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.

A complete and accurate list of the states, any one of which could contain the management institute that is not in State 6, would be _____.

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

Hint:

Rule out State 2 and State 3, which are tied up by technical institutes, and State 6, which already has the first management institute, then see what is left.

Answer 1

The Correct Option is D

The question asks for every state that is still free to receive the second management institute, the one that is not already sitting in State 6. The fastest way to answer this is to rule out the states that are blocked, rather than confirm the ones that work.

1. 1, 4: incomplete. Both State 1 and State 4 are genuinely open, but this list leaves out State 5, which is also open, so it is not the complete list the question asks for.
2. 2, 4: wrong. State 2 already holds a technical institute, since it had to take the second technical institute once the boundary rule ruled out State 1, State 4 and State 5, and State 6 already had the management institute. A state cannot hold both a management and a technical institute, so State 2 is blocked.
3. 4, 5: incomplete. Both are open, but State 1 is missing from this list, so again it is not the complete list.
4. 1, 4, 5: correct. State 2 and State 3 are blocked because each already holds a technical institute, and State 6 is blocked because it already holds the other management institute. That leaves exactly State 1, State 4 and State 5 open, and every one of them can pair a management institute with a medical institute without breaking any rule.

So the complete and accurate list is State 1, State 4 and State 5, since these are the only three states not already tied up by a technical institute or the first management institute.

Let's summarize:

• State 2 and State 3 are out because they hold the technical institutes, and technical and management cannot mix in one state.
• State 6 is out because it already holds the other management institute.
• State 1, State 4 and State 5 remain completely open, and each can take a management institute paired with a medical one.

The correct answer is the list 1, 4, 5.