Eight chairs are placed at a uniform distance from each other around a round table. C and D are equal distance away from both A and B; E sits between A and C; G and F sit opposite each other. Where does H sit ?
- (a)Adjacent to D
- (b)Adjacent to A
- (c)Adjacent to F
- (d)Adjacent to G
Correct — A, (a) Adjacent to D. Number the eight chairs 1 to 8 around the table. Because the chairs are uniformly spaced, chair n and chair n+4 face each other across the table, and 'equal distance from A and B' means a chair sits on the perpendicular bisector of the line joining A and B. Start with the clue that fixes the geometry. 'E sits between A and C' means there is exactly one chair between A and C, with E in it — so C is two places from A. Combine that with 'C is equal distance from both A and B': if C is two places from A and the same distance from B, then B must be two places from C on the other side. So B is four places from A, which on eight uniformly spaced chairs means A and B sit directly opposite each other. Now place everyone. Take A at chair 1; then B is at chair 5. The chairs equidistant from 1 and 5 are chairs 3 and 7 — the two midpoints of the arcs — and those are exactly the two seats the first clue reserves for C and D. Say C is at 3; then D is at 7, and E, sitting between A and C, takes chair 2. That leaves chairs 4, 6 and 8 for F, G and H. F and G must sit opposite each other, and among the three free chairs the only facing pair is 4 and 8 — chair 6 faces chair 2, which E already occupies. So F and G take chairs 4 and 8 in some order, and H is left with chair 6. Chair 6 sits between chair 5 (B) and chair 7 (D). H is therefore adjacent to D, and also adjacent to B. The arrangement is not fully determined — nothing says whether C is at 3 or at 7, and nothing says which of F and G takes which chair — but it does not need to be. Run the mirror case with C at 7 and D at 3: E goes to chair 8, F and G take the facing pair 2 and 6, and H lands on chair 4, between chair 3 (D) and chair 5 (B). H is adjacent to D again. Both readings give the same answer, which is what makes the question well posed despite the loose ends. In neither arrangement is H adjacent to A, to F or to G.
- (b)Adjacent to A — A's two neighbours are E on one side and one of F or G on the other, in both of the possible arrangements. H sits at the far side of the table from A — five chairs round in one direction, three in the other — and is never next to A. The option is attractive only because A is the letter the puzzle starts from.
- (c)Adjacent to F — F is one of the facing pair, and in both arrangements that pair occupies the chairs that flank H's neighbours rather than H itself. With C at 3, F and G are at 4 and 8 while H is at 6, whose neighbours are 5 and 7. With C at 7, F and G are at 2 and 6 while H is at 4, whose neighbours are 3 and 5. H is next to F in neither case.
- (d)Adjacent to G — Wrong for the same reason as option (c), and necessarily so: the puzzle never distinguishes F from G, since the only thing said about them is that they face each other. Any statement true of F's position is true of G's and vice versa, so (c) and (d) stand or fall together — and two options that cannot be separated cannot be the single correct answer. Noticing that symmetry eliminates both of them before any chair is drawn.
Circular arrangement puzzles are solved by fixing one person arbitrarily and building outwards, because only relative positions matter — the whole arrangement can be rotated without changing anything. With an even number of seats the useful extra structure is the facing pair: on eight uniformly spaced chairs, chair n faces chair n+4, and there are exactly four facing pairs. 'Equidistant from two people' is the other structural clue, and it is worth translating at once into geometry: a chair equidistant from A and B lies on the perpendicular bisector of AB, and on a regular polygon of eight vertices such a chair exists only when A and B are an even number of places apart. Clues of that kind fix the shape of the arrangement before any individual is placed, which is why they should be used first.
The EO/AO reasoning items are built to be solved with a diagram in the margin and nothing else, and this one is a good example of a puzzle that looks underdetermined and is not. Two things are genuinely left open — which of C and D takes which midpoint, and which of F and G takes which chair — and a candidate who insists on resolving them will burn the clock. The habit rewarded is checking whether the ambiguity actually touches the question asked: here it does not, because both branches put H between B and D.
- On eight uniformly spaced chairs, chair n and chair n+4 sit opposite each other; there are four facing pairs.
- 'E sits between A and C' fixes exactly one chair between A and C, so C is two places from A.
- A chair equidistant from A and B lies on the perpendicular bisector of AB; requiring two such chairs (C and D) forces A and B to be four places apart, that is, directly opposite.
- With A at chair 1 and B at chair 5, the equidistant chairs are 3 and 7, which C and D occupy.
- E takes the chair between A and C; the only facing pair among the three chairs then left is the pair that F and G must take.
- H is left with the remaining chair, which lies between B and D in both possible arrangements.
- The arrangement is not unique — C and D may be swapped, and F and G may be swapped — but H's neighbours are the same either way.
- Options (c) and (d) are interchangeable by the symmetry of F and G, so neither can be the unique answer.
- Trying to pin down every position when the question asks about only one person.
- Reading 'between A and C' as anywhere along the arc rather than directly between — the puzzle is unsolvable on the looser reading.
- Forgetting that a facing pair must be checked against chairs already occupied. Chair 6 faces chair 2, which E holds, so 6 cannot be part of F and G's pair in the first arrangement.
- Overlooking that F and G are interchangeable, which by itself rules out two of the four options.
Seating arrangement appears at least once in every reasoning block of this paper, usually with six or eight positions and three or four clues. The clues are always of the same types — opposite, adjacent, between, equidistant — and translating each into a geometric constraint before drawing anything is what keeps these to under two minutes.
No directly related past PYQ was found.
- practice — not a real PYQ
Six persons sit at equal distances around a circular table. P and Q are each at an equal distance from both R and S. Which one of the following must be true ?
- (a)P and Q sit opposite each other
- (b)R and S sit opposite each other
- (c)P sits adjacent to Q
- (d)R sits adjacent to S
Answer(a) P and Q sit opposite each other
- practice — not a real PYQ
Eight persons sit at a uniform distance from each other around a circular table. If P sits opposite Q, the number of persons sitting between them along either side of the table is :
- (a)2
- (b)3
- (c)4
- (d)6
Answer(b) 3