Which one of the following could be true?

Ben-Couse on October 14, 2020

What is the best way to solve this question?

Victoria on October 15, 2020

Hi @Kweb1016 and @Ben-Couse,

Happy to help!

You don't need to create a hypothetical for each answer choice, but it does require a bit more work at the start.

We know that the dealer will auction exactly one antique per day for six consecutive days. These items are: H, L, M, S, T, and V.

_ _ _ _ _ _
1 2 3 4 5 6

Now let's go through our conditions.

Rule 1 - S is not auctioned on June 1st

Rule 2 - if H is auctioned on an earlier day than L, then M is also auctioned on an earlier date than L

H > L --> M > L
M < L --> H < L

Rule 3 - S is auctioned earlier than both M and V

> M
S
> V

Rule 4 - T is auctioned earlier than H or V, but not both

T > H OR T > V

So, what can we conclude based on the rules above? Our most restrictive rule seems to be Rule 2, so let's diagram out a couple options based on this. We have two options: either H is auctioned earlier than L, or H is not auctioned earlier than L

Option 1: H is auctioned earlier than L
H
> L
> M
S
> V

Option 2: H is not auctioned earlier than L

> H
L
> M
S
> V

You don't need to create full hypotheticals for each scenario here. We can eliminate all of the options using the rules alone or our two possible order options we have diagrammed above.

We can immediately eliminate answer choice (C) because it violates Rule 3 which tells us that S is auctioned earlier than M.

We can eliminate answer choice (D) for the same reason. Rule 3 tells us that S is auctioned earlier than V.

Answer choice (E) cannot be true because Rule 3 tells us that S is auctioned earlier than both M and V. In this scenario, there is only one open date after S is auctioned.

_ _ _ S T _
1 2 3 4 5 6

Answer choice (A) cannot be true because of the second option we've outlined above.

Rule 4 tells us that T is auctioned earlier than H or V, but not both. We know that L must be auctioned earlier than H. This would mean that T must also be auctioned earlier than H.

Therefore, V would need to be auctioned earlier than T. However, this is impossible as there is only one date preceding T and we know from Rule 3 that S must be auctioned earlier than V.

Therefore, this cannot be true because S and V must both be auctioned before T but T is auctioned on the second day.

_ T L _ _ _
1 2 3 4 5 6

This means that answer choice (B) should be correct, but let's double-check using a hypothetical based on the second option we've outlined above. Remember, to prove that this could be true, we just need to show one possible case where it can be true.

L S V T M H
1 2 3 4 5 6

Notice that this meets all of our conditions.

1) S is auctioned on June 2nd
2) H is not auctioned earlier than L, so this rule is irrelevant.
3) S is auctioned earlier than both M and V
4) T is auctioned earlier than H and later than V

This is definitely a skill that comes with practice.

Keep up the great work and please let us know if you have any further questions!