The Weekend On-Call Rota
A hospital's four weekend on-call slots are Saturday AM, Saturday PM, Sunday AM and Sunday PM. Four registrars - Ines, Joel, Kira and Malik - each take exactly one slot. The rota rules are: (1) Ines does not work on Sunday. (2) Joel must take the slot immediately before Kira's. (3) Malik will work either Saturday AM or Sunday PM. (4) Kira will not work Saturday AM. The clinical director says the rules leave only one possible rota and asks you to confirm it before publication. A colleague objects that the director has misread rule (2), arguing that because Joel must precede Kira and Ines cannot work Sunday, Kira must be on Sunday morning. The rota must be published in two days, and no further constraints have been agreed.
1. Determine whether the four rules determine a unique rota. Show your reasoning and list every valid rota.
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Let the slots be 1 Sat AM, 2 Sat PM, 3 Sun AM, 4 Sun PM. Joel must occupy the slot immediately before Kira, so the possible (Joel, Kira) pairs are (1,2), (2,3) and (3,4). Ines cannot be in 3 or 4.
Pair (1,2): Joel 1, Kira 2. Ines must be in 1 or 2, but both are taken, so no valid rota.
Pair (2,3): Joel 2, Kira 3. Ines must be 1, the only non-Sunday slot left. Malik then takes 4, which satisfies rule 3. Valid: Ines 1, Joel 2, Kira 3, Malik 4.
Pair (3,4): Joel 3, Kira 4. Ines can be 1 or 2. If Ines is 1, Malik is 2, which violates rule 3. If Ines is 2, Malik is 1, which satisfies rule 3. Valid: Malik 1, Ines 2, Joel 3, Kira 4.
There are two valid rotas, not one. The director's claim is false.
2. A colleague argues: 'Joel must precede Kira, and Ines cannot work Sunday, so Kira must be on Sunday morning.' Evaluate the argument.
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The argument is invalid. It treats one arrangement that satisfies the premises as if it were the only one. The second valid rota has Kira on Sunday PM and Joel on Sunday AM, with Ines on Saturday PM. In that rota, Ines is not on Sunday and Joel immediately precedes Kira, so both stated premises hold while the conclusion 'Kira Sunday morning' is false.
The mistake is failing to check all cases, especially the Malik rule. If Kira is Sunday PM and Joel is Sunday AM, Malik can take Saturday AM, which rule 3 allows. If Ines took Saturday AM instead, Malik would be forced into Saturday PM, which rule 3 forbids. So the colleague's conclusion is not forced. Kira must work on Sunday in both valid rotas, but not necessarily Sunday morning. The correct inference is weaker than the one offered.
3. Propose one additional single constraint that makes the rota unique. Show which valid rota it eliminates, and explain why it is a genuine extra constraint rather than a restatement of the existing rules.
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Add: 'Ines works Saturday AM.' The two valid rotas are R1 (Ines Sat AM; Joel Sat PM; Kira Sun AM; Malik Sun PM) and R2 (Malik Sat AM; Ines Sat PM; Joel Sun AM; Kira Sun PM). The new constraint is true in R1 and false in R2, so only R1 remains. It is a genuine extra constraint because neither existing rule fixes Ines to Saturday AM: rule 1 only says she is not on Sunday, and in R2 she is on Saturday PM. It is also a single atomic condition, not a compound such as 'Ines Saturday AM and Malik Sunday PM'.
Other single constraints would work equally well, for example 'Malik works Sunday PM', 'Joel works Saturday PM' or 'Kira works Sunday AM'. Each selects R1 by eliminating R2. A constraint such as 'Ines does not work Sunday' would not work, because it is already rule 1 and both rotas satisfy it.