The Class VIII Examination Timetable That Would Not Settle
Five papers - Hindi (H), English (E), Mathematics (M), Science (S) and Social Science (SS) - must be held on five consecutive days, Monday to Friday, one paper per day, for the Class VIII terminal examination of a Bihar middle school. The examination committee has fixed four rules:
- Science must be on Wednesday. The laboratory apparatus is hired from a neighbouring school and is released only for that day.
- Hindi must be exactly two days before English (Monday→Wednesday, Tuesday→Thursday, Wednesday→Friday).
- Mathematics cannot be on Monday. The maths teacher is on block-level duty that day.
- Social Science cannot be on Friday. The Social Science teacher is the external observer at another centre.
Three drafts are on the table. Draft 1 (senior teacher): Mon SS, Tue H, Wed S, Thu E, Fri M. Draft 2 (new teacher): Mon H, Tue M, Wed S, Thu E, Fri SS. Draft 3 (clerk): Mon M, Tue SS, Wed S, Thu H, Fri E. The senior teacher claims Draft 1 is the only possible schedule; the new teacher insists several schedules are possible and proposes drawing lots.
1. Test each draft against the four rules and settle whether Draft 1 is genuinely the only possible schedule.
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Check the drafts.
- Draft 1: SS Mon (not Friday ✓), H Tue and E Thu are exactly two days apart ✓, S Wed ✓, M Fri (not Monday ✓). Valid.
- Draft 2: H Mon, E Thu - three days apart, not two. Fails rule 2. SS is also on Friday, failing rule 4. Invalid on two counts.
- Draft 3: M on Monday, failing rule 3; H Thu → E Fri is one day apart, failing rule 2. Invalid.
Now exhaust the possibilities instead of guessing. Science occupies Wednesday, so H, E, M and SS must fill Monday, Tuesday, Thursday and Friday. The only admissible Hindi–English pairs from that set are (Tue, Thu); (Mon, Wed) and (Wed, Fri) both need Wednesday, which is taken by Science. Therefore H = Tuesday and E = Thursday, forced. Monday and Friday remain for M and SS; Mathematics cannot be Monday, so M = Friday and SS = Monday.
Conclusion: Draft 1 is correct, and the new teacher's 'several schedules' claim is false. But the senior teacher's reasoning ('it looks right') is not the reason: uniqueness follows from the fact that only one of the three permissible Hindi–English pairs survives once Science is fixed on Wednesday. A schedule is proved, not chosen.
2. A teacher requests that Science be moved to Thursday. What does that do to the problem, and what should the committee do?
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Removing the Wednesday anchor destroys uniqueness. With Science free, all three Hindi–English pairs become available and each generates several completions:
- H Mon, E Wed: Tuesday, Thursday, Friday remain for S, M and SS. SS cannot be Friday, so SS takes Tuesday or Thursday (2 choices), and S and M fill the last two days in 2 orders → 4 schedules.
- H Tue, E Thu: Monday, Wednesday, Friday remain. M cannot be Monday. If SS = Monday, M is Wednesday or Friday (2). If SS = Wednesday, M must be Friday (1). SS = Friday is barred. → 3 schedules.
- H Wed, E Fri: Monday, Tuesday, Thursday remain. M is Tuesday or Thursday (2), and S and SS take the remaining two days in 2 orders → 4 schedules.
Total: 11 valid schedules. The committee would be choosing among 11, not one, and no rule yet discriminates between them.
The correct response is procedural, not mathematical: either keep Science on Wednesday (the apparatus matters more than the convenience of one teacher), or, if the move is unavoidable, add one explicit tie-break rule before drafting - for example, 'the paper with the largest number of candidates goes first' - so that the resulting timetable is defensible rather than arbitrary. Drawing lots would produce a valid schedule but an indefensible one if a parent asks why.
3. The clerk argues: 'Since Mathematics cannot be Monday, the safe choice is to put it on Friday.' Is that inference sound?
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The conclusion is right; the inference is not.
'Mathematics is not on Monday' rules out exactly one day out of five. On its own it leaves Tuesday, Wednesday, Thursday and Friday open - four possibilities, not one. The clerk's step from 'not Monday' to 'therefore Friday' is valid only because of the other three rules: Science consumes Wednesday, and the Hindi–English pair consumes Tuesday and Thursday, leaving Monday and Friday; of those two, Monday is barred. It is the combination that forces Friday.
This distinction has practical weight. If any one of the other rules is relaxed - as in the previous task, when Science moves off Wednesday - the clerk's shortcut silently becomes false, and a timetable built on it may violate nothing the clerk is checking yet still be wrong. The safe habit is to write down, for each rule, exactly which days it eliminates, and to stop only when every remaining day has exactly one surviving paper. 'It feels constrained' is not a proof; 'one admissible option remains' is.