Five Lectures, Five Days
A college has to schedule five guest lectures - P, Q, R, S, and T - one on each weekday from Monday to Friday, with no two lectures on the same day. The scheduling committee has fixed the following conditions: Q cannot be on Monday or Friday; R must be on the day immediately after P; S must be before Q; T cannot be on a day immediately adjacent to S; and P cannot be on Thursday. The committee has not yet decided the final order. A student claims that 'the schedule is completely unconstrained; any speaker can be on any day.' The coordinator replies that some placements are forced. Before publishing the timetable, the committee must determine which statements are necessarily true and whether a proposed schedule with P on Wednesday and Q on Thursday is valid.
1. Which speaker must be on Friday? Show your reasoning by testing the possible positions of P.
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P can only be Monday, Tuesday, or Wednesday: R must be the next day, and P cannot be Thursday, so P cannot be Friday either (no next day). Test each. If P=Mon, R=Tue; remaining Wed/Thu/Fri for Q,S,T. Q cannot Fri or Mon, so Q=Thu, S=Wed, T=Fri. If P=Tue, R=Wed; remaining Mon/Thu/Fri. Q cannot Mon or Fri, so Q=Thu, S=Mon, T=Fri. If P=Wed, R=Thu; remaining Mon/Tue/Fri. Q cannot Mon or Fri, so Q=Tue, S=Mon, T=Fri. In every valid arrangement T is on Friday. Therefore T is forced.
2. Is the proposed schedule 'P on Wednesday and Q on Thursday' valid? Explain.
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No. If P is Wednesday, the rule that R is immediately after P puts R on Thursday. Thursday is then occupied by R, so Q cannot also be on Thursday. This is confirmed by the full enumeration: when P=Wed, the only valid arrangement is S=Mon, Q=Tue, P=Wed, R=Thu, T=Fri. The proposal violates both the one-lecture-per-day rule and the Q placement that the constraints force.
3. If S is on Monday, what are the possible days for Q?
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With S=Mon, P cannot be Monday because S occupies it, so P is Tuesday or Wednesday. If P=Tue, R=Wed; remaining Thu/Fri for Q and T. Q cannot Friday, so Q=Thu and T=Fri. If P=Wed, R=Thu; remaining Tue/Fri for Q and T. Q cannot Friday, so Q=Tue and T=Fri. Thus Q can be Tuesday or Thursday. The student's 'any day' claim is false.