EPFO APFC 2023 General Studies · Question 38 of 120

Correct — D, (d) 6. Work in blocks rather than in seats. Step 1 — build the blocks. 'W is a neighbour of both Q and U' can only mean that W sits between Q and U, so Q-W-U is a solid block of three. 'T is a neighbour of R and V' likewise puts T between them, giving R-T-V, another block of three. 'P and S are neighbours' gives a block of two. Their sizes add to 3 + 3 + 2 = 8, which is the whole row, so the eight friends are exactly these three blocks laid end to end. All that is left to decide is the order of the blocks and whether each is read forwards or backwards. Step 2 — use the distance. 'Four people between P and V' means four seats separate them, so P and V stand five places apart. P is at one end of a two-block and V at one end of a three-block, and a separation of five can be achieved only with the three-person Q-W-U block lying between them. That forces the order P-S block, then Q-W-U, then R-T-V, with P at the very end of the row: P at seat 1, S at 2, the Q-W-U block at 3, 4 and 5 with W in the middle, and then V at 6, T at 7 and R at 8. Check it: between P at 1 and V at 6 stand S, and the three of Q, W and U — exactly four people. Step 3 — read off the answer. P is at seat 1 and R at seat 8, so the people between them are those in seats 2 to 7: S, the three members of the Q-W-U block, V and T. That is six people, which is option (d). The arrangement is not unique — reversing the entire row gives R, T, V, then the Q-W-U block, then S and P, and it satisfies every condition equally well. That mirror image is harmless here, because the question asks only how many people stand between two of them, and a count of that kind is unchanged by reflection. Recognising that a puzzle's two solutions are mirror images, and that the quantity asked for is symmetric, saves the time a candidate would otherwise spend hunting for a unique seating.

• (a)1 — This is what you get from an off-by-one on the distance clue combined with a different block order. Read 'four people between P and V' as 'P and V are four seats apart' and you can place P at one end, then the R-T-V block with V four seats away, which puts R just two seats from P and leaves a single person between them. The arrangement satisfies all three neighbour conditions, which is what makes the error invisible from inside. The safeguard is to write the count out rather than compute it: if n people stand between two positions, those positions are n + 1 seats apart, so four people between means a gap of five, not four.
• (b)3 — Three people between P and R is the count you get if the R-T-V block is placed immediately beside the P-S block — for instance P and S in seats 1 and 2 and then V, T and R in seats 3, 4 and 5, which puts R four seats from P. Every neighbour condition holds in that arrangement; what fails is the distance condition, since V would then be only two seats from P with a single person between them, not four. The option therefore catches a candidate who builds the blocks correctly and then arranges them by eye instead of using the one clue that fixes their order.
• (c)5 — Five people between P and R arises from the same off-by-one as option (a), applied in the block order that this question's distance clue actually rules in. Put S at seat 1 and P at seat 2, the Q-W-U block at 3 to 5 and V, T, R at 6, 7 and 8: R is then six seats from P with five people between. But V now sits four seats from P, with only three people between them, so the condition in the stem is not met. The arrangement is a near miss in the most literal sense — one seat's shift of the P-S block away from the end of the row — and it is the reason the stem's count of four must be converted into a gap of five before anything else is done.

Linear-arrangement puzzles are solved by building blocks and then placing them, not by testing seatings one at a time. Three ideas do the work. First, a condition of the form 'X is a neighbour of both Y and Z' is far stronger than an ordinary adjacency: it pins X between the other two and creates a rigid unit of three, which can then be moved and reversed but never broken. An ordinary 'X and Y are neighbours' creates a unit of two in the same way. Second, once the blocks are built, add their sizes: if the total equals the number of seats, the blocks tile the row exactly and the only remaining freedom is their order and orientation, which reduces the puzzle to a handful of cases. Third, distance clues must be converted before use — n people between two positions means the positions are n + 1 apart — and a distance clue is what usually selects the order of the blocks. A fourth observation applies to a single row facing one direction: any solution can be reflected end to end to give a second solution, so a well-set question asks for something that reflection preserves, such as a count between two people or the identity of the person at a particular distance from another, rather than for who sits at the left end.

Reasoning items sit alongside quantitative ones in this block of the paper, and they are among the highest-return questions on it, because they need no recalled knowledge at all — everything required is printed in the stem. What they do need is a method, since a candidate who starts drawing eight seats and filling them by trial will lose several minutes and probably still be wrong. The block method converts an eight-seat search into a choice among a few orderings, and it can be executed on the margin of the question paper in under a minute. Note also how this item is worded: the seating conditions are separated by semicolons rather than printed as a numbered list, which packs three constraints into one line and makes it easy to read past one of them. Extract each condition on to its own line before beginning, exactly as you would with a numbered statement list, and the packing loses its effect.

• A condition that one person is a neighbour of two named others places that person between them and creates a rigid block of three; here W lies between Q and U, and T lies between R and V.
• When the block sizes add up to the number of seats — 3 + 3 + 2 = 8 in a row of eight — the blocks tile the row exactly, and the only freedom left is the order of the blocks and the reversal of each.
• A distance clue must be converted before use: n people between two positions means the positions are n + 1 seats apart, so 'four people between P and V' means P and V are five seats apart.
• The unique block order consistent with that separation is the P-S pair at one end, the Q-W-U block in the middle and the R-T-V block at the other end, with P at the extreme end of the row, V adjoining the middle block, T next and R at the far end.
• P and R therefore occupy the two ends of the row, with six people between them; the whole arrangement can be reflected end to end without breaking any condition, and a count of people between two seats is unchanged by that reflection.

• Reading 'four people between' as a gap of four seats; the gap is five, and this single off-by-one produces two of the three wrong options in this item
• Breaking a three-person block: if W is a neighbour of both Q and U then W is between them, and no arrangement may separate the three
• Arranging the blocks by eye once they are built, instead of using the distance clue to fix their order — the wrong order satisfies every neighbour condition and looks entirely sound
• Hunting for a unique seating when the puzzle has a mirror-image pair of solutions; check first whether the quantity asked for is symmetric, in which case either solution answers it

This paper's reasoning items are drawn from a small set of families — linear seating, circular seating, ranking, coding, direction and blood relations — and the seating questions are almost always solvable by the block method. Expect the constraints to be packed into a single sentence separated by semicolons, expect one distance clue that fixes the order of the blocks, and expect the answer to be a count rather than a name, precisely because a count is immune to the reflection ambiguity that a row of this kind always carries.

No directly related past PYQ was found.

• practice — not a real PYQ

  Seven friends A, B, C, D, E, F and G are sitting in a row facing North. A and B are neighbours; D is a neighbour of both C and E; F and G are neighbours. If there are four people between A and C, how many people are there between B and E?

  • (a)2
  • (b)3
  • (c)4
  • (d)5

  Answer(b) 3 — D between C and E is a rigid block of three, A with B and F with G are blocks of two, and 3 + 2 + 2 fills the row of seven exactly. Four people between A and C means they are five seats apart, which forces the order B, A, the F-G pair, then E, D, C. Between B at one end and E stand A and the two of F and G, that is three people, and the count is unchanged if the row is reversed or if F and G exchange seats.

• practice — not a real PYQ

  In a row of people all facing the same direction, there are seven persons between X and Y. How many seats apart are X and Y?

  • (a)6
  • (b)7
  • (c)8
  • (d)9

  Answer(c) 8 — if n people stand between two positions, the positions are n + 1 seats apart, so seven people between X and Y means a separation of eight seats. This conversion is the single most common source of error in arrangement puzzles, and it should be done in writing before any seating is attempted.