Five-day radiographer roster: deduction under constraints
The radiography department must publish its weekly duty roster. Five radiographers - Meera, Naveen, Omar, Priya and Ravi - will each cover exactly one full day from Monday to Friday. The department works to the following rules. Priya must cover Wednesday because she is the only radiographer trained on the mammography unit that day. Ravi must work the day immediately after Priya so that the two can hand over the quality-control log. Meera must work on a day before Naveen. Naveen cannot work on Monday because he attends a postgraduate class that morning. Omar must cover Friday, when the department receives its contrast stock. The head of department also wants a contingency arrangement in case Omar has to swap with Meera at short notice, and she wants the reasoning written down so that the roster can be defended if it is challenged. The department operates 08:00–16:00, and each radiographer also has a separate administrative duty that week, but the administrative duties are outside this puzzle.
1. Determine the only roster that satisfies all five conditions, showing each deduction.
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Deduction 1. Priya is fixed on Wednesday.
Deduction 2. Ravi works immediately after Priya, so Ravi is Thursday.
Deduction 3. Omar is fixed on Friday.
That leaves Monday and Tuesday for Meera and Naveen.
Deduction 4. Naveen cannot work Monday, so Naveen is Tuesday and Meera is Monday.
Deduction 5 - check the ordering rule. Meera (Monday) is before Naveen (Tuesday), so the rule is satisfied.
Roster: Monday - Meera; Tuesday - Naveen; Wednesday - Priya; Thursday - Ravi; Friday - Omar.
There is no alternative. The fixed placements leave only two slots, and the Monday prohibition on Naveen resolves them uniquely. This is why the roster can be defended: it is not a preference, it is the only arrangement consistent with the stated rules.
2. Omar and Meera swap days. Is the resulting roster still valid? Give your reasoning.
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After the swap: Monday - Omar; Tuesday - Naveen; Wednesday - Priya; Thursday - Ravi; Friday - Meera.
Check each rule:
- Priya on Wednesday: satisfied.
- Ravi immediately after Priya: satisfied, Thursday follows Wednesday.
- Omar on Friday: violated, because Omar is now Monday and Meera is Friday.
- Naveen not on Monday: satisfied.
- Meera before Naveen: violated, because Meera is Friday and Naveen is Tuesday.
So the swap breaks two rules, not one. A roster change must always be re-tested against every constraint rather than only the constraint that prompted the change. If the swap is genuinely necessary, the department needs a different contingency - for example, moving the contrast stock acceptance to another named person - rather than simply exchanging the two names.
3. All contrast studies are supervised by a radiologist. Some emergency studies are contrast studies. What follows, and what does not?
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What follows: Some emergency studies are supervised by a radiologist.
The reasoning is a standard syllogism. The first premise places every contrast study inside the set of radiologist-supervised studies. The second premise says that at least one emergency study lies inside the contrast set. Whatever lies inside the contrast set must also lie inside the larger supervised set, so at least one emergency study is supervised.
What does not follow:
- That all emergency studies are supervised. The premises say only that some are contrast studies; the rest may be plain films read without a radiologist present.
- That some emergency studies are not supervised. Nothing in the premises asserts the existence of an unsupervised emergency study.
- That all supervised studies are contrast studies. That reverses the first premise.
In a departmental argument, this distinction matters: a rule about contrast studies cannot be used to justify a claim about all emergency work.