or M at a stop, she does not choose N at the next stop."

You were correct that "whenever" is a sufficient indicator so you could diagram that relationship as:

L
or N next stop
M

But let's think about what that rule is really saying--if we put it another way, it's really just saying that you can't have N at a stop right after L or at a stop right after M, right? When you have L, you can't have N right after it. When you have M, you can't have N right after it.

So a clearer way to diagram the rule is to use the Not Blocks that are in the diagram above.

LN and MN

It seems like you have a good handle on how to diagram those conditional statements. Now you just have to figure out when it's useful to use the conditional diagrams versus when a different visual representation (that shows the same relationship) would be more useful.

Hope this helps!

Best,
Kelsey

[ssnasir]

Hi ssnasir!

You actually correctly diagrammed the conditional statement, it's just not the clearest way to understand the relationship.

Let's break down the statement:

"Whenever she chooses either L or M at a stop, she does not choose N at the next stop."

You were correct that "whenever" is a sufficient indicator so you could diagram that relationship as:

L
or N next stop
M

But let's think about what that rule is really saying--if we put it another way, it's really just saying that you can't have N at a stop right after L or at a stop right after M, right? When you have L, you can't have N right after it. When you have M, you can't have N right after it.

So a clearer way to diagram the rule is to use the Not Blocks that are in the diagram above.

LN and MN

It seems like you have a good handle on how to diagram those conditional statements. Now you just have to figure out when it's useful to use the conditional diagrams versus when a different visual representation (that shows the same relationship) would be more useful.

Hope this helps!

Best,
Kelsey

Thank you so much!!