Scheduling Six Panchayat Verifications in a Block
A block development office has to verify six village panchayats - Amarpur (A), Bishanpur (B), Chandpur (C), Dhanhar (D), Ekma (E) and Phulwari (F) - across six working days, Monday to Saturday, exactly one village per day. The office has fixed these rules: (i) A must be verified before C; (ii) B and E must be verified on consecutive days, with B first; (iii) D cannot be on Monday or Saturday; (iv) C cannot be on Wednesday; (v) F cannot be on Thursday, because only half a day is available and F is the farthest. Distances from the block headquarters are: A 12 km, B 14 km, C 18 km, D 16 km, E 25 km and F 30 km. The vehicle's monthly diesel allocation covers 120 km of field travel, and these six round trips are the only field travel planned for that week. The accountant has already warned that the travel budget will be exceeded.
1. Construct one valid schedule and verify it against every condition.
Read the worked answer
One valid schedule is:
| Day | Village |
|---|---|
| Monday | B |
| Tuesday | E |
| Wednesday | A |
| Thursday | D |
| Friday | C |
| Saturday | F |
Verification, condition by condition.
- (i) A on Wednesday is before C on Friday - satisfied.
- (ii) B on Monday and E on Tuesday are consecutive, B first - satisfied.
- (iii) D on Thursday is neither Monday nor Saturday - satisfied.
- (iv) C on Friday is not Wednesday - satisfied.
- (v) F is on Saturday, not Thursday - satisfied.
Exactly one village occupies each of the six days, so no day is doubled. This is not the only possible schedule - for example, D could be on Wednesday or Friday - but it satisfies every stated rule, which is all the question asks. When constructing such a schedule, the efficient method is to fix the most tightly coupled pair (here B–E) first, then place the two order-linked villages (A before C), and only then fit the two remaining villages around the exclusions.
2. Which villages can never occupy the Saturday slot, and which of the remaining ones can? Justify each answer.
Read the worked answer
Cannot be on Saturday.
- A - rule (i) requires C after A, and there is no day after Saturday.
- B - rule (ii) requires E on the immediately following day, which would fall on Sunday.
- D - excluded directly by rule (iii).
Can be on Saturday.
- C: put B = Mon, E = Tue, A = Wed, D = Thu, F = Fri, C = Sat. C is not on Wednesday and A precedes it.
- E: put B = Fri and E = Sat; A, C, D, F fill Monday to Thursday, with D on Tuesday or Thursday and A before C.
- F: rule (v) excludes only Thursday, so Saturday is open.
So the complete answer is: A, B and D can never take Saturday; C, E and F can.
The trap is to assume only the village named in an explicit exclusion rule is barred. Two of the three barred villages are barred by inference from an ordering rule that needs a following day.
3. The accountant says the diesel allocation will be exceeded, and an officer suggests verifying B and E on the same day to save 14 km. Evaluate both claims.
Read the worked answer
Check the arithmetic first. Total distance = 12 + 14 + 18 + 16 + 25 + 30 = 115 km. Allocation = 120 km, so the plan is within budget by 5 km - the accountant's claim is wrong as stated, though the margin is so thin that one extra trip, a diversion or a detour on the F route would break it. Any fuel warning should be settled from odometer logs, not from impressions.
Reject the officer's proposal. Verifying B and E on the same day saves 14 km of travel but breaks two things at once: the one-village-per-day rule, and the requirement that B and E fall on consecutive days. It also buys trouble - one long day in the field, a thinner verification record and a likely objection at the next audit.
Defensible alternatives. Cluster the near villages: A (12 km) and B (14 km) are close, so scheduling them on Monday and Tuesday and staying overnight in that cluster removes one return leg. That frees roughly 12 km of margin for the F trip, which is the longest and the most likely to need a second visit.