Scheduling Four Tutoring Sessions
You are helping a student council schedule four peer-tutoring sessions during the week before exams. The sessions are Math, Reading, Writing, and Science. They will meet one per day, Monday through Thursday, in four different rooms: A, B, C, and D. The constraints are: Science is on Thursday in Room C. Math is not on Monday and is not in Room A. Writing is scheduled the day immediately before Reading. Reading is not in Room B. Room D is used on Tuesday. No room is used twice. A council member claims that Reading must be on Tuesday, but another member is not convinced. The council needs a complete schedule and a clear explanation of whether the claim is forced by the constraints.
1. Determine the complete schedule, showing how each constraint forces a placement. Is the claim that Reading must be on Tuesday correct?
Read the worked answer
Yes, Reading must be on Tuesday. Science is fixed on Thursday in Room C, so the Writing-before-Reading pair cannot be Wednesday-Thursday. It also cannot be Tuesday-Wednesday: if Writing were Tuesday, it would be in Room D because Room D is used Tuesday; then Reading would be Wednesday, leaving Math on Monday, but Math cannot be Monday. That leaves only Monday-Tuesday for Writing-Reading. So Writing is Monday and Reading is Tuesday. Tuesday uses Room D, so Reading is in Room D. Science is Thursday in C. Math must then be Wednesday. The remaining rooms for Monday and Wednesday are A and B. Math cannot be in A, so Math is Wednesday in B, and Writing is Monday in A. The complete schedule is Monday Writing in A, Tuesday Reading in D, Wednesday Math in B, Thursday Science in C.
2. Suppose the constraint that Math is not in Room A is removed. Does the complete schedule remain unique? Explain.
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No, the schedule would no longer be unique, but Reading would still be on Tuesday. Without the Math-not-in-A rule, the day pattern is still forced: Science is Thursday, and Writing-Reading must be Monday-Tuesday because Tuesday-Wednesday would force Math onto Monday. Reading is still Tuesday in Room D. The only change is that the remaining rooms A and B can be assigned to Writing and Math in two ways: Writing in A and Math in B, or Writing in B and Math in A. Both satisfy all remaining constraints. So removing that rule creates two complete schedules. This shows the difference between a forced conclusion about a day and a forced conclusion about a room.
3. Is the constraint 'Reading is not in Room B' needed to prove that Reading is on Tuesday? Explain.
Read the worked answer
No, it is not needed for that conclusion. The day of Reading is forced by the other constraints: Science occupies Thursday, Room D is used Tuesday, and the Writing-before-Reading pair cannot be Tuesday-Wednesday because that would put Math on Monday, which is forbidden. Therefore Writing is Monday and Reading is Tuesday. Since Tuesday's room is D, Reading is in Room D, which is automatically not Room B. The rule 'Reading is not in Room B' is consistent with the schedule but redundant for proving the Tuesday claim. A good explanation should separate redundant constraints from decisive ones rather than assuming every listed rule is necessary.