Weekend Cover Rota
The Riverside Clinic must cover four half-day slots this weekend: Saturday morning, Saturday afternoon, Sunday morning, and Sunday afternoon. Five staff are available: Aisha, Ben, Chloe, Dev and Elena. Each person can work at most one slot, and exactly four of the five will be used. Aisha cannot work on Sunday. Ben must take at least one morning slot. Elena can work afternoons only. Chloe and Dev must not work on the same day as each other. The clinic manager has already rejected one draft rota because it left Sunday afternoon uncovered, and wants a rota that satisfies every rule rather than a best-fit compromise.
1. Produce one rota that fills all four slots without breaking any rule, and show how you checked it.
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One valid rota is:
| Slot | Staff |
|---|---|
| Sat AM | Aisha |
| Sat PM | Chloe |
| Sun AM | Ben |
| Sun PM | Elena |
Checks: four distinct people, each working once; Aisha works Saturday only; Ben holds a morning slot; Elena works an afternoon; Chloe and Dev are not both rostered, so the same-day rule cannot be breached. Sunday afternoon is covered, which was the failure in the rejected draft. Because the rules only constrain pairs of slots, many rotas exist; this one is valid, not unique.
2. Which member of staff must appear in every valid rota? Justify your answer rather than testing rotas at random.
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Ben. The only rule that forces a named person is 'Ben must take at least one morning slot', so omitting Ben makes the rota invalid immediately. By contrast the others can be rested. A rota of Sat AM Aisha, Sat PM Dev, Sun AM Ben, Sun PM Chloe omits Elena. A rota of Sat AM Ben, Sat PM Chloe, Sun AM Dev, Sun PM Elena omits Aisha. Chloe or Dev can each be omitted because they never both need to work. Hence only Ben is indispensable, and that follows from a rule, not from checking cases.
3. Elena becomes unavailable. Either give a compliant rota or explain why none exists.
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A rota is still possible because Elena's rule restricts her to afternoons; it does not require her. Example: Sat AM Aisha, Sat PM Dev, Sun AM Ben, Sun PM Chloe. Four distinct people; Aisha is Saturday-only; Ben has a morning slot; Chloe and Dev work on different days. The binding constraints are Ben's morning slot and Aisha's Sunday ban, not Elena's availability. Losing Elena reduces the options, because Sunday afternoon must then fall to Chloe or Dev, but it does not make the problem infeasible.