Seating Puzzle with a Non-Unique Answer Key
A TGT is preparing a seating-arrangement worksheet for Class 7. Four teachers - A, B, C and D - are to be seated in four chairs numbered 1 to 4 from left to right. The draft gives four clues: (1) A sits at an end. (2) B sits immediately to the right of C. (3) C does not sit at an end. (4) D is not adjacent to A. The answer key states: 1-A, 2-C, 3-B, 4-D. Two students submitted 1-D, 2-C, 3-B, 4-A and were marked wrong. They have complained. The worksheet rules say there must be exactly one valid arrangement, no more than one clue may be added, and clues may use only 'left/right', 'immediately', 'end' and 'adjacent'. The teacher has one class period to resolve the complaint and reprint the worksheet.
1. Test both arrangements against all four clues. Is the students' complaint valid? Show the check explicitly.
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Yes, the complaint is valid. For the key arrangement 1-A, 2-C, 3-B, 4-D: A is at end 1; B is immediately right of C (2 to 3); C is not at an end; D is at 4, not adjacent to A at 1. All four clues hold. For the students' arrangement 1-D, 2-C, 3-B, 4-A: A is at end 4; B is immediately right of C; C is not at an end; D is at 1, not adjacent to A at 4. All four clues also hold. The key is therefore incomplete: the four clues admit at least two solutions, and the students' solution is one of them.
2. Propose one additional clue that makes the solution unique, and prove uniqueness by testing both arrangements.
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One suitable clue is: 'A sits to the left of C.' In the key arrangement, A is at 1 and C at 2, so A is left of C. In the students' arrangement, A is at 4 and C at 2, so A is not left of C. The added clue therefore eliminates the students' arrangement and keeps the key arrangement. To prove uniqueness, note that the original clues force C to chair 2: if C were at 3, B would be at 4, and A at an end would have to be at 1, leaving D at 2, which is adjacent to A and violates clue 4. So C is at 2, B at 3, and A can only be at 1 or 4, giving exactly the two arrangements already found. The new clue removes one, leaving exactly one.
3. How should the teacher handle marks and the key? Consider fairness and the worksheet's own instruction that exactly one arrangement exists.
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The teacher should award full credit to both students and revise the key. The item, not the students, was defective: the worksheet promised a unique solution but supplied clues that allowed two. Re-mark every script for this item, because students who gave the alternative arrangement may have been marked wrong. In class, show both arrangements and explain how the missing clue changes the solution. If the worksheet is reprinted, add the clue and test it again before use. The teacher should also record the defect so that the item is not reused unchanged. Penalising students for a valid arrangement would undermine the reasoning objective of the worksheet.