Six Counter Postings Before a Saturday Audit
The branch is reorganising its six service counters, numbered 1 to 6 from the entrance towards the manager's cabin, before a Saturday audit. Six clerks, Anil, Bina, Chetan, Divya, Esha and Farhan, must each be posted at exactly one counter, one clerk per counter. The branch head has fixed five conditions: Anil must sit at counter 3, because he is the only one trained on the new loan-enquiry module; Bina must sit immediately to the right of Farhan, as the two form the cash-verification pair; Chetan must not be adjacent to Anil, because the audit last year flagged that their counters were exchanging documents without a register entry; Divya must sit at one of the two end counters, since she also mans the emergency exit; and Esha must be posted somewhere to the right of Anil. The head also wants the cash-verification pair to be adjacent to the loan-enquiry counter before the arrangement is signed off.
1. Work out the complete posting order from counter 1 to counter 6, and show why the conditions permit only one arrangement.
Read the worked answer
Fixed point: counter 3 = Anil. Divya is at counter 1 or 6. Consider both.
Case A - Divya at counter 1. Free counters: 2, 4, 5, 6 for Bina, Chetan, Esha, Farhan. The Farhan-Bina block can only occupy (4,5) or (5,6), because (2,3) and (3,4) are blocked by Anil.
- Block (4,5): Farhan 4, Bina 5. Chetan and Esha take 2 and 6. Esha must be right of Anil, so Esha 6 and Chetan 2 - but Chetan at 2 is adjacent to Anil at 3. Violates the adjacency rule.
- Block (5,6): Farhan 5, Bina 6. Chetan and Esha take 2 and 4. Esha must be right of Anil, so Esha 4 and Chetan 2 - again adjacent to Anil. Violates. So Divya cannot be at counter 1.
Case B - Divya at counter 6. Free counters: 1, 2, 4, 5. The Farhan-Bina block can be (1,2) or (4,5).
- Block (4,5) leaves 1 and 2 for Chetan and Esha, both left of Anil, so Esha cannot be right of Anil. Invalid.
- Block (1,2): Farhan 1, Bina 2. Chetan and Esha take 4 and 5. Esha must be right of Anil, so Esha 4 and Chetan 5. Chetan at 5 is not adjacent to Anil at 3. Valid.
Unique order: 1 Farhan, 2 Bina, 3 Anil, 4 Esha, 5 Chetan, 6 Divya.
Notice the method: the end condition on Divya is the strongest lever, and the adjacency condition on Chetan is what kills two otherwise-valid branches. Always test the extreme-value clue first.
2. Because of a staff transfer the "Chetan must not be adjacent to Anil" condition is withdrawn. Is the posting still unique? If not, list every valid arrangement.
Read the worked answer
No, it is no longer unique. Re-run the two cases with that one condition removed; nothing else changes.
Case A - Divya at 1. Both sub-branches that previously failed now survive:
- Farhan 4, Bina 5, Chetan 2, Esha 6 → 1 Divya, 2 Chetan, 3 Anil, 4 Farhan, 5 Bina, 6 Esha
- Farhan 5, Bina 6, Chetan 2, Esha 4 → 1 Divya, 2 Chetan, 3 Anil, 4 Esha, 5 Farhan, 6 Bina
Case B - Divya at 6.
- Farhan-Bina at (1,2): counters 4 and 5 go to Chetan and Esha in either order, since Esha only has to be right of Anil and both 4 and 5 qualify:
- 1 Farhan, 2 Bina, 3 Anil, 4 Esha, 5 Chetan, 6 Divya
- 1 Farhan, 2 Bina, 3 Anil, 4 Chetan, 5 Esha, 6 Divya
- Farhan-Bina at (4,5) still fails, because Esha would have to sit at 1 or 2, left of Anil.
Total: four valid arrangements.
The practical point: removing a negative constraint usually multiplies the answer set rather than nudging it. If the branch head now says "just fix any valid order", the clerk should ask which arrangement he means, because the audit will be checking one specific plan.
3. The branch head adds that the cash-verification pair must be adjacent to the loan-enquiry counter. Does the original unique arrangement already satisfy this, and what should you report?
Read the worked answer
Check it, do not rebuild it. In the unique arrangement the pair is Farhan at counter 1 and Bina at counter 2, and the loan-enquiry counter (Anil) is counter 3.
- Bina at 2 and Anil at 3 are directly adjacent.
- Farhan at 1 is not adjacent to Anil, but the requirement speaks of the pair being adjacent to the loan counter, and an adjacent pair is adjacent if the pair sits next to the counter - which it does, through Bina.
Report: the original arrangement already satisfies the new requirement, so no reshuffle is needed and the signed-off order stays 1 Farhan, 2 Bina, 3 Anil, 4 Esha, 5 Chetan, 6 Divya.
Why this is the trap: candidates often re-solve the whole puzzle on hearing a new condition, lose time, and sometimes "find" a different valid-looking order and paste it over a correct one. The efficient habit is to test the new condition against the arrangement you already have; if it holds, stop. Only if it fails do you go back to the clue set.