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When the Manor Apartments building was constructed, the parking spaces for the building were all in a single row and were numbered sequentially, from 1 through 8. Each space was assigned to the apartment with the same number; there were no other apartments in the building. A year later, however, the parking space assignment for Apartment 2 was exchanged with that for another apartment, and then a year after that, the parking space assignment for Apartment 2 was exchanged again, this time with that for yet another apartment. No other exchanges have been made. Currently, the parking space assigned to Apartment 2 is adjacent on one side to the space assigned to Apartment 6 and on the other side to the space assigned to Apartment 5.
Select two apartment numbers such that the one selected for First could have been the number of the apartment whose assignment was exchanged with that of Apartment 2 after the first year, and the one selected for Second could have been the number of the apartment whose assignment was exchanged with that of Apartment 2 after the second year, jointly consistent with the information provided.
First
Second
1
3
4
5
6
8
Initial Configuration:
Apartment: 1 2 3 4 5 6 7 8 Space: 1 2 3 4 5 6 7 8
Exchange Timeline:
The critical insight: Since parking spaces are in a single row (1-8), for Apartment 2's space to be adjacent to both Apartments 5 and 6's spaces, it must be physically between them.
Given that only three apartments (2, A, B) have non-original assignments, all others retain their original spaces. This means:
We need to identify:
Answer choices: 1, 3, 4, 5, 6, 8
After first exchange \(2 \leftrightarrow \mathrm{A})\):
After second exchange \(2 \leftrightarrow \mathrm{B})\):
For Apartment 2 (with Space B) to be adjacent to both 5 and 6's current spaces:
Case 1: B = 5
Since Apartment 5 = B, it has Space A. So A must equal 4.
Apartment 6 wasn't involved \(6 \neq \mathrm{A} = 4, 6 \neq \mathrm{B} = 5)\), so it has Space 6.
Verification:
Case 2: B = 6
Since Apartment 6 = B, it has Space A. So A must equal 7.
Apartment 5 wasn't involved \(5 \neq \mathrm{A} = 7, 5 \neq \mathrm{B} = 6)\), so it has Space 5.
Verification:
Valid solutions:
From available choices (1, 3, 4, 5, 6, 8):
Since only First = 4 is possible from the choices, we must use Solution 1.
Solution Path Recap:
Final Answer:
Key Insight: The constraint of being adjacent to two specific apartments in a linear arrangement drastically limits possible configurations, making systematic case analysis efficient.