Seating for a Grievance Review with a Late Change
Six officers must sit around a hexagonal table for a grievance review. The seats are numbered 1 to 6 clockwise. The officers are the Collector (C), Joint Collector (JC), District Revenue Officer (DRO), Revenue Divisional Officer (RDO), Mandal Parishad Development Officer (MPDO) and Tahsildar (T). The following conditions are fixed: (i) C sits opposite DRO; (ii) JC sits next clockwise to C; (iii) RDO sits adjacent to C; (iv) MPDO sits opposite RDO; (v) T does not sit adjacent to C; (vi) DRO sits adjacent to T. After the arrangement is finalised, the Collector says he wants MPDO, not JC, next clockwise to him, and wants JC opposite RDO. The review is due to start in ten minutes and the seating cards are already printed.
1. Determine the original seating arrangement.
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Place C at seat 1. By (i), DRO is at seat 4. By (ii), JC is at seat 2. By (iii), RDO must be adjacent to C, but seat 2 is occupied, so RDO is at seat 6. By (iv), MPDO is opposite RDO, so MPDO is at seat 3. The only remaining seat is 5, so T is at seat 5. Check (v): T at 5 is not adjacent to C at 1, because the adjacent seats are 2 and 6. Check (vi): DRO at 4 is adjacent to T at 5. The arrangement clockwise is C, JC, MPDO, DRO, T, RDO. It is unique up to rotation and reflection.
2. Can the Collector's change be met while keeping all six original conditions? What should be done?
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No. Let C be at seat 1 and MPDO at seat 2, as the Collector now wants. By (iii), RDO must be adjacent to C; seat 2 is taken, so RDO is at seat 6. The Collector also wants JC opposite RDO, so JC must be at seat 3. By (i), DRO is at seat 4. The only remaining seat for T is 5, which satisfies (v) and (vi). But condition (ii) requires JC next clockwise to C, that is, seat 2, which is now MPDO. So at least one original condition must be relaxed. The minimal relaxation is (ii), because it is a seating preference, not a functional need such as the Tahsildar's adjacency. The officer should record the change, tell the Collector that only the JC condition is being relaxed, and reprint or hand-correct the cards.
3. Which original condition is redundant, and why?
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Condition (vi), that DRO sits adjacent to T, is redundant. Once conditions (i) to (v) are applied, C is at 1, DRO at 4, JC at 2, RDO at 6 and MPDO at 3. The only remaining seat is 5, so T must be at 5, and seat 5 is adjacent to DRO at 4. The adjacency follows from the other conditions; it does not independently restrict the arrangement. Identifying a redundant condition is useful because it shows which rule can be dropped if a late change creates a conflict. It also prevents a false claim that all six conditions have equal force when one is merely a consequence of the others.