Five Verification Calls, Four Rules, One Schedule
At a rural branch, the business correspondent must schedule five customer-verification calls-A, B, C, D and E-one on each working day from Monday to Friday. The calls cover KYC re-verification, a loan-utilisation check, insurance consent, pension verification and an SHG group visit. Only one call can be made per day. The branch follows four scheduling rules: C must be exactly two days after A; B must be on the day immediately before E; D cannot be scheduled on Monday or Friday; and E cannot be scheduled on Friday. The manager wants the final Monday-to-Friday order displayed on the notice board before the week starts. A clerk claims the four rules do not produce a single order and that the branch should wait for more information. You are asked to test that claim, prepare the schedule if one exists, and advise whether any rule can be removed without changing the order.
1. Prepare the only valid Monday-to-Friday order. Show the case check for A rather than guessing.
Read the worked answer
Let Monday = 1 and Friday = 5. Since C is exactly two days after A, A can only be 1, 2 or 3:
- A = 1, C = 3: remaining days are 2, 4 and 5. B immediately before E requires consecutive days, so the only pair is 4–5; but E cannot be Friday. Fail.
- A = 2, C = 4: remaining days are 1, 3 and 5. No two consecutive days remain for B and E. Fail.
- A = 3, C = 5: remaining days are 1, 2 and 4. B = 1, E = 2; D = 4. D is not Monday or Friday, so this is valid.
Final order: Monday B, Tuesday E, Wednesday A, Thursday D, Friday C. The clerk's claim is false: the rules do produce a single order.
2. The clerk wants to drop one rule to simplify the notice. Which rule is redundant, and does removing it create another valid order?
Read the worked answer
The rule 'D cannot be scheduled on Monday or Friday' is redundant. In the only valid placement, A = Wednesday and C = Friday use two days; B and E must occupy Monday and Tuesday; D is then forced to Thursday, which already satisfies the D rule. Removing it does not create a new order because the other three rules still force B/E to Monday/Tuesday and A/C to Wednesday/Friday. The rule is logically redundant only because the other constraints are strong; if those constraints changed, it could become necessary. Dropping it from the notice is safe, but keeping it is harmless.
3. A last-minute instruction bans E from Tuesday. Does a valid schedule still exist? Explain.
Read the worked answer
No. With A = Wednesday and C = Friday, B must be immediately before E. The only available consecutive pair is Monday/Tuesday, so E must be Tuesday. Banning Tuesday for E makes B/E impossible. The other A placements already fail for independent reasons, so no schedule satisfies all the rules. The branch must relax a rule-for example, allow E on Tuesday or move D-rather than publish a false schedule.