#9 - Local, Must Be True

[Administrator]

Complete Question Explanation
(The complete setup for this game can be found here: lsat/viewtopic.php?t=15461)

The correct answer choice is (B)

The question stem establishes that the 3-3-0 distribution is in effect. As discussed during the setup, under that distribution the middle shelf must always have three dishes stored on it, and so answer choice (B) must be true and is therefore correct.

[jfalbo32]

Question # 9 asking for one shelf to be empty forcing the 3-3-0 distribution. Answer choice is B is correct. How does this not go against rule #2? (dish 2 is stored at least one shelf above dish 6)

Why cant the bottom shelf be empty? I thought the "3-3-0" could go in any direction in no particular order

[Jon Denning]

Hey jfalbo,

Thanks for the question! This was a tricky game for a lot of people (mostly due to the distribution idea, I think), so let's walk through it together and hopefully I can clarify #9.

You're correct to note that of the three possible distributions-- 2-2-2, 3-2-1, 3-3-0 --question 9 requires the 3-3-0 option. So two shelves will have 3 dishes, and one shelf will be empty.

The reason that the middle shelf must be one of the 3-dish shelves (cannot be empty), is because of rule 3: dish 6 is either one shelf above or one shelf below dish 5. That means that no matter where we put that vertical block, at all times either dish 5 or dish 6 must be on the middle shelf.

Consider: if you put 5 on the top shelf, then 6 must be on the middle to keep them adjacent. Similarly, if you have either 5 or 6 on the bottom, the other must be on the middle. And of course if you start with either on the middle shelf that ensures one of the two is there, as well. So no matter what placement options you go with, either 5 or 6 will always be on the middle shelf!

So what that tells us then for the 3-3-0 distribution is that the middle shelf, with either 5 or 6, must be one of those 3s, and that's why answer choice (B) is correct.

The bottom shelf in this case could absolutely be empty: put 6 on the middle, and 2 and 5 on the top. Now as long as we keep 1 and 4 away from each other we're good!

Similarly, the top shelf could be empty. Put 2 and 5 on the middle, with 6 on the bottom. Again, as long as 1 and 4 aren't together we're okay.

So while we can't know the empty shelf, we can know that the middle shelf is never empty, which means for question 9 it must have 3 dishes.

Final note: this is an idea that gets tested frequently and something we cover extensively in our books and courses called "Hurdle the Uncertainty." In short, it means that just because you cannot know the exact placement or behavior of certain variables, does NOT mean that you can't make inferences at times! Here, we don't know where 5 and 6 will go, but because they form a block with somewhat limited placement options we can know that one must always be on the middle shelf.

I hope that helps to clarify things! Let me know if you have further questions!

Jon