Logic Challenge #26 - Pitch Meetings

[shadowclad17]

Link to the challenge

Initially, we have lots of "if, then" rules and even then, most of the hypothetical results speak of the relative positions of letters and don’t concretely snap anything into place. We have only one unconditional rule: G is not last. The key to this is thorough primary deductions. Once you’ve decoded what the rules are hinting at, this seemingly fluid game crystallizes into something relatively easy, and the questions can be rapidly answered like clockwork.

Obviously, we can symbolize our rules initially. Arrow is "if, then," dash is "is before," and tilde is "not." I like to number my rules because it makes them easier to refer to.

1) The screenwriter will meet with K before G if he meets with F before J.

Careful! The "if" aspect of this statement refers to the second half. It’s tempting, if you’re too rushed, to write this "K"”G"”>F"”J," but that would be incorrect. Understand what this rule really says:

1) F"”J"”>K"”G

If you’re having trouble telling whether a rule makes sense, insert words in the phrase to make the relationship clearer to yourself. Mammal, if dog is equivalent to "If dog, then mammal." I like to use animals because the relationships are clear and it’s easy to imagine counterscenarios. (For example, mammal"”>dog is false because a cat is a mammal but not a dog.)

On to rule 2...

2) The screenwriter will meet with G before H only if he meets with I before J.

This is, again, very, VERY sneaky. My compliments to the clever writer! "A only if B" means, essentially, "If not B, then not A," which is the contrapositive of "If A then B." Since we just did a rule that had "if" in the second half, it’s tempting to flip this one as well, but it is actually:

2) G"”H"”>I"”J

On to rule 3...

3) The screenwriter will meet with F before I if, and only if, he meets with F after J.

"If and only if" is classically symbolized with a "triple bar" in symbolic logic. This essentially means that the "if, then" relationship goes both ways. I will note both in my symbolization:

3) a. F"”I"”>J"”F

Again, be careful. Most of these rules read "before," so it’s tempting to write "F"”J," but the clever writer snuck in "after" in the third rule to make sure you’re paying attention!

3) b. J"”F"”>F"”I

4) The screenwriter cannot meet with G last.

4) G =/= 6

I go ahead and write ~G in my diagram to remind me of this:

_1_|_2_|_3_|_4_|_5_|_6_
___|___|___|___|___|~G

So far, we don’t know much besides that. So let’s try and make some deductions from these rules.

First of all, let’s derive the contrapositive of each. If you didn’t know, the contrapositive of any "p"”>q" is "~q"”>~p". So for rule #1, we know:

~(K"”G)"”>~(F"”J)

What good does that do us? Well, the great thing that this is a simple, one-at-a-time order game. No two elements can go at the same time. Thus, if K isn’t before G, and K can’t be with G, K must be after G.

~(K"”G)"”>G"”K

So, essentially, the contrapositive for each of these rules involves flipping the elements AND the order within those elements, and we don’t have to worry about "not."

So, here are the rules with their corresponding contrapositives:

1) F"”J"”>K"”G

1C) G"”K"”>J"”F

2) G"”H"”>I"”J

2C) J"”I"”>H"”G

3) a. F"”I"”>J"”F

3) ac. F"”J"”>I"”F

3) b. J"”F"”>F"”I

3) bc. I"”F"”>F"”J

Note that, for rule #3, the element "F" is in common to both the antecedent ("if" part) and the consequent ("then" part). Thus we could extend the chain for each consequent to include both elements:

3) a. F"”I"”>J"”F"”I

3) ac. F"”J"”>I"”F"”J

3) b. J"”F"”>J"”F"”I

3) bc. I"”F"”>I"”F"”J

Now, since two components can’t go at the same time, it logically follows that F will be either before or after I.

(F"”I) v (I"”F)

The same applies to F and J.

(F"”J) v (J"”F)

That little "v" symbol is symbolic logic for "either, or," but if it’s easier for you just imagine it’s a slash "/" mark.

We can do a constructive dilemma here and logically conclude, from either of these new premises, that we have only two possible scenarios:

(J"”F"”I) v (I"”F"”J)

So, essentially, we’ve established that F must be between I and J. If we know the relative position of two components, we can determine the relative position of the third. There’s something concrete for us to work with! Isn’t that a relief?

But it gets even better when we synthesize this with rules 1 and 2. Remember the contrapositive of rule 1:

G"”K"”>J"”F

We now know that J"”F means J"”F"”I, so we can add to this:

G"”K"”>J"”F"”I

Similarly, we can add to rule 2:

G"”H"”>I"”F"”J

Here we can clearly see that both K and H cannot be after G! Why? Because it would lead to a contradiction about the relative positions of I, F, & J! Thus, either H or K (possibly both) must be before G!

H/K"”G

If both H and K are before G, then we still know is that F must be between I and J. But if either H or K comes after G, it determines the relative order of I and J! Thus, we have three master scenarios:

SCENARIO X:

H&K before G, F between I&J

(*I would symbolize this much better on paper than in text)

SCENARIO Y: K"”G"”H & I"”F"”J

SCENARIO Z: H"”G"”K & J"”F"”I

Now we have a pretty clear idea will work, and what won’t! From here, the questions are MUCH easier to answer!

1) For B, G is last, so we can cross this off right away. For both C and E, G is before both H&K, which is impossible, so we can eliminate both. For D, F is after both I and J, so this scenario is impossible. The only option left is A; it’s correct. You can double-check if you like to make sure it complies with the rules. But perfect accuracy is useless if it takes all day, so I’d only go back and confirm if I had tons of extra time and nothing to do with it.

2) Obviously G cannot be first; it would have to be before both H&K, which we know is impossible. Thus the answer is E.

3) If I is 2nd, we must have an I"”F"”J scenario, since there’s not enough room before I for a J"”F"”I scenario. Thus K must be before G. Since F must be before J, F cannot be last. Since K must be before G, K cannot be last. G also cannot be last because the last rule prohibits it. I obviously can’t be last if I is 2nd. Therefore, only J or H can be last; D is the answer.

4) If K is first and H is last, we must have an I"”F"”J scenario. We have 4 open spaces and only 1 free agent: G. Wherever G goes determines where the other three must go. Thus, we have only 4 possible arrangements: G 2nd, G 3rd, G 4th, and G 5th. So C is the answer.

5) If K is 5th we cannot have a K"”G"”H scenario. G can’t be last so H&K can’t both be before G, either. Thus we have H"”G"”K, and thus J"”F"”I. Since we know that H is before G, it must be false that G is before H, so B is the answer.

6) The key to finding the most restrictive scenario is to find a pair that determines the relative positions of two elements that are components of one of our chains. Only D, G&H, does this. It’s our best bet, but let’s test it just to be sure. If G is 2nd and H is 3rd, we have K"”G"”H. Thus we know K must go first. Since we have K"”G"”H, we must have I"”F"”J, which go in spaces 4, 5, & 6, respectively. Thus, D is the answer.

[noah]

Looks like someone beat us to it in setting up this thread! Shadowclad17, your name is well-deserved!

Go ahead and submit your answers through the game's page http://www.atlaslsat.com/logic-games-practice.cfm, but if you have an explanation -- put that here (and if you want us to attach a diagram, e-mail it to logicmaster@atlaslsat.com and he shall do your bidding). Writing your own question and/or submitting a diagram definitely earns you brownie points.

Best explanation -- as chosen by the game's author -- wins a free Atlas LSAT Strategy Guide or $200 off a new, live class registration.

Good luck!

[kinberdan]

Note: "X">"Y" means that X comes before Y

Rules:

F>J -- > K>G
J>F <--> F>I
G>H --> I>J
G is not last

Analysis: The 1st two rules can offer a complete solution (F>J or J>F), so let’s try 2 templates.

Template 1: If F>J

1a: If G>H
K>G>H
I>F>J

Note: I>F b/c of ~(J>F) --> ~(F>I)

1b: If H>G
H&K>G
I>F>J

Template 2: If J>F

2a: If G>H

Doesn’t work!
J>F F>I but I >J

2b: If H>G
H>G K=floater
J>F>I

Now the questions:

(1) Correct Answer A: The only that matches a template, in this case 2b.

(2) Correct Answer E: Looking at the templates, G always has something preceding it.

(3) Correct Answer D: If I is 2nd then we’re dealing with Template 1a and 1b.
(F>J, G, H/K) have to come after it for the 4 remaining spaces. F and G and K are not last based on the templates and last rule. So J or H are last.

(4) Correct Answer C: Only template 1a works here. There are only 4 ways to arrange I>F>J and G between K-1st, and H-last.

(5) Correct Answer B: Only template 2b works here (template 1b would force G last). G cannot be before H.

(6) Correct Answer D: First look at template 1a, since it has 2 full sequences with relationships established. So it is likely capable of establishing 1 solution to the game with placing only 2 letters. Template 1a eliminates A & B. Try C _ no way to fit I, K & G in 2 spaces before F. Try D _ it works!

[ManhattanPrepLSAT2]

Terrific explanations -- great work to both of you!

This is a very challenging game, and the key is to have a correct initial set-up. One small mistake interpreting a conditional constraint and the whole thing will be a mess. But nail that initial set-up, as these two have done, and the questions are actually quite straightforward.

Thanks again to everyone for playing!

[chrisnxn]

I have a question about the 3rd question. If I is 2nd, does it not follow that K must be first? If not then why?

[ManhattanPrepLSAT2]

Hi -

Not sure exactly how you got to that inference, but here's one example that would not violate the rules, and doesn't place K first:

HIKFGJ

Hope that helps. Please follow up if you want to walk through the reasoning you used.

[siliconrs]

I missed it like the previous poster and I suspect that we both made the same blunder. Since I must be 2, we have to use the I-F-J construct. The key is that this is the necessary condition of the K-G-H scenario. On my first try, I applied the K-G-H -> I-F-J relationship as though it had to had to happen. The very clever author is preying upon our confusing the sufficient and necessary conditions. I checked off A and moved on with a mistaken confidence.

After I got 5/6 right, I reread the rule put forth in #3. It states that I-F-J must be true since I is #2 and that F and J can be any place from 3-6. Given this, you CAN (not must) apply the K-G-H rule if you want and if you do then K is #1.

Remembering that this scenario only required the use of chain 1's necessary condition, one can do numerous sequences that do not violate any of the rules, but prove that K does not have to be #1. Examples: G-I-H-K-F-J or H-I-K-F-G-J. As such, K in slot 1 is not a case of must be true, but case of could be true. What must be true is that J or H is last every time, and every valid scenario will end up that way.

[zagreus77]

Link to the challenge






l






SCENARIO Y: K"”G"”H & I"”F"”J

SCENARIO Z: H"”G"”K & J"”F"”I




answer.

Great work, but Scenario Z needs modification. There is no rule of inference to fix k if JFI obtains, , so K becomes a free variable in that scenario. I wanted to make the same inference to have a nice double arrow. But that inference is an incorrect reversal.

[shadowclad17]

SCENARIO Y: K"”G"”H & I"”F"”J

SCENARIO Z: H"”G"”K & J"”F"”I

Great work, but Scenario Z needs modification. There is no rule of inference to fix k if JFI obtains, , so K becomes a free variable in that scenario. I wanted to make the same inference to have a nice double arrow. But that inference is an incorrect reversal.

The first rule states that K precedes G if F precedes J. The contrapositive of that rule determines that K follows G if F follows J.

[zagreus77]

SCENARIO Y: K"”G"”H & I"”F"”J

SCENARIO Z: H"”G"”K & J"”F"”I

Great work, but Scenario Z needs modification. There is no rule of inference to fix k if JFI obtains, , so K becomes a free variable in that scenario. I wanted to make the same inference to have a nice double arrow. But that inference is an incorrect reversal.

The first rule states that K precedes G if F precedes J. The contrapositive of that rule determines that K follows G if F follows J.

The first rule does state that K precedes G if F precedes J, or more clearly if F precedes J, then K precedes G. (F--J -->K--G) The contrapositive is G--K --> J---F. This only makes G--K sufficient to infer J--F, not necessary for J--F to be the case.. We know J--F, if we know G--K, but not the reverse. So the second scenerio would still allow HKGJFI, for example. When J--F obtains, nothing in the rules determines the position of K.

[zagreus77]

Another way of seeing what's going on in this game is as follows:

rule 3: F--I <---> J--F

contra: I--F <---> F--J

This generates the following two orders as the only possible options of these three elements:

J -- F-- I

I -- F-- J

So, taking the contapositive of rule 2 (J--I -->H--G), we generate the base scenario J--F--I --> H--G

Similarly, taking rule 1 F--J --> K--G, we generate the base scenario I--F--J --> K--G

These two scenarios are the only possible orders among the elements listed.

The contrapositives of these base scenarios respectively are
{K} G--H --> I --F--J --> K--G
{H} G--K ---> J--F--I --> H--G

However, these are not only ordering possible for the base scenarios. In fact, in the I--F--J scenario H is a free variable, it it a sufficient condition when it follows G, but it is not necessary for the scenario.

Similarly, in the case of J--F--I, K is a free variable, although sufficient to generate the base scenario not necessary for it to obtain.

For example, HKGIFJ and KHGJFI are both valid scenarios which do not adhere to the ordering of the sufficient conditions of the contrapositives. That is, H can come before G and K before G in the second and create a valid order.

[ohthatpatrick]

I agree with everything the previous poster was saying. I think the confusion arose because Shadowclad initially outlined 3 master scenarios, and in scenario X, he acknowledged the flexibility that both K and H could come before G.

His scenarios Y and Z were made by arbitrarily starting with K--G--H and H--G--K. He wasn't saying that G was always stuck in between them in his original post.

But I agree that it's probably easiest to anchor our thinking in this game around 2 scenarios:
J--F--I or I--F--J

When we're in the J--F--I world, we also know H--G, and then K is a floater.

When we're in the I--F--J world, we also know K--G, and H is a floater.

Nice work all around.