Five Clerks, Five Counters: Testing a Proposed Duty Roster
A cooperative bank branch must assign five clerks-Amit, Bina, Chetan, Divya and Esha-to counters 1 to 5 for one day. Each counter gets exactly one clerk. The branch rules are: Divya must sit at counter 3; Chetan must sit immediately to the right of Bina; Amit must sit at an end counter; Bina cannot sit at counter 2; and Esha cannot sit at counter 2. Two rosters were proposed. Proposal 1: Bina at 1, Chetan at 2, Divya at 3, Esha at 4, Amit at 5. Proposal 2: Amit at 1, Esha at 2, Divya at 3, Bina at 4, Chetan at 5. The manager must decide which roster satisfies every rule, whether any clerk's counter is fixed, and what must change if Divya is later required at counter 4.
1. Test both proposals against all five rules. Which one is valid, and why does the other fail?
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Check Proposal 1: Bina 1, Chetan 2, Divya 3, Esha 4, Amit 5.
- Divya at 3: satisfied.
- Chetan immediately right of Bina: 1 → 2, satisfied.
- Amit at an end: counter 5, satisfied.
- Bina not at 2: at 1, satisfied.
- Esha not at 2: at 4, satisfied. Proposal 1 is valid.
Check Proposal 2: Amit 1, Esha 2, Divya 3, Bina 4, Chetan 5.
- Divya at 3: satisfied.
- Chetan immediately right of Bina: 4 → 5, satisfied.
- Amit at an end: counter 1, satisfied.
- Bina not at 2: at 4, satisfied.
- Esha not at 2: violated, because Esha is at counter 2. Proposal 2 fails only on the Esha rule.
2. Is any clerk's counter fixed? Derive the complete valid arrangement.
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Divya is fixed at counter 3 by the rule. Now place Bina and Chetan.
- Bina cannot be at 2, and counter 3 is taken by Divya.
- If Bina were at 4, Chetan would be at 5. The remaining counters 1 and 2 would go to Amit and Esha. Amit must be at an end, so Amit would be at 1 and Esha at 2 - but Esha cannot be at 2. So Bina cannot be at 4.
- Therefore Bina is at 1 and Chetan at 2.
- Remaining counters are 4 and 5 for Amit and Esha. Amit must be at an end, so Amit is at 5 and Esha at 4.
Complete arrangement: Bina 1, Chetan 2, Divya 3, Esha 4, Amit 5. Every clerk's counter is fixed by the rules.
3. If Divya is later required at counter 4, which rule must be relaxed, and what is the least disruptive change?
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Divya at counter 4 directly violates the rule 'Divya must sit at counter 3'. That is the only rule that must be relaxed.
Keeping the other rules:
- Bina cannot be at 2.
- Chetan must be immediately right of Bina.
- Amit must be at an end.
- Esha cannot be at 2.
Place Bina at 1 and Chetan at 2. Divya is at 4. Remaining counters are 3 and 5 for Amit and Esha. Amit must be at an end, so Amit goes to 5 and Esha to 3. Esha at 3 satisfies her restriction.
Least disruptive roster: Bina 1, Chetan 2, Esha 3, Divya 4, Amit 5. Only Divya's fixed-counter rule changes; no other clerk needs to break a rule.