Episode 8 — Aptitude and Reasoning / 8.23 — Arrangements

8.23.c Solved Examples -- Arrangements

Example 1: Simple Linear Arrangement

Problem: Five friends A, B, C, D, E sit in a row facing North.

  • B sits at the right end.
  • A is to the immediate left of B.
  • C is to the immediate left of D.
  • E sits at the left end.

Find the arrangement.

Solution:

Step 1: Fixed positions.

  E  __  __  __  B
  (1) (2) (3) (4) (5)

Step 2: A is to the immediate left of B -> A at position 4.

  E  __  __  A  B
  (1) (2) (3) (4) (5)

Step 3: C is to the immediate left of D. Remaining: C, D for positions 2, 3. C left of D -> C at 2, D at 3.

  E  C  D  A  B
  (1) (2) (3) (4) (5)

Step 4: Verify all conditions. All satisfied.

Answer: E C D A B (left to right)

Example 2: Linear Arrangement with "Between"

Problem: Six persons P, Q, R, S, T, U sit in a row facing North.

  • T sits at one end.
  • S is 3rd from the left.
  • Q is adjacent to S but not to T.
  • R is adjacent to T.
  • U is not adjacent to R.

Find the arrangement.

Solution:

Step 1: S is 3rd from left.

  __  __  S  __  __  __
  (1) (2) (3) (4) (5) (6)

Step 2: T sits at one end.

Case 1: T at position 1.

  T  __  S  __  __  __

Q adjacent to S but NOT to T -> Q at position 4 (adjacent to S at 3, not adjacent to T at 1). R adjacent to T -> R at position 2.

  T  R  S  Q  __  __

Remaining: P, U for positions 5, 6. U is not adjacent to R (R is at 2) -> U can be at 5 or 6 (both are fine). But no additional constraint to decide order. Let's say both P and U are possible at 5 and 6.

Actually, U is not adjacent to R (position 2). Positions 5 and 6 are not adjacent to 2, so either works. We need more info or Q might also be at position 2.

Wait: Q adjacent to S -> Q at 2 or 4. Q not adjacent to T (pos 1) -> Q not at 2. So Q at 4. Confirmed.

  T  R  S  Q  {P,U}  {P,U}

U not adjacent to R (pos 2): pos 5 and 6 are not adjacent to 2. Both work. Two possibilities:

  T  R  S  Q  P  U    or    T  R  S  Q  U  P

Case 2: T at position 6.

  __  __  S  __  __  T

Q adjacent to S but not to T -> Q at 2 or 4. Not adjacent to T(6) -> Q not at 5. Q at 2 or 4. R adjacent to T -> R at position 5.

If Q at 2:

  __  Q  S  __  R  T

Remaining: P, U for positions 1 and 4. U not adjacent to R(5) -> U not at 4. U at 1, P at 4.

  U  Q  S  P  R  T

If Q at 4:

  __  __  S  Q  R  T

Remaining: P, U for positions 1, 2. U not adjacent to R(5) -> fine at 1 or 2.

  P  U  S  Q  R  T    or    U  P  S  Q  R  T

Multiple valid arrangements exist. If the question asks for one specific:

Answer: Possible arrangements include T R S Q P U, U Q S P R T, etc.

Example 3: Circular Arrangement

Problem: Six people A, B, C, D, E, F sit around a circular table facing the center.

  • A is opposite D.
  • B is to the immediate right of A.
  • C is to the immediate left of D.
  • E is not adjacent to A.

Find the arrangement.

Solution:

Step 1: Fix A at the top (circular = fix one person).

         A
        / \
      ?     B      (B is to the immediate right of A = clockwise)
      |     |
      ?     ?
        \ /
         D         (A is opposite D)

Step 2: C is to the immediate left of D = anti-clockwise from D.

         A
        / \
      ?     B
      |     |
      C     ?
        \ /
         D

Step 3: Remaining: E, F for the two empty positions (left of A, right of B... wait, let me label properly).

  Position layout (clockwise from A):
  A -> B -> ? -> D -> C -> ?

Remaining: E, F. E is not adjacent to A. Adjacent to A = B (right) and position 6 (left). Position 6 (to A's left) is the last empty. If E is at position 3, E is not adjacent to A. Then F at position 6.

  Clockwise: A -> B -> E -> D -> C -> F
  
         A
        / \
      F     B
      |     |
      C     E
        \ /
         D

Verify: E adjacent to B and D. E NOT adjacent to A. Correct!

Answer: Clockwise from A: A, B, E, D, C, F

Example 4: Double Row Arrangement

Problem: Eight people A, B, C, D (Row 1, facing South) and P, Q, R, S (Row 2, facing North) sit in two rows facing each other.

  • A faces R.
  • B is to the immediate right of A.
  • S faces D.
  • Q is not at any end of Row 2.

Find the arrangement.

Solution:

Step 1: Setup (observer's perspective, left to right).

  Row 1 (facing South): __  __  __  __
                          |   |   |   |
  Row 2 (facing North): __  __  __  __
  Positions:             (1) (2) (3) (4)

Step 2: A faces R. Let's say A is at position X in Row 1, R is at position X in Row 2.

Step 3: B is to the immediate right of A. People facing South: their right (from their perspective) = observer's LEFT. So B is to A's immediate right = position to A's LEFT in the diagram?

Actually, let's clarify. Row 1 faces South. If you face South, your right hand points West. If the diagram has positions numbered left(1) to right(4):

When facing South: your LEFT = diagram's RIGHT, your RIGHT = diagram's LEFT.

So "B is to the immediate right of A" (from A's perspective facing South) = B is at the position to A's LEFT in the diagram.

Let's say A is at position 2. Then B (A's immediate right, facing South = diagram left) is at position 1.

If A is at position 2 and faces R, then R is at position 2 in Row 2.

  Row 1: B   A   __  __     (B at 1, A at 2)
          |   |   |   |
  Row 2: __  R   __  __

Step 4: S faces D. Remaining in Row 1: C, D at positions 3, 4.

If D at position 3, S at position 3 in Row 2. If D at position 4, S at position 4 in Row 2.

Step 5: Q is not at any end of Row 2. Ends of Row 2 = positions 1 and 4. Q must be at position 2 or 3. But position 2 is R. So Q at position 3.

If Q at position 3 in Row 2, and S faces D:

  • If D at position 3, S must be at position 3 too = conflict with Q. So D is NOT at position 3.
  • D at position 4, then S at position 4 in Row 2.
  Row 1: B   A   C   D      (C at 3, D at 4)
          |   |   |   |
  Row 2: __  R   Q   S      (Q at 3, S at 4)

Remaining in Row 2: P at position 1.

  Row 1: B   A   C   D
          |   |   |   |
  Row 2: P   R   Q   S

Step 6: Verify.

  • A(pos 2) faces R(pos 2). Correct.
  • B is to A's immediate right (facing South, right = left in diagram). B at 1, A at 2. Correct.
  • S(pos 4) faces D(pos 4). Correct.
  • Q at position 3 (not at end). Correct.

Answer:

  Row 1 (facing South): B  A  C  D
  Row 2 (facing North): P  R  Q  S

Example 5: Linear with Negative Constraints

Problem: Seven friends 1, 2, 3, 4, 5, 6, 7 sit in a row.

  • 4 sits at the exact center.
  • 1 and 7 sit at the two ends (not necessarily in this order).
  • 2 is not adjacent to 1.
  • 3 is adjacent to 4.
  • 6 is not adjacent to 4.

Find possible arrangements.

Solution:

Step 1: 4 at center (position 4 in a row of 7). 1 and 7 at ends.

  {1/7}  __  __  4  __  __  {7/1}
   (1)   (2) (3) (4) (5) (6)  (7)

Step 2: 3 is adjacent to 4 -> 3 at position 3 or 5.

Step 3: 6 is not adjacent to 4 -> 6 NOT at position 3 or 5.

Step 4: 2 is not adjacent to 1 -> if 1 is at position 1, then 2 is NOT at position 2. If 1 is at position 7, then 2 is NOT at position 6.

Case A: 1 at position 1, 7 at position 7.

  • 2 not at position 2.
  • 3 at position 3 or 5.

Sub-case A1: 3 at position 3.

  1  __  3  4  __  __  7

6 not at 3 or 5 -> 6 at position 2 or 6. 2 not at position 2 -> 2 at 5 or 6.

If 6 at 2: remaining 2, 5 at positions 5, 6.

  1  6  3  4  {2,5}  {5,2}  7
  -> 1  6  3  4  2  5  7    or    1  6  3  4  5  2  7

If 6 at 6: remaining 2, 5 at positions 2, 5. 2 not at 2 -> 2 at 5, 5 at 2.

  1  5  3  4  2  6  7

Sub-case A2: 3 at position 5. Similar analysis.

Case B: 1 at position 7, 7 at position 1. Similar.

Answer: Multiple valid arrangements, including: 1 6 3 4 2 5 7, 1 6 3 4 5 2 7, 1 5 3 4 2 6 7, etc.

Example 6: Circular with "Not Adjacent"

Problem: 5 people A, B, C, D, E sit around a circular table facing center.

  • A is not adjacent to B or C.
  • D is to the immediate right of B.

Find the arrangement.

Solution:

Step 1: Fix B. D is to B's immediate right (clockwise).

         ?
        / \
      ?     ?
      |     |
      B     D
        \ /
         ?

Wait, let me redraw. Fix B at top. D to B's right = clockwise from B.

         B
        / \
      ?     D
      |     |
      ?     ?
        \ /
         ?

Actually, 5 people:

  Clockwise: B, D, ?, ?, ?

Step 2: A is not adjacent to B or C. Adjacent to B: D (right) and position 5 (left). A must NOT be at D's position or position 5 or position adjacent to B.

Let me label clockwise: B(1), D(2), ?(3), ?(4), ?(5).

A not adjacent to B -> A not at positions 2 or 5. But D is at 2. So A not at 5. A at position 3 or 4.

A not adjacent to C: we need to figure out where C is.

Remaining: A, C, E for positions 3, 4, 5. A not at 5. So A at 3 or 4.

If A at 3: A's neighbors are D(2) and position 4. A not adjacent to C -> C not at 4. C at 5. E at 4.

  Clockwise: B, D, A, E, C

Check: A adjacent to D and E. A not adjacent to B (ok) or C (ok -- C is at 5, A at 3, not adjacent). Valid!

If A at 4: A's neighbors are position 3 and position 5. A not adjacent to C -> C not at 3 or 5. Remaining spots for C: only 3 and 5 are available! Contradiction.

So only one arrangement works.

Answer: Clockwise from B: B, D, A, E, C

Example 7: "Persons Between" in Linear

Problem: 8 persons sit in a row facing North. There are 3 persons between A and B. A sits at one end. C is to the immediate right of B. D is 2nd to the left of C.

Solution:

Step 1: A at one end, 3 persons between A and B -> B at position 5 from A.

Case 1: A at position 1. B at position 5.

  A  __  __  __  B  __  __  __
  (1) (2) (3) (4) (5) (6) (7) (8)

C is to the immediate right of B. Facing North: right = position to the right in diagram. C at position 6.

  A  __  __  __  B  C  __  __

D is 2nd to the left of C. C at 6 -> D at position 4.

  A  __  __  D  B  C  __  __

Case 2: A at position 8. B at position 4.

  __  __  __  B  __  __  __  A
  (1) (2) (3) (4) (5) (6) (7) (8)

C immediate right of B -> C at position 5. D 2nd to left of C -> D at position 3.

  __  __  D  B  C  __  __  A

Answer: Two possible arrangements (partial). Case 1: A _ _ D B C _ _. Case 2: _ _ D B C _ _ A.

Example 8: Square/Rectangular Arrangement

Problem: 8 people sit around a square table, 2 on each side, all facing the center.

  • A sits at a corner.
  • B is opposite A.
  • C is adjacent to A.
  • D is not adjacent to B.
  • E sits at a corner.

Solution:

  Square table with 8 seats:
  
      (1)  (2)
  (8)          (3)
  (7)          (4)
      (6)  (5)
  
  Corners: 1, 3, 5, 7 (or 2, 4, 6, 8 depending on convention)
  Let's say corners = positions 1, 3, 5, 7.

Step 1: A at a corner. Say A at position 1. B is opposite A -> B at position 5.

      A    (2)
  (8)          (3)
  (7)          (4)
      (6)   B

Step 2: C is adjacent to A. A at 1, adjacent = positions 2 and 8.

Step 3: D is not adjacent to B. B at 5, adjacent = positions 4 and 6. D not at 4 or 6.

Step 4: E at a corner. Corners: 1(A), 3, 5(B), 7. E at 3 or 7.

This gives multiple possibilities. The key takeaway is the approach.

Answer: Multiple valid arrangements depending on remaining constraints.

Example 9: Circular with Opposite and Adjacent

Problem: 6 friends P, Q, R, S, T, U sit around a circular table facing center.

  • P is opposite R.
  • Q is to the immediate left of P.
  • T is not adjacent to R.
  • S is to the immediate right of R.

Solution:

Step 1: Fix P at top. R opposite P.

         P
        / \
      Q     ?      (Q is to P's immediate left = anti-clockwise from P)
      |     |
      ?     ?
        \ /
         R

Step 2: S to R's immediate right (clockwise from R).

         P
        / \
      Q     ?
      |     |
      S     ?
        \ /
         R

Clockwise from P: P, ?, ?, R, S, Q.

Wait, let me reconsider. Clockwise from P: P -> (right of P = clockwise) -> ? -> R(opposite) -> ...

6 seats clockwise: P, X, Y, R, S (right of R = clockwise from R), Q (left of P = anti-clockwise = position 6 clockwise).

Hmm, let me be more careful. In a 6-person circle facing center:

  • Immediate right = clockwise.
  • Immediate left = anti-clockwise.

Fix P at position 1. Clockwise: 1(P), 2, 3, 4, 5, 6. R opposite P: R at position 4. Q immediate left of P: Q at position 6 (anti-clockwise from P). S immediate right of R: S at position 5 (clockwise from R).

  1(P) -> 2(?) -> 3(?) -> 4(R) -> 5(S) -> 6(Q)

         P(1)
        /     \
     Q(6)     (2)?
      |         |
     S(5)     (3)?
        \     /
         R(4)

Remaining: T, U for positions 2 and 3. T not adjacent to R(4). Position 3 is adjacent to R. So T not at 3. T at 2, U at 3.

         P(1)
        /     \
     Q(6)     T(2)
      |         |
     S(5)     U(3)
        \     /
         R(4)

Verify: P opposite R (yes). Q immediate left of P (yes). S immediate right of R (yes). T not adjacent to R -- T at 2, R at 4, they are not adjacent (yes).

Answer: Clockwise from P: P, T, U, R, S, Q

Example 10: Complex Double-Row with 6+6

Problem: 12 people: A-F in Row 1 (facing South), P-U in Row 2 (facing North).

  • B is at the right end of Row 1 (from B's perspective).
  • S faces B.
  • R is to the immediate left of S (from R's perspective).
  • A is not at any end.
  • T faces A.
  • P is at the left end of Row 2 (from P's perspective).

Solution:

Step 1: Facing South, right hand points West. If we draw left-to-right (West-to-East):

Row 1 facing South: Their right = our left. "B at right end from B's perspective" = B at the leftmost position in our diagram.

  Row 1 (facing South): B  __  __  __  __  __
                          |   |   |   |   |   |
  Row 2 (facing North): __  __  __  __  __  __
  Observer positions:    (1) (2) (3) (4) (5) (6)

Actually, this is getting confusing. Let me use observer positions and convert.

Convention: Positions 1-6 from LEFT to RIGHT from an observer facing the arrangement. Row 1 is the top row (facing South), Row 2 is bottom (facing North).

Row 1 faces South: their LEFT = observer's RIGHT (position 6), their RIGHT = observer's LEFT (position 1).

B at RIGHT end of Row 1 (from B's perspective) = observer's LEFT end = position 1.

Row 2 faces North: their LEFT = observer's LEFT (position 1), their RIGHT = observer's RIGHT (position 6).

P at LEFT end of Row 2 (from P's perspective) = observer's LEFT = position 1.

  Row 1: B(1)  __(2)  __(3)  __(4)  __(5)  __(6)
          |      |      |      |      |      |
  Row 2: P(1)  __(2)  __(3)  __(4)  __(5)  __(6)

Step 2: S faces B. B at position 1 in Row 1. S at position 1 in Row 2. But P is at position 1 in Row 2! Contradiction.

Let me re-examine: Maybe "right end from B's perspective" facing South means position 6?

Facing South: imagine standing and facing South. Left hand points East, right hand points West. If the row goes from position 1 (West) to position 6 (East):

  • Facing South, RIGHT = West = position 1 side.

Hmm, this is the standard confusion. Let me resolve it definitively.

Stand facing South. Your left = East, your right = West. If positions are labeled 1 (leftmost from observer facing North) to 6:

  • Position 1 is on the WEST side, position 6 is on the EAST side.
  • Person facing South: their RIGHT = West = toward position 1.

So B at right end (facing South) = position 1. This creates a conflict with P.

OR, the problem means the rightmost position from a neutral observer's view. Let's assume B is at position 6 (rightmost in the diagram).

  Row 1: __(1)  __(2)  __(3)  __(4)  __(5)  B(6)
          |      |      |      |      |      |
  Row 2: P(1)  __(2)  __(3)  __(4)  __(5)  __(6)

S faces B(6) -> S at position 6 in Row 2.

R to the immediate left of S (from R's perspective, facing North). Facing North: left = West = toward position 1. So R at position 5 in Row 2.

  Row 1: __(1)  __(2)  __(3)  __(4)  __(5)  B(6)
          |      |      |      |      |      |
  Row 2: P(1)  __(2)  __(3)  __(4)  R(5)   S(6)

A not at any end -> A at positions 2-5 in Row 1. T faces A.

This is a partial solution showing the approach. The remaining people would be placed based on additional constraints.

Answer (partial): Row 1: __, __, __, __, __, B. Row 2: P, __, __, __, R, S.

Examples 11-20: Quick-Solve Format

Example 11: "Who is 3rd to the left of D?"

Given arrangement: A B C D E F G (left to right). 3rd to the left of D(position 4) = position 1 = A.

Example 12: "How many people sit between B and F?"

Given: A B C D E F G B at position 2, F at position 6. Between them: C, D, E = 3 people.

Example 13: Circular -- "Who is opposite C?"

6 people clockwise: A B C D E F. Opposite of C(3) = position 3+3 = 6 = F.

Example 14: "Who sits second to the right of E?"

Given linear: P Q R S T E U V E at position 6. Second to right = position 8 = V.

Example 15: Double Row -- "Who faces C?"

Row 1: A B C D. Row 2: W X Y Z. C at position 3 faces Y at position 3. Y.

Example 16: Linear -- "A is 5th from left and 3rd from right. How many people total?"

Total = 5 + 3 - 1 = 7.

Example 17: "If A and B interchange, who is to the right of A's new position?"

Original: P A Q B R. After swap: P B Q A R. Right of A(position 4) = R. R.

Example 18: Circular -- "3 seats clockwise from A?"

8 people: A B C D E F G H (clockwise). 3 seats CW from A = D.

Example 19: "Who is at the midpoint between A and E?"

Linear: A B C D E F G. A at 1, E at 5. Midpoint = position 3 = C.

Example 20: Double Row -- "Who is diagonally opposite to P?"

  Row 1:  A  B  C  D
  Row 2:  P  Q  R  S

P at Row2-position1. Diagonally opposite = Row1-position2 = B.

Example 21: Full Puzzle Set (Banking Style)

Problem: Eight people A, B, C, D, E, F, G, H sit in a row facing North.

  1. D sits 4th from the right end.
  2. A is to the immediate right of D.
  3. F sits at the left end.
  4. There are two people between F and G.
  5. B is not adjacent to D.
  6. C sits to the immediate left of H.
  7. E is not at any end.

Solution:

Step 1: D is 4th from right = position 5 (in row of 8). A is immediate right of D = position 6.

  __  __  __  __  D  A  __  __
  (1) (2) (3) (4) (5) (6) (7) (8)

Step 2: F at left end = position 1.

  F  __  __  __  D  A  __  __

Step 3: 2 people between F and G. F at 1 -> G at 4 (1+3=4).

  F  __  __  G  D  A  __  __

Step 4: C immediate left of H. B not adjacent to D (positions 4 or 6). E not at any end.

Remaining: B, C, E, H for positions 2, 3, 7, 8. B not adjacent to D(5) -> B not at 4 (taken) or 6 (taken). But B is for positions 2,3,7,8 anyway. B not at any position adjacent to D: well, 4 and 6 are taken. So no additional restriction from adjacency unless we miscounted. Actually B not adjacent to D means B not at position 4 or 6. Both are taken. So B can be at 2, 3, 7, or 8.

E not at any end -> E not at position 8 (position 1 is taken by F). E at 2, 3, or 7.

C immediate left of H: [C H] as a block. Possible at (2,3), (3,4-taken), (7,8). So [C H] at positions (2,3) or (7,8).

Sub-case A: C at 2, H at 3. Remaining: B, E at 7, 8. E not at end(8) -> E at 7, B at 8.

  F  C  H  G  D  A  E  B

Sub-case B: C at 7, H at 8. Remaining: B, E at 2, 3. E not at end -> fine at 2 or 3.

  F  B  E  G  D  A  C  H    or    F  E  B  G  D  A  C  H

All three are valid based on the given constraints.

Answer: Three possible arrangements: F C H G D A E B, F B E G D A C H, F E B G D A C H.

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