Five Remedial Slots, Five Constraints
A small school must run five remedial sessions - A, B, C, D and E - on Monday to Friday, one per day. The constraints are: A must be before B; C can be held only on Wednesday because the guest counsellor comes that day; D must be the day immediately after E because the two teachers share a school vehicle; B cannot be on Friday because Bhavna has block duties; and A cannot be on Monday because Anu takes assembly. The principal will not move C or the D-after-E pairing. Bhavna's Friday block duty is contractual and cannot be changed. Anu's Monday assembly is a school custom that another teacher can cover. You must propose a timetable or show why none exists.
1. Determine whether a schedule satisfying all five constraints exists. Show your case reasoning, not just a final answer.
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No schedule satisfies all five. Fix C on Wednesday. Since D must be immediately after E, the possible pairs are (Mon, Tue), (Tue, Wed), (Wed, Thu) and (Thu, Fri). The pair cannot include Wednesday because C already occupies it, so only (Mon, Tue) or (Thu, Fri) remain.
Case 1: E on Monday, D on Tuesday. Then A and B must take Thursday and Friday. A before B forces A on Thursday and B on Friday. But B on Friday violates Bhavna's block duty. Case 1 fails.
Case 2: E on Thursday, D on Friday. Then A and B must take Monday and Tuesday. A before B forces A on Monday and B on Tuesday. But A on Monday violates Anu's assembly constraint. Case 2 fails.
Both possible placements of the E–D pair fail, so the five constraints are jointly inconsistent. The failure is not a matter of trying harder; it follows from the Wednesday and adjacency constraints narrowing the pair to two options.
2. Which single constraint should you ask to relax, and what fallback timetable do you propose? Justify your choice by distinguishing fixed resources from preferences.
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I would ask to relax 'A cannot be Monday'. C is fixed by the counsellor's only available day; the D-after-E pairing is fixed by the shared vehicle; B's Friday block duty is contractual. Anu's Monday assembly is a school custom that another teacher can cover, so it is the least costly change and the only one that does not damage a real resource commitment.
Fallback: Monday A, Tuesday B, Wednesday C, Thursday E, Friday D. Check: A is before B; C is Wednesday; D is immediately after E; B is not Friday. Only the Monday-assembly preference is broken.
If the principal refuses even that, the next least harmful request is to move B's Friday duty, but that is contractual and would need Bhavna's agreement. I would not ask to move C or split E–D, because those would break the counsellor's visit or the shared transport and would create a worse operational problem than covering one assembly.