Duty Roster at a Two-Platform Junction
The station manager of a small junction must post four MTS staff - P, Q, R and S - to cover one day's duties. The morning announcement-and-cleaning duty needs two persons, the evening enquiry duty needs one, and the night parcel-lockup duty needs one; each person does exactly one duty. Four local rules apply: (i) P and Q must not be posted together, because they have an unresolved complaint against each other; (ii) R has a medical certificate against early-morning work, so R cannot take the morning duty; (iii) S cannot take the night duty because the last bus leaves before the lockup closes; (iv) the night duty must not go to Q, who is the only person trained on the new parcel software and is needed in the evening. The manager has already decided that P will take the night duty, and wants the roster pinned up before the shift begins.
1. Complete the roster: who takes the morning, evening and night duties? Show every deduction, and confirm that no rule is broken.
Read the worked answer
- Night is fixed: P. This immediately satisfies rule (iv), since Q is not on night.
- Because P is on night, P cannot be in the morning pair. The morning pair must therefore be two people from Q, R and S.
- Rule (ii) removes R from the morning.
- So the only possible morning pair is Q and S. Rule (i) is safe because P is not in that pair.
- The one person left is R, who takes the evening duty.
Final roster: Morning - Q and S; Evening - R; Night - P.
Rule check: P and Q are in different shifts (i); R is not on morning (ii); S is not on night (iii); Q is on evening, not night (iv). Every rule holds, and this is the only assignment that satisfies all four.
2. On the day, S reports sick. A substitute U is available, but U cannot do the morning duty. Can all four rules still be met with P on night? Explain, and state the smallest change that produces a workable roster.
Read the worked answer
No. The people available are P, Q, R and U. The morning duty needs two of them, but R cannot do mornings (rule ii) and U cannot do mornings. That leaves only P and Q as morning-capable - and rule (i) forbids posting P and Q together.
So the morning pair cannot be filled under the existing rules, whatever is done with the night duty. This is not a shortage of one person; it is that the two remaining morning-capable staff are the two who must be kept apart.
The smallest effective change is to relax exactly one constraint, and the options are not equal:
- Best: call for a second substitute who can work the morning, keeping all safety-related rules intact.
- Acceptable fallback: keep P and Q in the same shift but add a third person or a supervisor to the morning duty, so rule (i) is met in spirit while the shift is covered.
- Worst: send R to the morning against the medical certificate, or put the untrained person on the evening parcel software. The judgement is to change the least costly rule, not the most convenient one.
3. Suppose the manager had not already fixed P on the night duty. Would the roster still be unique? Explain what that tells you about the value of the manager's decision.
Read the worked answer
Without fixing P, the night duty can be held only by P or R, because rule (iii) excludes S and rule (iv) excludes Q.
- If night is P: morning is Q + S, evening R - one solution.
- If night is R: the morning pair must be drawn from P, Q, S without using P and Q together, giving two possibilities - (P + S) with Q on evening, or (Q + S) with P on evening.
So three valid rosters exist, and they are not equivalent in practice: the two with R on night put R alone at the lockup after a full day, while the version with P on night puts the software-trained Q on evening as rule (iv) intended.
The manager's decision does real work - it turns a three-way choice into a single answer and protects the evening parcel work. In reasoning terms, an added condition can be what makes a puzzle determinate.