Science Fair Volunteer Assignment With a Late Parent Request
A science fair coordinator must assign four volunteers-Asha, Bilal, Chitra and Dev-to four stations: Registration, Physics demo, Chemistry demo and Crowd control. Each volunteer takes exactly one station, and each station gets exactly one volunteer. The constraints are: Asha cannot be at the Chemistry demo; Bilal must be at either Registration or Crowd control; Chitra must be at the Physics demo if Dev is at Crowd control; Dev cannot be at Registration; and if Asha is at Registration, Bilal must be at Crowd control. The coordinator also has a late request: Dev’s parent wants him not to be at the Chemistry demo because of a chemical sensitivity, but this request has not yet been approved. The printed programme must go out tomorrow, and the coordinator wants to know whether the five base constraints already settle the assignment or whether a further decision is needed.
1. How many assignments satisfy the five base constraints? Show the enumeration logic.
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Use Bilal’s two allowed stations as the cases.
Case 1: Bilal at Registration. Registration is occupied, so the “if Asha is at Registration” clue is not triggered. Remaining stations for Asha, Chitra and Dev are Physics, Chemistry and Crowd control. Dev can be at any of these because he is not at Registration.
- Dev at Physics: Asha must avoid Chemistry, so Asha = Crowd control, Chitra = Chemistry. Valid.
- Dev at Chemistry: Asha and Chitra fill Physics and Crowd control. Asha = Physics, Chitra = Crowd control is valid; Asha = Crowd control, Chitra = Physics is also valid.
- Dev at Crowd control: then Chitra must be at Physics, leaving Asha at Chemistry, which violates Asha’s restriction. Invalid.
That gives 3 assignments.
Case 2: Bilal at Crowd control. Remaining stations for Asha, Chitra and Dev are Registration, Physics and Chemistry. Dev cannot be at Registration.
- Asha at Registration: Chitra and Dev fill Physics and Chemistry. Dev at Physics, Chitra at Chemistry is valid; Dev at Chemistry, Chitra at Physics is valid.
- Asha at Physics: then Dev cannot be at Registration, so Dev = Chemistry and Chitra = Registration. Valid.
- Asha at Chemistry is forbidden.
That gives 3 more assignments.
Total: 6 valid assignments.
2. If Dev’s request not to be at the Chemistry demo is approved, which assignments survive? Does that settle the programme?
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Remove every assignment with Dev at Chemistry from the six. The survivors are:
- Asha = Crowd control, Bilal = Registration, Chitra = Chemistry, Dev = Physics.
- Asha = Registration, Bilal = Crowd control, Chitra = Chemistry, Dev = Physics.
So the approved request reduces the field from six assignments to two, not one. The programme is still not uniquely determined. The coordinator must either make a defensible choice between the two or obtain one more constraint. Publishing either one without acknowledging the remaining choice would be a judgement call, not a deduction from the stated rules.
3. Propose one additional constraint that would make the assignment unique, and justify it in terms of the fair’s operational needs.
Read the worked answer
One workable additional constraint is: Asha must be at Crowd control. Under the approved medical restriction, only two assignments survive, and this constraint selects the first one: Asha = Crowd control, Bilal = Registration, Chitra = Chemistry, Dev = Physics.
The justification should be operational rather than arbitrary. Crowd control needs someone who can be stationed at the entrance and manage queues; if Asha has that brief, the assignment becomes defensible and auditable. The coordinator should record the added constraint and its reason, because otherwise the assignment looks like a preference rather than a rule. A different valid constraint-for example, “Bilal must be at Registration”-would select a different single assignment, so the coordinator should choose the constraint that matches the actual task requirements, not merely the one that produces a convenient answer.