Watch Roster Constraints
A duty sergeant must assign four soldiers-Pte Alia, Cpl Ben, Pte Chen and Sgt Devi-to four consecutive watch periods: 06:00-12:00 (shift 1), 12:00-18:00 (shift 2), 18:00-24:00 (shift 3) and 00:00-06:00 (shift 4). Each soldier takes exactly one shift and each shift has exactly one soldier. The standing orders impose five rules: (1) Sgt Devi must take shift 1 or shift 4; (2) Cpl Ben must be immediately before Pte Chen, meaning Ben's shift ends when Chen's begins; (3) Pte Alia must not take shift 4; (4) Pte Chen must not take shift 3; (5) if Sgt Devi takes shift 1, then Pte Alia must take shift 3. A proposed roster lists Alia on shift 1, Ben on shift 2, Chen on shift 3 and Devi on shift 4. The sergeant must determine whether any valid roster exists, identify every rule the proposal breaks, and advise on a qualification rule: only soldiers who are sergeant-qualified may take shift 4.
1. Determine the only roster that satisfies all five rules. Show the elimination reasoning.
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Use the Ben-Chen pair from rule (2). The possible adjacent pairs are (1,2), (2,3) and (3,4).
- Pair (2,3): Chen would be on shift 3, which breaks rule (4). Reject.
- Pair (3,4): Chen is on shift 4. That leaves shifts 1 and 2 for Alia and Devi. Devi must take shift 1 or 4, and shift 4 is taken by Chen, so Devi must take shift 1. Rule (5) then requires Alia on shift 3, but shift 3 is already Ben's and Alia would be left with shift 2. Contradiction. Reject.
- Pair (1,2): Ben is shift 1 and Chen is shift 2. Remaining shifts are 3 and 4 for Alia and Devi. Devi must take shift 1 or 4, and shift 1 is taken, so Devi takes shift 4. Alia takes shift 3. Rule (3) is satisfied because Alia is not on shift 4, and rule (5) is not triggered because Devi is not on shift 1.
Only roster: Ben shift 1, Chen shift 2, Alia shift 3, Devi shift 4.
2. The proposed roster is Alia on shift 1, Ben on shift 2, Chen on shift 3, Devi on shift 4. Identify every rule it breaks.
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Check each rule in turn:
- Rule (1): Devi is on shift 4. Satisfied.
- Rule (2): Ben is on shift 2 and Chen on shift 3. Shift 2 ends at 18:00 and shift 3 begins at 18:00, so Ben is immediately before Chen. Satisfied.
- Rule (3): Alia is on shift 1, not shift 4. Satisfied.
- Rule (4): Chen is on shift 3. This is expressly forbidden. Broken.
- Rule (5): Devi is not on shift 1, so the condition is not triggered. Satisfied.
The proposal breaks only rule (4). It is therefore close to valid but cannot be used as written; Chen must be moved off shift 3, which forces a different roster altogether.
3. Evaluate these conclusions: (a) 'Only sergeant-qualified soldiers may take shift 4. Sgt Devi is on shift 4, so Devi is sergeant-qualified.' (b) 'Pte Alia is not sergeant-qualified, so Alia cannot take shift 4.' (c) 'All sergeants are qualified for shift 4.'
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(a) Valid. 'Only sergeant-qualified soldiers may take shift 4' means: if a soldier takes shift 4, that soldier is sergeant-qualified. Devi takes shift 4, so Devi is sergeant-qualified.
(b) Valid. The same rule gives the contrapositive: if a soldier is not sergeant-qualified, that soldier cannot take shift 4. Alia is not sergeant-qualified, so she cannot take shift 4.
(c) Not valid. The rule restricts who may take shift 4; it does not say every sergeant is qualified for that shift. A sergeant could be unqualified for night duty for medical or other reasons. The conclusion reverses the direction of the rule.