When the Manor Apartments building was constructed, the parking spaces for the building were all in a single row and...
GMAT Two Part Analysis : (TPA) Questions
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.
OWNING THE DATASET
Visual Representation
Initial Configuration:
Apartment: 1 2 3 4 5 6 7 8 Space: 1 2 3 4 5 6 7 8
Exchange Timeline:
- Year 0: Original assignments
- Year 1: Apartment 2 exchanges with Apartment A
- Year 2: Apartment 2 exchanges with Apartment B \(\mathrm{B} \neq \mathrm{A})\)
- Current: Apartment 2's space is adjacent to spaces of Apartments 5 & 6
Key Constraints
- Exactly 2 exchanges, both involving Apartment 2
- Different apartments in each exchange \(\mathrm{A} \neq \mathrm{B})\)
- Final result: Apt 2's space is between spaces of Apts 5 & 6
- Only apartments 2, A, and B have non-original spaces
SEEKING THE CRITICAL INSIGHT
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:
- If neither 5 nor 6 is A or B, they'd have spaces 5 and 6 (adjacent with no space between)
- Therefore, at least one of \{5, 6\} must be in \{\mathrm{A}, \mathrm{B}\}
UNDERSTANDING THE QUESTION
We need to identify:
- First: Apartment that exchanged with Apartment 2 in year 1 (A)
- Second: Apartment that exchanged with Apartment 2 in year 2 (B)
Answer choices: 1, 3, 4, 5, 6, 8
PROCESSING THE SOLUTION
Tracking the Exchanges
After first exchange \(2 \leftrightarrow \mathrm{A})\):
- Apartment 2 has Space A
- Apartment A has Space 2
After second exchange \(2 \leftrightarrow \mathrm{B})\):
- Apartment 2 has Space B
- Apartment B has Space A
- Apartment A still has Space 2
Finding Valid Configurations
For Apartment 2 (with Space B) to be adjacent to both 5 and 6's current spaces:
Case 1: B = 5
- Apartment 2 has Space 5
- Space 5 is adjacent to spaces 4 and 6
- Need Apartments 5 and 6 to have spaces 4 and 6
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:
- Apt 2: Space 5 (adjacent to 4 and 6) ✓
- Apt 5: Space 4 ✓
- Apt 6: Space 6 ✓
Case 2: B = 6
- Apartment 2 has Space 6
- Space 6 is adjacent to spaces 5 and 7
- Need Apartments 5 and 6 to have spaces 5 and 7
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:
- Apt 2: Space 6 (adjacent to 5 and 7) ✓
- Apt 5: Space 5 ✓
- Apt 6: Space 7 ✓
Checking Against Answer Choices
Valid solutions:
- Solution 1: First = 4, Second = 5
- Solution 2: First = 7, Second = 6
From available choices (1, 3, 4, 5, 6, 8):
- For First: Only 4 is available (7 is not in the list)
- For Second: Both 5 and 6 are available
Since only First = 4 is possible from the choices, we must use Solution 1.
FINAL SOLUTION SYNTHESIS
Solution Path Recap:
- Recognized that Apt 2's final space must be physically between spaces of Apts 5 & 6
- Deduced that one of \{5, 6\} must have exchanged with Apt 2
- Worked through cases where B = 5 or B = 6
- Found two valid configurations: \(\mathrm{A}=4, \mathrm{B}=5)\) and \(\mathrm{A}=7, \mathrm{B}=6)\)
- Selected the only configuration matching available answer choices
Final Answer:
- First: 4
- Second: 5
Key Insight: The constraint of being adjacent to two specific apartments in a linear arrangement drastically limits possible configurations, making systematic case analysis efficient.