A Seating Puzzle with Four Valid Answers
A reasoning faculty member at a B.Ed institute in Gorakhpur drafts a linear seating puzzle for a 40-item mock test: six students P, Q, R, S, T and U sit in a row facing north. The clues are: (i) P sits third to the left of U. (ii) R sits immediately to the right of P. (iii) S sits at one of the ends. (iv) T sits to the right of Q. In a trial with 30 students, 19 wrote one arrangement (S-P-R-Q-U-T), 8 wrote a different arrangement (Q-P-R-T-U-S), and 3 wrote that the puzzle has no unique solution. The faculty member's own key, prepared in a hurry, lists only S-P-R-Q-U-T. The mock is to be printed the next morning; the printer needs the final file by 8 p.m. He has 20 minutes and must decide whether the item stays, changes, or is dropped, knowing that the 3 students' objection will reach the batch WhatsApp group within the hour.
1. Test the item. Are the 19 students right, the 8, or the 3?
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Enumerate. Clues (i) and (ii) force the block P-R with U three places to the right of P, so P can be at 1, 2 or 3 and U at 4, 5 or 6. Testing each case against (iii) S at an end and (iv) T somewhere to the right of Q gives four valid rows: S-Q-P-R-T-U; Q-P-R-T-U-S; S-P-R-Q-U-T; P-R-Q-T-U-S. Check one: in P-R-Q-T-U-S, P is at 1 and U at 4, so U is three places to the right of P; R is immediately right of P; S is at the right end; T at 5 is right of Q at 3.
So the item is not unsolvable - it is under-determined. The 19 produced one valid row and the 8 produced another; both are correct on the clues given. The 3 who wrote 'no unique solution' are the only ones who diagnosed the item's actual defect, even though they produced no arrangement.
Conclusion: the key listing only S-P-R-Q-U-T is indefensible, because it marks one of four correct outcomes as the answer.
2. Which single added clue would make the item unique, and how would you verify that it does not destroy the intended answer?
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Method: work from the full solution set, not from the intended answer. Take the four rows and test any proposed clue against all four; a clue is useful only if exactly one row survives.
Example of a bad fix: 'Q is not adjacent to R' is true in Q-P-R-T-U-S and in S-Q-P-R-T-U, and false in the intended row - so it kills the setter's own answer and still leaves two. This is the classic error of writing clues backwards from a preferred answer.
A good fix: 'U sits fourth from the left' leaves only P-R-Q-T-U-S. If a one-clue fix feels forced, add two and record both: 'S is at the right end and Q sits immediately to the left of U' also leaves exactly one row. Write the verification in the item file so the reviewer can repeat it rather than trust it.
3. How should the three students be marked, and what does the episode reveal about what the item was actually testing?
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Marking: any row that satisfies all four clues earns full credit - four rows qualify, so the 19 and the 8 both deserve it and the printed key cannot be applied. The 3 who wrote 'no unique solution' should get credit for the reasoning if the item was meant to have a unique answer, because spotting under-determination is the higher-order skill the item accidentally tested; record their response and use it in class.
What it reveals: as drafted, the item tested guessing the setter's intended arrangement, not deduction. A candidate with sound reasoning could be marked wrong for a correct row.
Action before 8 p.m.: replace the item in the file with the corrected version, tell the batch before the WhatsApp message circulates, and add 'verify uniqueness against the full solution set' to the faculty's item-review checklist.