#8 - Local, Could be True

[alluv001]

Hello everyone,

One of the rules of the game states

"Lalitha performs third only if Norton performs fifth".

I diagrammed it as

L(3rd)--------->N(5th)

due to the "only if" wording which would qualify the Norton as 5th.

Subsequently, I arrived at the natural contrapositive (-N(5th)--------->-L(3rd).

However, the explanation for #8 states that the contrapositive is a Mistaken Reversal, hence the correct answer being C. I'm having trouble seeing why it is that is a Mistaken Reversal, if its the contrapositive of the rule. Unless I diagrammed the rule wrong from the beginning?

[Nikki Siclunov]

Hi alluv001,

Thanks for your question.

Your diagram of the rule is undoubtedly correct, and so is the contrapositive:

L3 N5
~N5 ~L3

In question 8, you're required to identify what could be true if N performs fifth. It would be a Mistaken Reversal of the rule to infer that L must perform third. Students who make this Mistaken Reversal immediately eliminate answer choice (C), believing - mistakenly - that L cannot perform fourth. Of course, nothing prevents L from performing fourth, as shown in the two local diagrams for this question.

Does this clear things up?

Thanks!

[Adam Tyson]

Your initial diagram of the rule looks perfect to me, but your contrapositive has a typo in it - it should be N5 -> L3. Looks like you may have forgotten to negate the second term? Remember, to get the contrapositive you must do two things: 1) reverse the order of the conditions and 2) negate them both (which also means taking any negatives in the original and turning them to positives).

Many students will make a mistaken reversal in this game and assume that any time N is 5th, L must be 3rd, and Answer C to Q8 capitalizes on that - it looks like a bad answer to anyone who has made that error, when in fact it's entirely possible that N is 5th and L is 4th, since N5 is not sufficient to make l3.

I hope that helps!

[Adam Tyson]

Ah, I see that tiny negative sign in front of your second term in the contrapositive now. Getting old, need to start wearing my reading glasses! As Nikki and I were saying, the question is about what can happen, and the correct answer looks bad only if you make a mistaken reversal.

[ddion8206]

'm currently reviewing Linear Games from lesson 3 and I am currently stuck on one concept. It occurred on Game #4: December 1997. It was problem #8. The answer was C: Lalitha is scheduled for the fourth performance. I don't understand why the mistaken reversal happened especially because of the wording. The rule states L is third only if Norton Performs fifth. The question ask If Norton is fifth which could be true? I am not understanding this concept. Any help is appreciated.

[Emily Haney-Caron]

Hi ddion,

If N is 5th, we know P has to be 2nd, and M therefore will have to be 1st. The KJ block will either be in 3-4 or 6-7. The other two spots will be filled by L and O.

A cannot be the correct answer, because K has to come right before J, and N is 5th.
B cannot be correct because P is 2nd.
D cannot be correct because M is 1st (because has to be before P).
E cannot be correct because M is 1st.

C can be correct. IF L is 3rd, THEN N has to be 5th. The contrapositive of that is, IF N is NOT 5th, THEN L is NOT 3rd. It doesn't tell us anything about what happens to L if N IS 5th. So if N is 5th, L can be in any open space. Thinking that N being 5th means L has to be 3rd would be a mistaken reversal.

[kkauf11]

I cannot get the correct answer for this problem. The book says the correct answer is C) Lalitha is scheduled for the fourth performance, but I do not understand how that can be true if Lalitha performs third only if Norton performs fifth. This is the order that I keep coming up with:

M P L O N K J

I get this because if N=5, L=3. Therefore, P=2. M must go before P, so M=1. K performs immediately before J, so they fall into 6 & 7, which leaves O in slot 4.

I know I'm doing something wrong, but I cannot figure out what it is. Could someone please explain to me where I am making a mistake?

Thanks!

[Emily Haney-Caron]

Hello!

The problem you're having here is a mistaken reversal; we know that L is in 3 only if N is in 5, but we don't have that L HAS to be 3 if N is 5. So we have L3 N5 and a contrapositive of N5 L3
What you are suggesting, that L has to be in 3 if N is in 5, would look like this, and is NOT represented by the rules: N5 L3

So what we actually have is:
If N is in 5, P has to be in 2. And M has to be in 1, because M has to be before P. We also know we have a KJ block. That gives us two possible templates; in one, the KJ block is in 3-4, and in the other the KJ block is in 6-7. Whichever of those spots the KJ block does not occupy will be taken up by L and O (but we don't know what order they will be in).

So, if the KJ block is in 6-7, L and O will each be in either 3 or 4, making C correct.

Does that help?

[kkauf11]

Yes, thank you!

[Jon Denning]

Was just asked about this question by a student, so although it's been discussed at length in this thread I thought I'd post my reply to her for anyone still a bit unclear on #8.

Let's start with the third rule: L 3rd only if N 5th. That's conditional, of course, and can be shown as L3 --> N5, meaning the contrapositive is NOT N5 --> NOT L3. In other words if N is anywhere but the fifth spot, then L cannot go in 3.

The fastest way to trigger this contrapositive is with P from the final rule, since P is always either 2 or 5 (if P isn't 2 then P is 5, meaning N is not 5 and L is then not 3). But other things could initiate it as well, of course.

In question 8, we're told that N is 5. Now looking at the diagram above, with L3 --> N5, what does N5 tell us? What's it pointing to? Nothing. To think that forces L3 would be to go backwards against the arrow and commit the Mistaken Reversal. N5 merely *allows* L to go anywhere, including 3.

At this point we could move to the answers--jump to the discussion below if you want to--but before I do I'll outline the two possible arrangements we could have if we put N 5th. Here's what you'd get, 1-7:

1. M P K J N (L / O)
2. M P (L / O) N K J

Those "/" marks indicate possible movement between variables, so for #1 L or O could be 6 or 7. Notice that L can go in 3 (#2 has that option), but doesn't have to.

As I said though, with just a broad understanding of the game and a specific understanding of the condi-tional implications of N5 you could go right to the answer choices and get to work. For this could be true question then four answers are impossible, and one could occur. Let's go through them:

(A) cannot happen because J is in a block with K (as KJ), and with N 5 J cannot be right after it in 6.
(B) cannot happen because if N is 5 then P must be 2 (from the last rule).
(C) CAN happen! Again, to assume L must go 3 and can't go elsewhere would be to go backwards against the arrow. L is free to go anywhere that isn't taken by someone else already (like P in 2, for instance). L3? Fine. L4? Fine. L 6 or 6? Fine.
(D) cannot happen because M is before P, and P is in 2 here. So M must be first.
(E) cannot happen because as discussed above for (D) M must be in 1.

And that's it. If you struggled with (C) it's almost certainly a Mistaken Reversal issue. Take another look at it after reading this and make sure it's clear--this trap appears all the time on the test so the sooner you can start consistently avoiding it the better.