Question

Six square states of equal area are arranged in two columns running north to south in a country. State 1, State 3, and State 5 lie on the western side, from north to south, and State 2, State 4, and State 6 lie on the eastern side, from north to south, so the states directly across the street from each other are State 1 and State 2, State 3 and State 4, and State 5 and State 6. Two states share a boundary if they sit next to each other in the same column (north-south) or directly across from each other (east-west). Within these six states there are exactly eight institutions in total: four medical institutes, two management institutes, and two technical institutes, placed according to these rules:

1. No institution is located in more than one state.
2. No state contains more than one management institute, and no state contains more than one technical institute.
3. No state contains both a management institute and a technical institute.
4. Every management institute is located in a state that also contains at least one medical institute.
5. The two technical institutes are located in states that do not share a common boundary.
6. State 3 contains a technical institute, and State 6 contains a management institute.

If one of the states contains exactly two medical institutes and exactly one technical institute, then which combination of three states might contain no medical institute?

• A. 1, 3, 5
• B. 1, 4, 5
• C. 2, 3, 5
• D. 2, 4, 6

Hint:

The state with 2 medical institutes and 1 technical institute must be State 2 or State 3, since only those two ever hold a technical institute; work out both cases.

Answer 1

The Correct Option is A

This question layers an extra condition onto the base puzzle: one state has exactly 2 medical institutes and 1 technical institute together. Since only State 2 and State 3 ever hold a technical institute, this "one state" has to be State 2 or State 3, so the problem splits into two cases, and each option needs to be checked against both.

In both cases, State 6 keeps its fixed 1 management and 1 medical institute. Whichever of State 2 or State 3 is the described state takes 2 medical institutes, which uses up 3 of the 4 medical institutes in total, 2 there and 1 in State 6. That leaves exactly 1 medical institute and 1 management institute still to place, and they must land together in whichever one of State 1, State 4 or State 5 takes the second management institute, since a management institute always needs a medical partner in the same state. The other two of State 1, State 4, State 5 then get nothing.

1. 1, 3, 5: this matches the case where State 2 is the one with 2 medical institutes. State 3 then has only its technical institute and no medical institute, and if the leftover management-and-medical pair goes to State 4, then State 1 and State 5 also end up with no medical institute. All three conditions line up, so this combination is possible.
2. 1, 4, 5: this would need all three of State 1, State 4 and State 5 to lack a medical institute at once. But the leftover management institute always has to sit with a medical institute in one of these three states, so one of them always has a medical institute. This combination can never happen.
3. 2, 3, 5: for State 2 to have no medical institute, we need Case 2, where State 3 holds the 2 medical institutes. But then State 3 has 2 medical institutes, contradicting the requirement that State 3 also has no medical institute in this option. The two conditions cancel each other out, so this combination is impossible.
4. 2, 4, 6: State 6 always has at least 1 medical institute, since its management institute needs one by rule 4. Any option that claims State 6 has no medical institute is automatically wrong.

Only the first combination survives both cases cleanly, so State 1, State 3 and State 5 are the trio that might contain no medical institute.

Let's summarize:

• The "2 medical plus 1 technical" state can only be State 2 or State 3, since those are the only two states that ever hold a technical institute.
• Whichever case applies, exactly 1 medical and 1 management institute remain, and they always pair up in the same state among State 1, State 4 and State 5.
• State 6 can never be medical-free, so any option listing State 6 is ruled out immediately.

The correct combination is State 1, State 3 and State 5.