Six-Day Gate Duty Roster
A school has six cadets - Aman, Bhavya, Chetan, Divya, Esha and Farhan - who must take gate duty on six consecutive days from Monday to Saturday, one cadet per day. The rules are:
- Chetan's duty is immediately after Divya's.
- Aman is neither on Monday nor on Saturday.
- Esha is exactly two days after Bhavya.
- Farhan is on Wednesday.
- Divya is not on Tuesday. The house master wants a complete roster and also wants to know whether every rule is actually needed. A visitor claims the roster cannot be solved without rule 5. Test that claim rather than accepting it.
1. Determine the complete Monday-to-Saturday duty schedule. Show how you eliminate at least two possible placements.
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Farhan is fixed on Wednesday. Chetan must follow Divya on consecutive days, so the possible pairs are (Mon, Tue), (Tue, Wed), (Thu, Fri) and (Fri, Sat); (Wed, Thu) is impossible because Wednesday is Farhan.
Test (Tue, Wed): Chetan would be on Wednesday, but Farhan is there. Reject. Test (Thu, Fri): remaining days for Aman, Bhavya and Esha are Mon, Tue, Sat. Esha must be two days after Bhavya, which needs either Tue→Thu or Thu→Sat. Thu is occupied by Divya, so both fail. Reject. Test (Fri, Sat): remaining days are Mon, Tue, Thu. The only Bhavya→Esha fit is Tue→Thu. That leaves Monday for Aman, but Aman cannot be on Monday. Reject.
Only (Mon, Tue) works: Divya Monday, Chetan Tuesday, Farhan Wednesday. Remaining days Thu, Fri, Sat. Bhavya→Esha must be Thu→Sat, leaving Aman on Friday. Final roster: Mon Divya, Tue Chetan, Wed Farhan, Thu Bhavya, Fri Aman, Sat Esha.
2. Is rule 5 (Divya is not on Tuesday) necessary? Justify by testing what happens if it is removed.
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Rule 5 is not necessary. Without it, the possible Divya-Chetan pairs are still (Mon, Tue), (Tue, Wed), (Thu, Fri) and (Fri, Sat). The pair (Tue, Wed) is already impossible because Farhan occupies Wednesday. The pairs (Thu, Fri) and (Fri, Sat) fail for the reasons in the main solution. The only surviving pair is (Mon, Tue). Therefore rule 5 adds no new restriction; the puzzle has the same unique solution without it. The visitor's claim is false.
3. If Farhan and Aman swap days, which rules remain true and which become false?
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Original: Farhan Wednesday, Aman Friday. After swapping: Aman Wednesday, Farhan Friday.
Rule 1: Divya Monday, Chetan Tuesday - still true. Rule 2: Aman is on Wednesday, which is neither Monday nor Saturday - still true. Rule 3: Bhavya Thursday, Esha Saturday - still true. Rule 4: Farhan is on Wednesday - false, because Farhan is now on Friday. Rule 5: Divya is not on Tuesday - still true.
Only rule 4 breaks.