Seating Arrangement for a Watch Bill
Five sailors - A, B, C, D, and E - are to be assigned to a five-hour night watch, one per hour, hours numbered 1 to 5. Constraints: A cannot take hour 1. B must take an hour immediately after C. D must take hour 5. E must take an hour earlier than A. The watch officer drafts a roster: hour 1 = C, hour 2 = B, hour 3 = E, hour 4 = A, hour 5 = D. A second sailor proposes swapping A and E. The officer wants to know whether the draft is valid, whether the swap is valid, and whether any other valid roster exists.
1. Test the officer's draft roster against every constraint.
Read the worked answer
Draft: 1=C, 2=B, 3=E, 4=A, 5=D.
- A not hour 1: A is at 4 ✓
- B immediately after C: C at 1, B at 2 ✓
- D at hour 5 ✓
- E earlier than A: E at 3, A at 4 ✓ The draft is valid.
2. Test the proposed swap of A and E, and determine whether any other valid roster exists.
Read the worked answer
Swap gives: 1=C, 2=B, 3=A, 4=E, 5=D.
- A not hour 1 ✓
- B after C ✓
- D at 5 ✓
- E earlier than A: E at 4, A at 3 → E is later than A ✗ Swap is invalid.
Now check for other valid rosters. Fix D=5. Remaining slots 1–4 for A,B,C,E with B right after C, E before A, A≠1. Possible (C,B) adjacent pairs in slots 1–4: (1,2) or (2,3) or (3,4).
- (1,2): remaining slots 3,4 for E,A with E<A → E=3,A=4 ✓ → roster C,B,E,A,D (the draft).
- (2,3): remaining 1,4 for E,A with E<A → E=1,A=4 ✓ (A≠1 ok) → E,C,B,A,D. Valid.
- (3,4): remaining 1,2 for E,A with E<A → E=1,A=2 ✓ → C? wait C=3,B=4, so E,C,B? Recheck: C=3,B=4, E=1,A=2 → E,A,C,B,D. E<A ✓, A≠1 ✓. Valid. Two additional valid rosters exist: E,C,B,A,D and E,A,C,B,D.