A seating puzzle with two valid answers
A district practice test for 120 B.Ed aspirants contains this seating-arrangement item: six students P, Q, R, S, T and U sit in a row facing north. (i) P is third to the left of U; (ii) Q is at one end; (iii) R sits next to S; (iv) T is not adjacent to U; (v) S is second to the right of P. The publisher's key says the order is Q, P, S, R, T, U, with no explanation. After the test, one trainee produces two different arrangements that satisfy all five clues, and three others show that the key's order violates clue (i). Of the 120 candidates, 18 chose the key's option, 46 chose one of the two valid arrangements, 22 marked two options, and 34 left it blank. The item carries 2 marks and there is a 0.5 negative mark for wrong answers. The coordinator must decide the scoring within 24 hours and announce it before the next practice test. A parent has already called to ask whether 'these publisher keys are reliable'.
1. Determine whether the item has a unique answer. Show your reasoning and at least one valid arrangement.
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The item is not unique. From clue (i), P must be third to the left of U, so the possible (P, U) position pairs are (1,4), (2,5) and (3,6). Clue (v) then fixes S two places right of P: (3), (4) and (5) respectively.
- P1, U4, S3: Q must be at an end, so Q6. R must be next to S, so R2 (position 4 is U). That leaves T5, which is adjacent to U - clue (iv) fails.
- P2, U5, S4: If Q1, then R3 and T6, but T6 is adjacent to U5 - fails. If Q6, then R3 and T1, and T1 is not adjacent to U5 - valid: T, P, R, S, U, Q.
- P3, U6, S5: Q must be at position 1. R must be next to S, so R4, leaving T2, which is not adjacent to U6 - valid: Q, T, P, R, S, U.
So two arrangements satisfy all five clues. The publisher's key Q, P, S, R, T, U is invalid: P at position 2 is four places left of U at position 6, not third, and S at position 3 is only one place right of P. The trainees are right, and the key cannot be defended.
2. Recommend a scoring decision for the 120 candidates and justify it against the 2-mark/0.5-negative scheme.
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Because the item has two correct answers and the printed key is invalid, no candidate should lose marks for it. My recommendation is: accept either T, P, R, S, U, Q or Q, T, P, R, S, U for 2 marks; cancel the 0.5 negative for anyone who marked two options; give 0 to blanks.
Reasoning: the 46 candidates who chose a valid arrangement clearly solved the puzzle and must be rewarded. The 22 who marked two options detected the ambiguity the setter missed; penalising them would punish better reasoning. The 18 who chose the publisher's order made a genuine error, so they do not get the 2 marks merely because a key existed - the key was never shown to them during the test. The 34 blanks attempted nothing, so they stay at zero; giving them full marks would reward non-attempt and make the item meaningless.
I would announce this in writing, show both arrangements on the board, and state that the publisher's key was wrong. If this were a high-stakes selection test rather than practice, I would drop the item entirely and give 2 marks to all who attempted it, because defending a flawed item under negative marking creates more unfairness than the lost measurement precision. Here, with practice stakes and a teachable moment, accepting both valid answers is the better balance.
3. How would you prevent such an item from entering a test in future? Give a concrete three-step check.
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Step 1 - Independent blind solving. Two people solve the item without seeing the key and write down every arrangement they find. If they differ, the item is not ready. This alone would have caught the publisher's error, because neither solver would produce Q, P, S, R, T, U from clue (i).
Step 2 - Ambiguity audit of relational words. Check every phrase that can be read two ways: 'next to' versus 'immediately next to', 'third to the left' versus 'third from the left', 'at an end' versus 'at the extreme end'. In this item, 'R sits next to S' is broad enough to allow either side, which is legitimate, but the setter must know that and test for uniqueness. If uniqueness is intended, add one clue such as 'R is immediately to the left of S' and re-solve.
Step 3 - Pre-test and item analysis. Try the item on 10–15 candidates, record the distribution of options, and discard or revise any item where a substantial group selects a second defensible answer. Require a written one-line explanation with every key, and never accept a publisher's key without solving it yourself. The cost is about 20 minutes per item; the cost of a wrong key in a 120-candidate test is far higher.