Four Constables and the Night-Beat Roster
You are preparing the night-beat roster for four constables - Ali, Barman, Chatterjee and Das - for Monday to Thursday. There are four beats: Market, Station Road, River Ghat and Bus Stand. Each constable is to be assigned exactly one night and one beat, and each beat is to be covered exactly once. The following conditions apply: (i) Das cannot take the Market beat. (ii) Ali is on Monday and is not on the Bus Stand. (iii) Barman is on Wednesday at Station Road. (iv) The Market beat is on Tuesday. (v) The River Ghat beat must fall on an earlier night than the Bus Stand beat. The Inspector also wants a short written note showing why the roster is the only one that satisfies all five conditions, because a constable has challenged the fairness of the allocation.
1. Determine the full roster and show why no other roster satisfies all five conditions.
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Barman is fixed on Wednesday at Station Road, and Market is fixed on Tuesday. Ali is on Monday, so he cannot take Tuesday Market or Wednesday Station Road; he must take River Ghat or Bus Stand. He is not on Bus Stand, so Ali is Monday at River Ghat. Tuesday Market cannot be Das, cannot be Barman (Wednesday) and cannot be Ali (Monday), so it is Chatterjee. That leaves Das for Thursday. The only remaining beat for Thursday is Bus Stand. Check condition (v): River Ghat on Monday is earlier than Bus Stand on Thursday, so the roster is: Monday - Ali, River Ghat; Tuesday - Chatterjee, Market; Wednesday - Barman, Station Road; Thursday - Das, Bus Stand. Each constraint is satisfied, and the deduction is forced at every step, so the roster is unique.
2. Suppose Barman and Das exchange their nights and beats. Which conditions fail?
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The exchange gives Barman Thursday at Bus Stand and Das Wednesday at Station Road. Check each condition. Das is not on Market, so (i) holds. Ali remains on Monday and not on Bus Stand, so (ii) holds. But Barman is no longer Wednesday at Station Road, so (iii) fails. Market remains on Tuesday, so (iv) holds. River Ghat remains on Monday and Bus Stand on Thursday, so (v) holds. Only condition (iii) is broken; the exchange is therefore invalid unless the Inspector formally changes the fixed posting.
3. If condition (ii) were changed to 'Ali is on Monday and is not on the River Ghat', how many rosters would satisfy the conditions? Show the alternatives.
Read the worked answer
Zero rosters. If Ali is on Monday and not on River Ghat, then, as before, he cannot take Tuesday Market or Wednesday Station Road, so he must be Monday at Bus Stand. Tuesday Market must then be Chatterjee, because Das cannot take Market, Barman is fixed on Wednesday, and Ali is on Monday. Barman remains Wednesday at Station Road, leaving Das for Thursday at River Ghat. But then River Ghat (Thursday) is later than Bus Stand (Monday), which contradicts condition (v). No assignment can satisfy all five conditions, so the amended condition makes the roster impossible; the Inspector would have to relax another condition.