Six tasks, six days: a scheduling puzzle with a constraint that will not fit
On Monday morning the VDO of Devgarh Gram Panchayat finds six pending tasks and exactly six working days, Monday to Saturday, with one task per day. D is a block-level review meeting that the Block Development Officer has fixed for Wednesday; it cannot be moved. A is cleaning and disinfection of the village drinking-water tank, and C is collection of water samples for bacteriological testing at the district laboratory; the tank must be cleaned exactly two days before sampling, so that exactly one working day falls between them. E is a ration-card verification camp, and B is the gram sabha, which the sarpanch has agreed to hold on the very next day after the camp so that objections raised at the camp can be placed before the sabha. F is a site inspection of the proposed cattle shed, and the animal husbandry officer, who does not work on Mondays or Saturdays, can accompany the VDO only on a day that is neither Monday nor Saturday. The VDO must also keep C after D, because the laboratory will not accept samples from a tank that has not been inspected at the review.
1. Fix the day for each of the six tasks, showing how the constraints force the answer.
Read the worked answer
Day numbers: Mon 1, Tue 2, Wed 3, Thu 4, Fri 5, Sat 6. D = 3.
- C must be after D, so C ∈ {4, 5, 6}. Also C = A + 2.
- If C = 4, then A = 2. Days left for E, B, F are 1, 5, 6. B = E + 1 forces E = 5, B = 6, leaving F = 1 - but F cannot be Monday. Rejected.
- If C = 5, then A = 3, which is D's day. Rejected.
- Therefore C = 6 and A = 4. Days left are 1, 2, 5. B = E + 1 forces E = 1, B = 2, and F = 5, which is neither Monday nor Saturday and is therefore allowed.
Schedule: Mon E (ration-card camp), Tue B (gram sabha), Wed D (block review), Thu A (tank cleaning), Fri F (cattle-shed inspection), Sat C (water sampling).
The order is unique, and every constraint is satisfied exactly once, which is the check to run before publishing it.
2. A ward member demands that the gram sabha be held on Saturday, after the review. Is a valid schedule still possible?
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No - the demand makes the set of constraints infeasible, and the VDO should say so rather than quietly break a rule.
Try B = 6, so E = 5. D = 3 is fixed. Days left for A, C, F are 1, 2, 4. The pair A, C with one day between can only be (2, 4) here, because (1, 3) collides with D and (4, 6) collides with B. So A = 2, C = 4, leaving F = 1 - a Monday, which the animal husbandry officer does not permit.
So the gram sabha on Saturday forces the cattle-shed inspection to Monday, which is barred. The VDO must place this before the sarpanch and ward member and ask them to relax one condition explicitly: either keep the sabha on Tuesday as scheduled, or accept that F moves to a day the officer is unavailable and the inspection is done by the VDO alone. Silently dropping the F condition would mean an invalid inspection report later.
3. If the block review (D) is shifted off Wednesday and may fall on any single day, does a schedule still exist, and is it unique?
Read the worked answer
A schedule exists, but uniqueness is lost - the VDO gains flexibility and must now justify whichever day is chosen.
One valid arrangement: Mon A, Tue D, Wed C, Thu F, Fri E, Sat B. Checks: A (1) is two days before C (3), with one day between - yes; C (3) is after D (2) - yes; B (6) is the day after E (5) - yes; F (4) is neither Monday nor Saturday - yes.
Other arrangements are also possible (for example, keeping E and B consecutive on a different pair of days), so the answer is not a single unique schedule.
The practical judgement: when the fixed anchor disappears, do not present the first workable order as if it were the only one. Publish the order along with the reason for it - officer availability, lab days, sabha notice period - so that a ward member cannot credibly claim a different order was possible and was denied to him.