General Intelligence and Reasoning - Fixing a Seating Plan for a Single-Window Verification Drive
The manager of an ESIC branch must seat five clerks - A, B, C, D and E - in a row of five chairs, numbered 1 to 5 from left to right, all facing north, for a single-window verification drive. The seating has to satisfy the office's supervision rules: (i) C must occupy one of the two end chairs. (ii) A must sit second to the left of E. (iii) B must not sit next to C. (iv) D must not sit next to E. The manager's assistant claims the rules fix a unique arrangement, but the union representative argues that some rules are unnecessary and can be dropped without changing the outcome. Before the drive begins, the manager wants to know whether every rule is actually pulling its weight, because each additional rule makes the daily roster harder to prepare and harder to defend if it is challenged.
1. Work out the seating arrangement that satisfies all four rules.
Read the worked answer
Use rule (i) to split the problem. C is at chair 1 or chair 5.
Case C = 1. Rule (ii) allows (A, E) at (2, 4) or (3, 5); the pair (1, 3) is impossible because chair 1 is taken.
- (A, E) = (2, 4): chairs 3 and 5 go to B and D. Rule (iv) says D is not next to E, but chairs 3 and 5 are both adjacent to chair 4, so D has nowhere to sit. Rejected.
- (A, E) = (3, 5): chairs 2 and 4 go to B and D. Rule (iii) keeps B off chair 2, so B = 4 and D = 2. D is not next to E (2 vs 5). Accepted.
Case C = 5. Rule (ii) allows (1, 3) or (2, 4).
- (A, E) = (1, 3): chairs 2 and 4 go to B and D; rule (iii) forces B = 2, D = 4, but D = 4 is next to E = 3. Rejected.
- (A, E) = (2, 4): chairs 1 and 3 go to B and D; either placement puts D next to E = 4. Rejected.
Only arrangement: C – A – D – B – E (chairs 1 to 5).
2. If rule (iv) is dropped, how many different arrangements satisfy rules (i) to (iii)? List them and state what this shows about rule (iv).
Read the worked answer
Enumerate systematically by the position of C and the pair (A, E).
C = 1 (pair can be (2,4) or (3,5)):
- (A, E) = (2, 4): chairs 3, 5 for B, D → C A D B E, C A B D E.
- (A, E) = (3, 5): chairs 2, 4 for B, D, and rule (iii) keeps B off chair 2 → C A D B E (already counted).
C = 5 (pair can be (1,3) or (2,4)):
- (A, E) = (1, 3): chairs 2, 4 for B, D; rule (iii) forces B = 2, D = 4 → A B E D C.
- (A, E) = (2, 4): chairs 1, 3 free in either order → B A D E C, D A B E C.
Total: 5 arrangements - C A D B E, C A B D E, A B E D C, B A D E C, D A B E C.
What this shows. Dropping rule (iv) takes the manager from one valid plan to five, so rule (iv) is doing real work and must be retained. A rule is only redundant if removing it leaves the solution unchanged - popularity or apparent obviousness is not the test.
3. Test rules (i) and (iii) the same way. Is any single rule redundant, and what should the manager tell the union representative?
Read the worked answer
Drop rule (iii). With (i), (ii) and (iv) only:
- C = 1, (A, E) = (2, 4): D cannot sit at 3 or 5, both next to chair 4 → impossible.
- C = 1, (A, E) = (3, 5): D = 2, B = 4 → C D A B E.
- C = 5, (A, E) = (1, 3): D = 4, B = 2 → A B E D C.
- C = 5, (A, E) = (2, 4): D = 1, B = 3 → D A B E C. Three arrangements survive, so rule (iii) is essential.
Drop rule (i). Leaving C free produces at least eight arrangements, for example A B E C D, A C E B D, D A B E C, D A C E B, D B A C E, D C A B E, B D A C E, C D A B E - all consistent with (ii), (iii) and (iv). Rule (i) is essential.
Conclusion. No single rule is redundant, and the assistant is right that the four rules together give exactly one arrangement, C A D B E. The manager can tell the representative that each rule has been tested by removal, with the surviving counts as evidence: dropping (i) gives 8 plans, dropping (iii) gives 3, dropping (iv) gives 5. Keeping all four is the cheapest way to a defensible roster.