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.

Which one of the following could be true?

• A. State 1 contains exactly one technical institute
• B. State 1 contains exactly one medical institute
• C. State 2 contains exactly one management institute
• D. State 5 contains exactly one technical institute

Hint:

Work out which state must hold the second technical institute, using the boundary rule and the fact that State 6 already has a management institute.

Answer 1

The Correct Option is B

This question checks whether you can lock in the few facts that are forced by the rules, then test each choice one at a time. Start by fixing the grid: State 1, State 3, State 5 run north to south on the west side, and State 2, State 4, State 6 run north to south on the east side, with State 1 facing State 2, State 3 facing State 4, and State 5 facing State 6. A state borders the state above and below it in its own column, and the state directly opposite it.

State 3 already holds a technical institute. Its neighbours are State 1 (north), State 4 (opposite) and State 5 (south). Since the two technical institutes cannot share a boundary, none of State 1, State 4 or State 5 can hold the second technical institute. That leaves State 2 or State 6. State 6 already holds a management institute, and a state can never hold both a management and a technical institute together, so the second technical institute has to sit in State 2. State 6, holding a management institute, must also hold at least one medical institute, since every management institute needs a medical institute in its own state.

1. State 1 contains exactly one technical institute: false. State 1 borders State 3, and the two technical institutes cannot be in bordering states.
2. State 1 contains exactly one medical institute: possible. State 1 has no institute fixed yet, and nothing in the rules stops it from being given a medical institute when the remaining institutes are placed.
3. State 2 contains exactly one management institute: false. State 2 already carries the second technical institute, and no state can carry both a management and a technical institute.
4. State 5 contains exactly one technical institute: false. State 5 borders State 3, the same reason that rules out State 1.

Only the second choice, State 1 contains exactly one medical institute, survives all four checks, so it is the one statement that could be true.

Let's summarize:

• The second technical institute is forced into State 2, since State 6 already has a management institute and the rules forbid a technical-and-management state.
• State 1 and State 5 are ruled out for technical institutes because both border State 3.

So the correct choice is that State 1 contains exactly one medical institute.