Reconstructing a Seating Note Before a Grievance Hearing
Seven officers - A, B, C, D, E, F and G - must be seated in a single row of seven chairs, all facing north, for a grievance hearing. The seating note left by the clerk records six conditions but no final arrangement: (i) A sits third to the left of E. (ii) Exactly two officers sit between B and F. (iii) C sits immediately to the right of G. (iv) D does not sit at either end. (v) F sits second to the right of A. (vi) G sits at one of the two ends. The clerk has since resigned, and the officer-in-charge must reconstruct the arrangement from these conditions alone, and be satisfied that the note is not under-determined before the chairs are fixed. Two chairs in the hall have already been labelled, so any ambiguity will produce a visible error in front of the parties attending the hearing.
1. Who sits second to the left of the officer who is third from the right end, and what is the general rule for converting 'nth from the right' into a chair number in a seven-chair row?
Read the worked answer
Convert to absolute chair numbers first; relative descriptions are where errors creep in.
Rule. In a row of N chairs, the position that is kth from the right is chair N − k + 1. Here N = 7, k = 3, so the officer third from the right is in chair 7 − 3 + 1 = chair 5, which is D.
Apply the displacement. Second to the left of chair 5 is chair 5 − 2 = chair 3, which is B.
So the answer is B.
A useful self-check: 'left' and 'right' in these puzzles are the officers' left and right, and since all seven face north they coincide with the reader's left and right. Had they faced south, the same phrase would have flipped, and converting via chair numbers makes that flip explicit rather than intuitive - which is exactly the step candidates skip under time pressure.
2. Before the chairs are fixed, the officer-in-charge asks whether the note is reliable on its own. Test what happens if condition (v) were missing, and state the practical instruction you would issue.
Read the worked answer
Remove (v) and re-run the deduction. G = 1 and C = 2 still follow. From (i), A ≤ 4, so A ∈ {3, 4}.
Branch A = 4: E = 7. Free chairs are 3, 5, 6 for B, F, D. Condition (ii) forces F = 6 and B = 3, leaving D = 5. One arrangement: G C B A D F E.
Branch A = 3: E = 6. Free chairs are 4, 5, 7 for B, F, D. Condition (ii) permits the pair {4, 7} in either order, giving:
- F = 4, B = 7, D = 5 → G C A F D B E
- B = 4, F = 7, D = 5 → G C A B D F E
Both satisfy (iv) because D sits in chair 5.
Result. Without (v) there are three valid arrangements, not one. The note is therefore under-determined, and (v) is doing essential work - it is not a redundant clue.
Practical instruction. Do not fix chairs from the note as it stands. Recover the missing condition from the file, or add a seventh discriminating condition, and re-derive. As a fallback, ask which chairs are occupied by the same officer in all three surviving arrangements - C is always in chair 2 and D always in chair 5 - and label only those pending confirmation of the rest.