Ambiguous seating-arrangement item at press stage
The examination cell of a district-level recruitment board is finalising a reasoning paper. A setter has framed a seven-person linear arrangement item: A, B, C, D, E, F and G occupy seven seats numbered 1 to 7 from left to right and all face north. Five clues are given:
- A sits at the extreme left end.
- B sits at the extreme right end.
- C sits second to the right of A.
- E is an immediate neighbour of A.
- D sits second to the left of B. The setter's answer key records the order A E C F D G B and marks the item as having a unique solution. A reviewer has flagged the item. Reprinting costs ₹40,000 and the paper goes to press in 48 hours. The board's rules allow an item to be dropped with full marks to all candidates, or a wrong key to be corrected, but no new clue may be added after the proof stage.
1. Test whether the five clues really yield a unique arrangement. Enumerate the possibilities and show your working.
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Step 1 - Fix the seats forced directly.
- Clue 1: A = seat 1.
- Clue 2: B = seat 7.
- Clue 3: C is second to the right of A, so C = seat 3.
- Clue 4: E is an immediate neighbour of A; seat 2 is the only neighbour available, so E = seat 2.
- Clue 5: D is second to the left of B, so D = seat 5.
Step 2 - Fill the remainder. Occupied: 1, 2, 3, 5, 7. Free: seats 4 and 6, to be taken by F and G in either order.
Step 3 - Both orders satisfy all five clues:
- Arrangement I: A E C F D G B
- Arrangement II: A E C G D F B
Check clue 5 in both: D is at 5, B at 7 - second to the left holds in each. No clue refers to F's or G's position, so neither order is excluded.
Conclusion: the clues give two valid solutions, not one. The setter's key happens to record Arrangement I, but Arrangement II is equally correct, so the item is ambiguous and any candidate who marked Arrangement II would be wrongly penalised.
2. Which of the board's two permitted remedies is appropriate here, and why is the other one not available?
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Drop the item and award full marks to all candidates. The defect is ambiguity, not a clerical slip in the key. There is no single correct answer to correct the key to: Arrangement I is not the right answer and Arrangement II the wrong one - both follow from the given clues. Changing the key from I to II would simply shift the injustice to candidates who reasoned correctly to I.
Why 'correct the key' is unavailable: that remedy works where the question has one demonstrably right answer and the key misprints it. Here the question itself under-determines the answer.
Why 'add a clue' is unavailable: the board's rule bars new clues after the proof stage - the paper is already typeset, and a post-proof clue would also privilege whoever saw the earlier version.
Practical handling:
- Record the two arrangements and the omission in the item-review register with both reviewers' signatures.
- Mark the item as deleted in the scoring scheme and confirm that the remaining items still total the printed maximum; if not, apply the board's moderation formula.
- If the board's rules allow accepting both arrangements as correct for an already-printed paper, that is the equivalent of a drop and should be notified before the exam, not after objections arrive.
Cost of the decision (₹40,000 reprint vs. litigation after the result) is not the deciding factor - fairness is.
3. If the setter had been allowed one extra clue, would 'F sits immediately to the right of C' make the item unique? Demonstrate.
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Yes - it forces a single arrangement.
Step 1 - Apply the extra clue to the two surviving arrangements:
- Arrangement I: A E C F D G B - F is at seat 4, C is at seat 3, so F is immediately to the right of C. Satisfied.
- Arrangement II: A E C G D F B - the seat immediately right of C (seat 3) is seat 4, occupied by G, not F. Violated.
Step 2 - Result: only Arrangement I survives, so the item becomes unique.
Step 3 - Note on minimality. Any clue that separates seats 4 and 6 works equally: 'G sits to the right of D' (G at 6 satisfies it, G at 4 does not) or 'F is not adjacent to D' would fail here, since both seats 4 and 6 are adjacent to D at seat 5 - a reminder to test a proposed clue against both candidate arrangements before accepting it.
Step 4 - Process lesson. The setter should have run the item against the two-arrangement check before sending it to proof. A useful review rule: count the persons, count the clues, and verify that every person is constrained by at least one clue; here F and G were never mentioned, which is exactly why the item leaked a second solution.