Eight Seats, One Suspect Clue
Eight colleagues - A, B, C, D, E, F, G and H - occupy eight equally spaced seats around a circular table in the bank's training room, all facing the centre. The facilitator dictated six clues:
(i) A sits third to the left of B. (ii) C sits second to the right of A. (iii) F sits opposite C. (iv) G sits second to the left of H. (v) D and E are immediate neighbours. (vi) E sits adjacent to F.
She then admitted that clue (vi) was carried over from an older version of the exercise and may be a misprint. Two teams submitted answers that both satisfied clues (i) to (v) but differed: in one, D sat next to F; in the other, E sat next to F. Each team insists its version is the only correct one. You are asked to arbitrate. Use the standard convention that for a person facing the centre, 'left' means clockwise and 'right' means anticlockwise, and that 'opposite' means four seats away.
1. Fix the seating from clues (i)–(vi) and state who sits immediately to the right of A. Show each deduction.
Read the worked answer
Number the seats 1–8 clockwise and place A at seat 1. Facing the centre, left = clockwise.
- (i) A is third to the left of B, so three clockwise steps from B must land on seat 1: B = seat 6.
- (ii) C is second to the right (anticlockwise) of A: C = seat 7.
- (iii) F is opposite C, four seats away: F = seat 3.
- Seats 2, 4, 5, 8 are left for D, E, G, H. From (iv), G = H + 2 clockwise. Testing the free seats: H = 8 gives G = 2; H = 2 gives G = 4; H = 4 or 5 gives seats that are already taken.
- If H = 2 and G = 4, then D and E must take seats 5 and 8, which are not adjacent - (v) fails. Hence H = 8, G = 2, and D, E occupy 4 and 5.
- (vi) E is adjacent to F at seat 3, so E = 4, D = 5.
Clockwise order: A, G, F, E, D, B, C, H. The immediate right of A is the anticlockwise neighbour, seat 8: H sits to the right of A.
2. The facilitator withdraws clue (vi) as a misprint. How many seating arrangements now satisfy clues (i)–(v), and what exactly changes between them?
Read the worked answer
With (vi) gone, steps 1–5 stand unchanged: A1, B6, C7, F3, H8, G2, with D and E in seats 4 and 5 in either order. Both versions satisfy (i)–(v), because D and E remain neighbours and no surviving clue distinguishes them.
So exactly two arrangements exist:
- A, G, F, E, D, B, C, H (clockwise)
- A, G, F, D, E, B, C, H (clockwise)
Both teams were therefore reasoning correctly from the clues they used. The team that placed D next to F had the second arrangement; the team that placed E next to F had the first. Only the withdrawn clue breaks the tie. The lesson for the exam is that a single unverified clue, not faulty deduction, was the source of the conflict - when a puzzle admits two answers, check whether any clue is redundant or contradicted before assuming an arithmetic slip.
3. Suppose the facilitator reverses the instruction: all eight now face outward, with 'left' and 'right' taken from each person's own perspective. Derive the arrangement and explain its relationship to the earlier one.
Read the worked answer
Facing outward, the directions invert: left = anticlockwise, right = clockwise. Every positional clue therefore mirrors.
- (ii) C is second to the right (clockwise) of A: C = 3.
- (i) A is third to the left (anticlockwise) of B: B = 4.
- (iii) F = 7. Free seats: 2, 5, 6, 8.
- (iv) G = H − 2. H = 8 → G = 6, but then D and E would take 2 and 5, which are not adjacent - rejected. H = 2 → G = 8, with D and E at 5 and 6 (adjacent).
- (vi) E adjacent to F at 7 forces E = 6, D = 5.
Clockwise order: A, H, C, B, D, E, F, G - the mirror image of the centre-facing solution. Every neighbour pair survives; only left and right swap. A candidate who assumes the facing direction can satisfy all six clues and still be wrong, so the direction must be stated and used, never guessed.