Four-Club Timetable Puzzle
A school has four clubs: Science, Drama, Debate, and Art. Each club must meet on exactly one day from Monday to Thursday, and no two clubs may meet on the same day. The school imposes four rules. First, Science must meet on an earlier day than Drama. Second, Art must meet on the day immediately after Science. Third, Debate cannot meet on Monday or Thursday. Fourth, Drama cannot meet on Monday. The student council must publish a timetable that satisfies all four rules. A new member asks whether the timetable is unique and what would happen if the Art club could not meet on Tuesday because its room is being painted. The council also wants to know which rules would still be needed if the head teacher fixed Science on Monday.
1. Find the unique timetable. Explain each deduction.
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Rule 2 restricts Science and Art to three pairs: (Mon, Tue), (Tue, Wed), or (Wed, Thu).
- (Tue, Wed): Debate and Drama would take Mon and Thu. Rule 3 forbids Debate on either day, so this is impossible.
- (Wed, Thu): Debate and Drama take Mon and Tue. Debate cannot be Mon, so Debate = Tue and Drama = Mon. But rule 4 forbids Drama on Monday. Impossible.
- (Mon, Tue): Debate and Drama take Wed and Thu. Debate cannot be Thu, so Debate = Wed and Drama = Thu. Rule 4 is satisfied because Drama is not Monday. Therefore the only timetable is: Monday Science, Tuesday Art, Wednesday Debate, Thursday Drama.
2. If Art cannot meet on Tuesday, does a valid timetable still exist? Explain.
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No. If Art cannot be Tuesday, the (Mon, Tue) Science–Art pair is impossible. The (Tue, Wed) pair is also impossible because it leaves Monday and Thursday for Debate and Drama, and Debate can meet on neither. The (Wed, Thu) pair leaves Monday and Tuesday for Debate and Drama; Debate must then be Tuesday and Drama Monday, which rule 4 forbids. Every possible Science–Art pair fails, so no valid timetable exists under the added restriction.
3. If the head teacher fixes Science on Monday, which of the four rules become unnecessary for producing the same unique timetable? Justify.
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With Science fixed on Monday, rule 2 (Art immediately after Science) still places Art on Tuesday. The remaining days are Wednesday and Thursday for Debate and Drama; rule 3 (Debate not Monday or Thursday) still forces Debate to Wednesday and Drama to Thursday. Rule 1 (Science before Drama) is automatically true because Monday is before Thursday. Rule 4 (Drama not Monday) is also automatically true because Drama is on Thursday. So rules 1 and 4 become unnecessary; rules 2 and 3 are still needed.