Understanding the Sitting Arrangement Puzzle

This question asks us to determine the person sitting immediately to the left of D, given two statements about the linear arrangement of six people (A, B, C, D, E, F) facing north.

In a linear arrangement facing north, "immediate left" means the person sitting in the seat position immediately lower than the person in question. For example, if D is at position 4, the person at position 3 is to the immediate left of D.

Let's analyze each statement separately first, and then combine them.

Analyzing Statement I for Sitting Arrangement

Statement I provides three clues:

1. A sits second from one of the extreme ends.
2. Only two people sit between A and B.
3. E sits to the immediate right of C.

Let the positions be 1 to 6 from left to right.

• Clue 1: A is at position 2 or position 5.
• Clue 2: If A is at position 2, with two people between A and B, B must be at position $2+2+1 = 5$. Arrangement starts as _ A _ _ B _.
• Clue 2: If A is at position 5, with two people between A and B, B must be at position $5-2-1 = 2$. Arrangement starts as _ B _ _ A _.
• Clue 3: E is immediately to the right of C, meaning C and E sit together as CE.

Let's explore the possibilities from Statement I:

• Case 1.1: A is at position 2, B is at position 5. Arrangement: _ A _ _ B _. The remaining people (C, D, E, F) must fill positions 1, 3, 4, 6.
  • The pair CE must sit together. Possible spots for CE: (3,4).
  • If CE are at 3,4: _ A C E B _. Remaining people are D, F to fill positions 1, 6.
    • Arrangement can be D A C E B F or F A C E B D.
    • If D A C E B F: D is at position 1. There is no one to the immediate left of D.
    • If F A C E B D: D is at position 6. Immediate left of D is B (at position 5).
• Case 1.2: A is at position 5, B is at position 2. Arrangement: _ B _ _ A _. The remaining people (C, D, E, F) must fill positions 1, 3, 4, 6.
  • The pair CE must sit together. Possible spots for CE: (3,4). (Cannot be (1,2) because B is at 2; cannot be (4,5) because A is at 5; cannot be (5,6) because A is at 5).
  • If CE are at 3,4: _ B C E A _. Remaining people are D, F to fill positions 1, 6.
    • Arrangement can be D B C E A F or F B C E A D.
    • If D B C E A F: D is at position 1. There is no one to the immediate left of D.
    • If F B C E A D: D is at position 6. Immediate left of D is A (at position 5).

From Statement I alone, the person to the immediate left of D could be B, A, or nobody (if D is at an end). Therefore, Statement I alone is not sufficient.

Analyzing Statement II for Sitting Arrangement

Statement II provides three clues:

1. C sits third from the left end.
2. Only one person sits between C and F.
3. E sits third to the right of F.

Let's explore the possibilities from Statement II:

• Clue 1: C is at position 3. Arrangement: _ _ C _ _ _
• Clue 2: One person sits between C (pos 3) and F. F can be at position 1 or position 5.
  • Possibility 2.1: F is at position 1. F _ C _ _ _
  • Possibility 2.2: F is at position 5. _ _ C _ F _
• Clue 3: E is third to the right of F.
  • In Possibility 2.1 (F at pos 1): Third to the right of F (pos 1) is position $1+3 = 4$. So, E is at position 4. Arrangement: F _ C E _ _
  • In Possibility 2.2 (F at pos 5): Third to the right of F (pos 5) is position $5+3 = 8$. This position is outside the line of 6 people. So, this possibility is not valid.

So, Statement II forces the arrangement F _ C E _ _. The people at positions 1, 3, 4 are F, C, E respectively. The remaining people (A, B, D) must fill positions 2, 5, 6.

Let's list some possible arrangements and who is left of D:

• F A C E B D: D is at pos 6, left is B (pos 5).
• F B C E A D: D is at pos 6, left is A (pos 5).
• F D C E A B: D is at pos 2, left is F (pos 1).
• F A C E D B: D is at pos 5, left is E (pos 4).

From Statement II alone, the person to the immediate left of D could be B, A, F, or E. Therefore, Statement II alone is not sufficient.

Combining Statements I and II

Statement II gives us the structure F _ C E _ _. C is at 3, E is at 4, F is at 1.

Statement I tells us: A is 2nd from an end (pos 2 or 5), two people between A and B, E is immediate right of C (which is consistent with C at 3, E at 4 from Statement II).

Now we use the A/B rule from Statement I (A 2nd from end, two between A and B) within the F _ C E _ _ structure from Statement II. The remaining people are A, B, D for positions 2, 5, 6.

• From Statement I, A is at position 2 or 5.
  • Possibility A: A is at position 2. The structure becomes F A C E _ _. Remaining people are B, D for positions 5, 6.
    • According to Statement I, two people sit between A (pos 2) and B. If A is at 2, B must be at position $2+2+1 = 5$.
    • Placing B at 5: F A C E B _. The remaining person D must go at position 6.
    • Arrangement: F A C E B D.
    • Check if this arrangement satisfies both statements:
      • I: A 2nd from end (yes, left end). Two between A (2) and B (5) (yes, C, E). E immediate right of C (yes, C at 3, E at 4). (Statement I satisfied)
      • II: C 3rd from left (yes, at 3). One between C (3) and F (1) (yes, A at 2). E 3rd right of F (yes, F at 1, E at 4). (Statement II satisfied)
    • In this valid arrangement (F A C E B D), D is at position 6. The person to the immediate left of D is B (at position 5).
  • Possibility B: A is at position 5. The structure becomes F _ C E A _. Remaining people are B, D for positions 2, 6.
    • According to Statement I, two people sit between A (pos 5) and B. If A is at 5, B must be at position $5-2-1 = 2$.
    • Placing B at 2: F B C E A _. The remaining person D must go at position 6.
    • Arrangement: F B C E A D.
    • Check if this arrangement satisfies both statements:
      • I: A 2nd from end (yes, right end). Two between A (5) and B (2) (yes, C, E). E immediate right of C (yes, C at 3, E at 4). (Statement I satisfied)
      • II: C 3rd from left (yes, at 3). One between C (3) and F (1) (yes, B at 2). E 3rd right of F (yes, F at 1, E at 4). (Statement II satisfied)
    • In this valid arrangement (F B C E A D), D is at position 6. The person to the immediate left of D is A (at position 5).

We found two distinct valid arrangements that satisfy all conditions from both statements:

In the first valid arrangement, B is immediately to the left of D. In the second valid arrangement, A is immediately to the left of D.

Since combining both statements leads to more than one possible answer for the person sitting immediately to the left of D, the data in statements I and II together is not sufficient to answer the question.

Conclusion on Data Sufficiency

Statement I alone is not sufficient. Statement II alone is not sufficient. Statements I and II together are not sufficient because they lead to multiple possible valid seating arrangements, each giving a different person to the immediate left of D.

Revision Table: Analyzing Sitting Arrangement Statements

Additional Information on Linear Sitting Arrangements

Linear sitting arrangement problems are common in logical reasoning sections of exams. Key terms to understand include:

• Facing North/South: Determines left/right orientation. North means left is towards lower position numbers, right is towards higher. South is the opposite.
• Immediate Left/Right: The very next person in that direction.
• 'N' persons between X and Y: If X is at position p, Y will be at position p + N + 1 or p - N - 1, assuming they are on the same side of X relative to direction.
• 'N' to the Left/Right: If X is at position p facing North, the person N positions to the right is at $p+N$, and N positions to the left is at $p-N$.
• From the End: Refers to counting from the first or last position in the line. Second from the left end is position 2. Second from the right end in a 6-person line is position 5.

When solving data sufficiency questions involving arrangements, it is crucial to check if the combined information leads to a unique arrangement or a unique answer to the specific question asked. If multiple valid arrangements provide different answers to the question, the data is not sufficient.