Settling a Seven-Chair Seating Puzzle Before the File Closes
You are helping an examination superintendent at a centre in Agra. Seven candidates - P, Q, R, S, T, U and V - must be seated in a single row of seven chairs, numbered 1 to 7 from the left, all facing north. The seating file records five clues: T sits at the extreme left end; Q sits third to the left of V; P sits immediately to the right of R; exactly one person sits between Q and R; S sits immediately to the right of U. Before the file is closed, a candidate objects that the answer key of a practice set claims 'S sits at position 6', and argues that the key is unjustified because nothing fixes S. The superintendent has ten minutes and wants a decision he can defend on file, not just an answer. You must settle the arrangement, test what happens if one clue is removed, and decide whether the key's claim is defensible.
1. Deduce the full seating order, numbering chairs 1 to 7 from the left, and state which clue does the decisive work.
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Step 1 - T is fixed at chair 1. Step 2 - Q is third to the left of V, so V − Q = 3. Possible pairs: (2,5), (3,6), (4,7). Chair 1 is taken by T, so Q cannot be 1, and V cannot exceed 7. Step 3 - Test each pair against P = R + 1 and |Q − R| = 2.
- Q = 2, V = 5: R must be 4 (R = 0 invalid), so P = 5, which clashes with V. Reject.
- Q = 3, V = 6: R = 1 (taken by T) or R = 5 giving P = 6, which clashes with V. Reject.
- Q = 4, V = 7: R = 2 or 6. R = 6 gives P = 7, clashing with V, so R = 2 and P = 3. Step 4 - Chairs 5 and 6 remain for U and S, and S sits immediately right of U, so U = 5, S = 6.
Final order: 1 T, 2 R, 3 P, 4 Q, 5 U, 6 S, 7 V.
The decisive clue is the distance clue 'exactly one person sits between Q and R'; without it, R is not pinned. The Q–V clue only shortlists candidates.
2. If the clue 'exactly one person sits between Q and R' is deleted, how many valid arrangements remain? Does 'S is at position 6' still follow necessarily?
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With T = 1 and Q = 4, V = 7 still forced by the Q–V clue, only P = R + 1 remains as a constraint on R.
Possible R values:
- R = 2 gives P = 3, leaving chairs 5 and 6 for U and S, and S = U + 1 forces U = 5, S = 6.
- R = 3 gives P = 4, which clashes with Q. Reject.
- R = 5 gives P = 6, leaving chairs 2 and 3 for U and S, so U = 2, S = 3.
- R = 6 gives P = 7, which clashes with V. Reject.
So exactly two arrangements survive: (T R P Q U S V) and (T U S R P Q V).
In the first, S is at 6; in the second, S is at 3. Therefore 'S sits at position 6' is not a necessary conclusion once the clue is deleted. The answer key's claim is conditional on that clue being present in the question paper. The reasoning to record is that the deleted clue is load-bearing, not decorative.
3. Is the answer key's claim 'S sits at position 6' defensible? State the condition under which it is correct and the condition under which it must be rejected.
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It is defensible only if the question paper actually printed all five clues. Under the full clue set the order is unique - T, R, P, Q, U, S, V - and S is at chair 6, so the key is correct and the candidate's objection fails.
It must be rejected if the practice set omitted or misprinted the distance clue. In that case two arrangements remain and S can be at 6 or at 3, so a single-answer key is over-determined and should be treated as a defective item.
The defensible file note is therefore: verify the clue list against the printed question paper; if all five clues appear, retain the key with the deduction above; if any clue is missing, drop the item from scoring rather than penalising candidates who answered 3. The judgement being tested is that a correct answer to a full puzzle does not validate the answer key of a defective puzzle.