Five Workshops: A Constrained Weekday Schedule
Five speakers-Alvarez, Bhat, Chen, Dubois, and Eze-will each give one workshop on a different weekday from Monday to Friday. The organizer has fixed the following conditions: Bhat must speak immediately before Chen. Alvarez cannot speak on Monday or Friday. Dubois must speak after Chen. Eze cannot speak on the day immediately before or immediately after Alvarez. No other restrictions apply. The organizer has not yet assigned the days and asks for every schedule that satisfies the conditions. Each workshop lasts the full day, so no two speakers can share a day. The venue is available all five days. A schedule is a complete Monday-to-Friday ordering of the five speakers. The organizer is particularly concerned about travel costs, so she wants to know which speakers could appear on which days.
1. List all schedules that satisfy the conditions.
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Two schedules work. Reason from the Bhat-Chen block. It cannot be Thursday-Friday because Dubois must come after Chen. If Bhat-Chen are Monday-Tuesday, Alvarez must be Wednesday (Thursday would force Eze next to Alvarez on either Wednesday or Friday, and Friday is barred for Alvarez). Then Dubois Thursday and Eze Friday. If Bhat-Chen are Tuesday-Wednesday, Alvarez must be Thursday, Dubois Friday, and Eze Monday. If Bhat-Chen are Wednesday-Thursday, Alvarez must be Tuesday; then Dubois Friday and Eze Monday, but Eze would be immediately before Alvarez, violating the adjacency rule. So the only schedules are: (1) Mon Bhat, Tue Chen, Wed Alvarez, Thu Dubois, Fri Eze; (2) Mon Eze, Tue Bhat, Wed Chen, Thu Alvarez, Fri Dubois.
2. If Eze speaks on Monday, who speaks on Thursday?
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If Eze is Monday, the second schedule is forced: Mon Eze, Tue Bhat, Wed Chen, Thu Alvarez, Fri Dubois. The first schedule has Eze on Friday, so it is excluded. Thursday is therefore Alvarez.
3. Suppose the organizer adds a condition that Alvarez must speak on Wednesday. What is the complete schedule?
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If Alvarez is Wednesday, the Bhat-Chen block must be Monday-Tuesday: Tuesday-Wednesday would put Chen on Wednesday, and Wednesday-Thursday would leave no valid day for Alvarez. Dubois must follow Chen and cannot take Wednesday, so Dubois is Thursday; Eze takes the remaining Friday. Complete schedule: Mon Bhat, Tue Chen, Wed Alvarez, Thu Dubois, Fri Eze.