Fitting five activities into five days
The head teacher of a government senior secondary school must fix one activity for each of the five working days, Monday to Friday, of a Sadbhavana Saptah. Five activities remain: a quiz (Q), a poster competition (P), a Hindi debate (H), a cleanliness rally (R) and a science model show (S). The staff meeting agreed four constraints: (i) the debate must be on Wednesday, because the Block Education Officer is visiting that day; (ii) the poster competition must be on the day immediately after the quiz, because quiz results are used to choose poster themes; (iii) the rally cannot be on Monday (assembly is short) or on Friday (half day); (iv) the science show must be before the debate. A newly posted teacher proposes the simple order Q, P, H, R, S from Monday onwards and asks whether anything is violated. The Hindi teacher has since applied for leave on Wednesday.
1. Test the new teacher's proposed order against the four constraints, then find the order that satisfies all four and show that it is the only one.
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Testing Q, P, H, R, S: Q on Monday, P on Tuesday satisfies (ii); H on Wednesday satisfies (i); R on Thursday satisfies (iii); but S on Friday falls after the debate, so constraint (iv) is violated - exactly one rule, which is why the order looks plausible.
Finding the valid order. H is fixed on Wednesday. The pair Q–P must be adjacent, so it can only be (Monday, Tuesday) or (Thursday, Friday) - (Tuesday, Wednesday) and (Wednesday, Thursday) are excluded because Wednesday is taken.
- Suppose Q = Monday, P = Tuesday. Then R and S fill Thursday and Friday; R cannot be Friday, so R = Thursday and S = Friday. But S must precede H on Wednesday, which fails.
- So Q = Thursday, P = Friday. Then R and S fill Monday and Tuesday; R cannot be Monday, so R = Tuesday and S = Monday, which does precede H.
Result: Monday S, Tuesday R, Wednesday H, Thursday Q, Friday P - and the contradiction in the first branch proves it is the only arrangement.
2. The Hindi teacher is on leave on Wednesday. Can all four constraints still hold? Give your judgement and, if the schedule must change, the arrangement you would adopt.
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First judgement: the leave is a staffing problem, not a scheduling problem. Constraint (i) exists because of an external official's visit; a leave can be covered by a substitute teacher or by having the debate anchored and judged by a colleague, with the Hindi teacher preparing the topics in advance. Institutional commitments outrank personal leave, so the first answer is to keep the original schedule and arrange cover.
If no cover is possible, H must move off Wednesday and (i) must be renegotiated with the BEO. H cannot be Monday, because nothing can precede it for S. H cannot be Thursday: then Q–P would have to be (Monday, Tuesday), leaving R and S on Wednesday and Friday, R cannot be Friday, so S lands on Friday - after the debate. So H = Tuesday is the clean repair: with S on Monday (before H), R on Wednesday (allowed), Q on Thursday and P on Friday, all remaining rules hold and the arrangement is again unique.
Moving H to Friday also works but produces several valid orders, so Tuesday is better because one clearly announced schedule is easier to run.
3. Is constraint (iv), the science show before the debate, actually necessary? Prove your answer by removing it.
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Without (iv) the puzzle has two solutions, so (iv) is what makes the plan unique - and in a school timetable uniqueness is the point.
Remove (iv) and keep (i), (ii), (iii). H stays on Wednesday, and Q–P must again be either (Monday, Tuesday) or (Thursday, Friday).
- Q = Monday, P = Tuesday: R and S take Thursday and Friday; R cannot be Friday, so R = Thursday and S = Friday. Valid.
- Q = Thursday, P = Friday: R and S take Monday and Tuesday; R cannot be Monday, so R = Tuesday and S = Monday. Valid.
Two arrangements now satisfy every stated rule, and nothing in the brief chooses between them. The office would have to invent a tie-breaker - for example, preferring the science show early so that models are displayed all week - which is simply a new constraint in disguise.
Conclusion: (iv) is not needed for feasibility, but it is essential for a single, communicable schedule. Keep it and say so in the notice, so no one later 'improves' the order.