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 one of the states contains exactly two medical institutes and exactly one technical institute, then which combination of three states might contain no medical institute?

• A. 1, 3, 5
• B. 1, 4, 5
• C. 2, 3, 5
• D. 2, 4, 6

Hint:

The state with 2 medical institutes and 1 technical institute must be State 2 or State 3, since only those two ever hold a technical institute; work out both cases.

Answer 1

The Correct Option is A

Step 1: Work out which state is being described.
The question describes a state with exactly one technical institute and exactly two medical institutes in it. Only two states hold a technical institute at all: State 3 (given) and State 2 (forced earlier, since it is the only state left once the boundary rule and the management clash rule are applied).
So the state described in the question is either State 2 or State 3, and it holds 2 of the 4 medical institutes on top of its technical institute. We check both possibilities.

Step 2: Case 1, State 2 holds 2 medical institutes and 1 technical institute.
State 6 already holds 1 management institute and 1 medical institute, fixed from the base rules. State 2 now holds 2 medical institutes.
That uses up 3 of the 4 medical institutes, 2 in State 2 and 1 in State 6, leaving 1 medical institute, and it uses up 1 of the 2 management institutes, leaving 1 management institute.
The last management institute must go into State 1, State 4 or State 5, since State 2, State 3 and State 6 are all unavailable to it, and it needs a medical institute in the same state (rule 4).
So the last management institute and the last medical institute both land in the same one of State 1, State 4 or State 5, say State 4. State 1 and State 5 then receive nothing at all, so they hold no medical institute either.
In this case, State 1, State 3 and State 5 all end up with no medical institute: State 3 only ever holds its technical institute in this case, and State 1 and State 5 get nothing since the last pair of institutes both went to State 4.

Step 3: Case 2, State 3 holds 2 medical institutes and 1 technical institute.
By the same counting, State 6 holds 1 management plus 1 medical, State 3 holds 2 medical plus 1 technical, using 3 of the 4 medical institutes and 1 of the 2 management institutes.
Again the last management institute and last medical institute land together in one of State 1, State 4 or State 5. State 2, holding only its technical institute, gets no medical institute in this case.
Any combination of three no-medical states in this case would have to include State 2, since State 2 never gets a medical institute here.

Step 4: Match the case results against the options.
Option A, 1, 3, 5: matches Case 1 exactly, where State 4 absorbs the last management and medical institute, leaving State 1, State 3 and State 5 with none. This is possible.
Option B, 1, 4, 5: not possible in Case 1, since one of State 1, State 4, State 5 always receives the last medical institute together with the management institute, so this exact trio cannot all be empty at once.
Option C, 2, 3, 5: includes State 2, which only lacks medical institutes in Case 2, but Case 2 has State 3 holding 2 medical institutes, contradicting "State 3 has no medical institute" in this option. Not possible.
Option D, 2, 4, 6: includes State 6, which always holds a medical institute by rule 4. Not possible.

Final Answer:
Only the combination State 1, State 3 and State 5 can be the trio with no medical institute. \[ \boxed{1,\ 3,\ 5} \]