Building a defensible exam timetable from five constraints
The regional office of a rural bank must hold six recruitment-related tests - Reasoning (R), Quantitative Aptitude (Q), English (E), Computer Knowledge (C), Banking Awareness (B) and Hindi (H) - on six consecutive days, Monday to Saturday, one test per day. Five constraints were circulated to the exam committee: (i) H must be on Saturday; (ii) Q must be held on the day immediately before H; (iii) B must be held on the day immediately after R; (iv) C cannot be held on Monday or Wednesday; (v) E must be held before R. The exam coordinator has booked the Computer lab vendor for Tuesday and argues that the timetable is therefore settled. The committee chair disagrees and wants every valid timetable listed before any booking is confirmed. No test may share a day, and no constraint may be relaxed.
1. Enumerate every timetable that satisfies constraints (i) to (v). Show the deduction step by step and state which position is forced in all solutions.
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Step 1: (i) and (ii) together fix H on Saturday and Q on Friday. Step 2: R, B, E, C must now fill Monday to Thursday, with B immediately after R. R can begin only on Tuesday or Wednesday - R on Monday leaves no earlier day for E, and R on Thursday pushes B to Friday, which Q occupies. Step 3: if R is Tuesday, B is Wednesday, E must be Monday (its only earlier day) and C takes Thursday, which is legal because C is barred only on Monday and Wednesday. Step 4: if R is Wednesday, B is Thursday, and E and C take Monday and Tuesday; since C cannot be Monday, C is Tuesday and E is Monday. Two timetables survive: (1) Mon E, Tue C, Wed R, Thu B, Fri Q, Sat H; (2) Mon E, Tue R, Wed B, Thu C, Fri Q, Sat H. E on Monday is common to both, so any proposed timetable that does not start with E is wrong without further checking.
2. If the vendor booking is treated as binding, how many of the timetables survive, and what does that tell the chair about the coordinator's reasoning?
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Only timetable (1) survives. In timetable (2) the Computer test sits on Thursday; forcing C to Tuesday pushes R to Thursday and therefore B to Friday, which is already occupied by Q. R cannot move to Monday, because E must precede R. So the vendor date converts a two-solution puzzle into a single answer: Mon E, Tue C, Wed R, Thu B, Fri Q, Sat H. The chair should note that the coordinator reached the right answer through an invalid step. When the five constraints were issued, the vendor date was not one of them, so the timetable was not settled at that point; it became settled only once the Tuesday booking was declared binding. Recording that as a formal sixth condition matters, because if the vendor shifts, both timetables reopen and the whole exercise must be repeated.
3. A sixth condition arrives: E and C must not be held on consecutive days. Which timetable remains, and what would it mean if a later condition eliminated that one too?
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Test the new condition against each surviving arrangement. Timetable (1) has E on Monday and C on Tuesday, which is consecutive, so it is invalid. Timetable (2) has E on Monday and C on Thursday, which is not consecutive, so it is valid and is the unique answer: Mon E, Tue R, Wed B, Thu C, Fri Q, Sat H. Method point: apply a new condition to the surviving solutions rather than rebuilding the puzzle from scratch, which is faster and less error-prone under time pressure. If a further condition knocked out this arrangement as well, the correct conclusion would be that no timetable exists under the full constraint set - the chair must then ask which constraint to relax or which constraint was mis-stated, not invent an extra day or quietly drop a test.