Five settlement files, one working week: testing a mid-week direction against office rules
Five settlement files, F1 to F5, must be examined in one working week, one on each day from Monday to Friday. The office rules are: (i) F2 must be taken up before F4; (ii) F5 must be taken up on Wednesday; (iii) F1 must be taken up on the day immediately after F4; (iv) F3 cannot be taken up on Monday or on Friday. Mid-week a Registrar's circular arrives requiring F3 to be heard on Thursday, and the same week has to be retained. The head clerk says the circular can be ignored because the office rules are older. The senior assistant says the whole schedule must be redrawn from scratch. There is no spare day, no power to add a day, and every file must be heard within the week.
1. Derive the only schedule that satisfies rules (i) to (iv). Show the elimination, not just the answer.
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Fix F5 on Wednesday.
Rule (iii) gives F1 immediately after F4. Possible adjacent pairs are Monday-Tuesday, Tuesday-Wednesday, Wednesday-Thursday and Thursday-Friday. The middle two are blocked because Wednesday is taken by F5. So either F4 is Monday and F1 Tuesday, or F4 is Thursday and F1 Friday.
Try the first: F4 Monday, F1 Tuesday. Thursday and Friday remain for F2 and F3. Rule (iv) bars F3 from Friday, so F3 is Thursday and F2 is Friday. But rule (i) requires F2 before F4, and Friday is after Monday. Contradiction, so this branch fails.
So F4 is Thursday and F1 is Friday. Monday and Tuesday remain for F2 and F3. Rule (iv) bars F3 from Monday, so F3 is Tuesday and F2 is Monday. Check rule (i): Monday is before Thursday, satisfied.
Schedule: Monday F2, Tuesday F3, Wednesday F5, Thursday F4, Friday F1. It is unique because every other placement of the pair was eliminated.
2. Can F3 be heard on Thursday with rules (i) to (iv) all intact? Find the minimum change.
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With F3 on Thursday and F5 on Wednesday, Monday, Tuesday and Friday are left for F2, F1 and F4. Rule (iii) forces F4 Monday and F1 Tuesday, because no other adjacent pair fits the remaining days. F2 is then left with Friday, which breaks rule (i), since Friday is not before Monday.
So no schedule satisfies all four rules together with the circular. One rule must give, and the question is which one costs least.
Rule (ii) is a fixed date and the circular is a direction from a higher authority, so neither can be moved by the office. That leaves (i), (iii) and (iv). Rule (iii) is a strict adjacency that also pins F4, and rule (iv) is an availability rule; relaxing either opens several schedules. Relaxing only rule (i) gives Monday F4, Tuesday F1, Wednesday F5, Thursday F3, Friday F2, which satisfies (ii), (iii) and (iv) and puts every file inside the week. So the minimum change is to drop rule (i) alone and to record the reason.
3. Who is right, the head clerk or the senior assistant? What does this case show about the constraints?
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Neither. The head clerk is wrong because a circular from the Registrar is a binding instruction from a higher authority, not an old office convention that can be ignored on seniority of age. The senior assistant is wrong because a single changed date does not require the whole schedule to be rebuilt; the exercise is to find the smallest breach, not the largest.
The case turns on separating hard constraints from soft ones. Fixed dates on the calendar, and directions from a higher authority, are hard: they cannot be varied by the office. Internal sequencing rules are soft: they express priority and can be relaxed with a recorded reason. The solution keeps every hard constraint, keeps three of four soft rules, and breaches exactly one, and it still places all five files within the week, which was the real requirement all along.