Sequencing Five Consignments from a Currency Chest
A currency chest must dispatch five consignments - A, B, C, D and E - to five link branches, one on each working day from Monday to Friday. Each moves in a single van and no day carries more than one consignment. The dispatch register has to satisfy five standing conditions: C goes on Wednesday; A is dispatched on the day immediately before B; D is not dispatched on Monday or Friday; E is dispatched before C; and D must move before A. The branch receiving B has a Friday-evening cash requirement, which is why the order is not a matter of convenience. Each consignment carries 60 packets of Rs 500 notes, that is Rs 30 lakh, and the van log, seal register and branch acknowledgements are checked against the register at the end of the week.
1. Determine the day on which each consignment is dispatched, showing how the alternatives are eliminated.
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Step 1 - place C. C is fixed on Wednesday (slot 3), so no other consignment can use that slot.
Step 2 - A immediately before B. Possible consecutive pairs are (Mon, Tue) and (Thu, Fri). The pairs (Tue, Wed) and (Wed, Thu) are impossible because Wednesday is taken by C.
Step 3 - test (Mon, Tue). D must be dispatched before A, and there is no slot before Monday. This pair fails.
Step 4 - fix (Thu, Fri). So A = Thursday, B = Friday. This also satisfies the Friday-evening requirement of B's branch.
Step 5 - place D. D must precede A, so D is Monday or Tuesday; the register bars Monday, so D = Tuesday.
Step 6 - place E. E must precede C; the only free slot left is Monday.
| Day | Mon | Tue | Wed | Thu | Fri |
|---|---|---|---|---|---|
| Consignment | E | D | C | A | B |
Every condition is satisfied: C on Wednesday, A immediately before B, D on Tuesday (neither Monday nor Friday), E before C, D before A. The solution is unique.
2. A colleague proposes shifting D to Monday and E to Tuesday, leaving C, A and B where they are. Is the proposal acceptable?
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Check the proposal against each condition rather than against intuition.
| Condition | Proposed (D Mon, E Tue, C Wed, A Thu, B Fri) | Verdict |
|---|---|---|
| C on Wednesday | C is on Wednesday | Satisfied |
| A immediately before B | Thu then Fri | Satisfied |
| D not on Monday or Friday | D is on Monday | Violated |
| E before C | Tue before Wed | Satisfied |
| D before A | Mon before Thu | Satisfied |
The proposal breaks exactly one rule - the bar on Monday dispatch for D - and that rule is not cosmetic: Monday is the day the chest reconciles the previous week's seals and acknowledgements, and a Monday move of the largest consignment would land it in the middle of that reconciliation. Reject the proposal. Because the original arrangement is unique, there is no valid alternative to fall back on; if D genuinely cannot move on Tuesday, one of the standing conditions has to be formally relaxed first, in writing.
3. The Wednesday consignment is postponed to Saturday and no other condition may be broken. Is a valid schedule still possible? Give it or prove it impossible.
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Now there are six slots (Mon to Sat) for five consignments, so one day is free. Re-run the same elimination.
Step 1 - A immediately before B. The consecutive pairs available are (Mon, Tue), (Tue, Wed), (Thu, Fri) and (Fri, Sat).
Step 2 - apply 'D before A' with D not on Monday.
- (Mon, Tue): no slot exists before Monday for D - fails.
- (Tue, Wed): D would have to be Monday, which is barred - fails.
- (Thu, Fri): D can sit on Tuesday - works.
- (Fri, Sat): D could sit on Tuesday or Wednesday - works in principle, but then E and C must both be placed before or after them without breaking 'E before C', which needs two ordered slots; only Monday is left before D on Tuesday, so E and C cannot both be placed before C's deadline - fails.
Step 3 - the surviving arrangement is A = Thursday, B = Friday, D = Tuesday, C = Saturday, E = Monday, with Wednesday free.
Check: E (Mon) before C (Sat) ✓; D (Tue) not Monday or Friday ✓; D before A ✓; A immediately before B ✓.
So the schedule survives, and the free Wednesday is the useful output of the change: it is the natural slot for the weekly seal reconciliation that the Monday rule was protecting.