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

Step 1: Recall what is already fixed.
From the base rules, State 3 and State 2 hold the two technical institutes. State 3 is given directly, and State 2 is the only state left once the boundary rule and the management-technical clash rule are applied to State 6. State 6 holds one management institute plus at least one medical institute.

Step 2: Work out which states can take the second management institute.
The rules say no state can hold both a management institute and a technical institute. State 2 and State 3 already hold technical institutes, so neither of them can take the second management institute.
That leaves State 1, State 4 and State 5 as the only states still open. Nothing in the rules blocks any of these three from holding a management institute, as long as that state also gets at least one medical institute (rule 4), which is always doable since there are four medical institutes to place in total.

Step 3: Check that all three, and only these three, work.
State 1: free of any fixed institute, can take the management institute together with a medical institute. Works.
State 4: free of any fixed institute, same reasoning. Works.
State 5: free of any fixed institute, same reasoning. Works.
No other state qualifies, since State 2 and State 3 are blocked by the technical institute clash, and State 6 already holds the other management institute.

Final Answer:
The complete list of states that could hold the second management institute is State 1, State 4 and State 5. \[ \boxed{1,\ 4,\ 5} \]