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 each of the six states contains at least one of the eight institutions, then which one of the following must be true?

• A. There is a management institute in State 1
• B. There is a medical institute in State 2
• C. There is a medical institute in State 3
• D. There is a medical institute in State 4

Hint:

Count how many medical and management institutes are left after fixing State 2, State 3 and State 6, then see how they must be shared among State 1, State 4 and State 5.

Answer 1

The Correct Option is D

The extra condition here, that every one of the six states must hold at least one institution, is the key that turns a "could be true" question into a "must be true" question. Before this condition, State 1, State 4 and State 5 were free to be empty. After it, none of them can be.

Count what is left to distribute. The 8 institutions are 4 medical, 2 management and 2 technical. State 2 and State 3 already use up the 2 technical institutes. State 6 already uses up 1 management institute and, by the rule that a management institute needs a medical institute in its own state, 1 medical institute too. That leaves 1 management institute and 3 medical institutes still to be placed, and they can only go into State 1, State 4 and State 5, since State 2, State 3 and State 6 are already accounted for.

Since each of State 1, State 4 and State 5 must now get at least one institution, and there are exactly 4 institutions left for exactly 3 states, one state gets 2 institutions, the leftover management institute plus a medical institute, since management needs a medical partner, and the other two states get 1 medical institute each. Every possible way of doing this still gives all three states a medical institute.

1. Management institute in State 1: only holds in the one arrangement where State 1 is chosen to carry the leftover management institute, not in the other two, so it does not always hold.
2. Medical institute in State 2: cannot hold, State 2's institution slot is already taken by a technical institute, and using it for medical too would strand one of State 1, State 4 or State 5 without any institution.
3. Medical institute in State 3: cannot hold, for the same reason as State 2.
4. Medical institute in State 4: always holds, because every possible way of sharing out the leftover management institute and 3 medical institutes among State 1, State 4 and State 5 gives State 4 at least one medical institute.

So the statement that always holds, no matter how the remaining institutes are shared out, is that State 4 has a medical institute.

Let's summarize:

• State 2 and State 3 are full with technical institutes, so they cannot take on a medical institute without breaking the "every state has something" rule.
• The last management institute and the last 3 medical institutes must be shared among State 1, State 4 and State 5, each of which needs at least one institution.
• Every possible sharing gives all three of State 1, State 4 and State 5 a medical institute, so this is guaranteed for State 4 specifically.

The correct answer is that there is a medical institute in State 4.