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

Step 1: Map out the grid and the adjacency rule.
The six states sit in two north-south columns. State 1, State 3 and State 5 form the western column from north to south, and State 2, State 4 and State 6 form the eastern column from north to south.
So State 1 sits next to State 3 (same column) and across from State 2. State 3 sits next to State 1 above it, State 5 below it, and across from State 4. State 5 sits next to State 3 and across from State 6.
Two states share a common boundary only if they are next to each other in this grid, either in the same column or directly across the street.

Step 2: Place the two technical institutes.
We are told State 3 has a technical institute. The rule says the two technical institutes sit in states that do not share a boundary.
State 3's boundary neighbours are State 1, State 4 and State 5, so none of these three can hold the second technical institute.
That leaves only State 2 and State 6 as candidates. But State 6 already has a management institute, and no state can hold both a management and a technical institute (rule 3).
So the second technical institute must sit in State 2. State 1, State 4 and State 5 are now confirmed to have no technical institute.

Step 3: Fix what State 6 must contain.
State 6 has a management institute, and every management institute needs at least one medical institute in the same state (rule 4).
So State 6 holds a management institute and at least one medical institute. State 6 cannot hold a technical institute, as already ruled out.

Step 4: Test each option against these fixed facts.
Option A, State 1 has a technical institute: ruled out in Step 2, since State 1 borders State 3. This cannot be true.
Option B, State 1 has exactly one medical institute: nothing blocks this. State 1 has no fixed institute yet, and it is free to receive a medical institute later when the remaining institutes are placed. This can be true.
Option C, State 2 has a management institute: State 2 already holds a technical institute, and rule 3 forbids a state from holding both. This cannot be true.
Option D, State 5 has a technical institute: ruled out in Step 2, since State 5 borders State 3. This cannot be true.

Final Answer:
Only option B survives every rule, so State 1 can contain exactly one medical institute. \[ \boxed{\text{Option B}} \]