Scheduling a Mixed-Level Studio Showcase
A community studio has 38 dancers aged 9-17 split across three levels: 14 beginners, 15 intermediates, and 9 advanced. The annual showcase runs 95 minutes including one 10-minute intermission, and the venue requires a hard stop. Each beginner piece runs 3 minutes, intermediate pieces 4 minutes, and advanced pieces 5 minutes. The director wants every dancer on stage at least twice, wants the show to open and close with advanced numbers, and has been told by the stage manager that changeovers between levels cost 2 minutes each while same-level transitions cost 1 minute. Two advanced dancers are also in a guest duet that must go immediately before intermission. The director is considering cutting one intermediate piece to make room.
1. Using the timing constraints, determine whether the current plan can fit within the 95-minute limit if the studio programs 5 beginner, 5 intermediate, and 4 advanced pieces plus the duet. Show your arithmetic and identify the binding constraint.
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Performance time: beginners 5 x 3 = 15; intermediates 5 x 4 = 20; advanced 4 x 5 = 20; duet 2. Total performance = 57 minutes. Available stage time = 95 - 10 intermission = 85 minutes. That leaves 85 - 57 = 28 minutes for transitions. With roughly 15 pieces plus duet (16 items) there are 15 gaps. If even half are level changes at 2 minutes, transitions run about 22-24 minutes, which fits but only barely. The binding constraint is transitions, not performance time. The director should map the running order so same-level pieces cluster, converting 2-minute changes into 1-minute transitions, and reserve the duet slot directly before intermission. Cutting an intermediate piece frees 4 minutes of stage time plus at least one transition, giving comfortable margin.
2. The director wants every dancer on stage at least twice. Evaluate whether the proposed piece counts achieve that for each level, and recommend the minimum change.
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Beginners: 14 dancers across 5 pieces means average cast of 2.8 per piece; achievable if pieces are ensemble-based, but a director must verify no beginner appears only once. Intermediates: 15 dancers across 5 pieces, average 3 per piece, workable but tight if any piece is a solo or duet. Advanced: 9 dancers across 4 pieces plus a 2-person duet. The duet counts as a stage appearance for those two, but the other 7 advanced dancers need two appearances from the 4 advanced pieces, meaning casts must overlap heavily. The minimum change is not adding pieces but re-casting: fold the duet dancers into at least one advanced ensemble piece and ensure every intermediate appears in two of the five intermediate works. If a piece is a solo, it consumes a slot without helping the twice-on-stage goal, so convert it to a small-group work.
3. A parent complains that the beginner dancers get less stage time than advanced dancers. How should the director reason about fairness given the constraints?
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Stage time per dancer is the relevant metric, not total minutes per level. Beginners: 15 minutes of beginner repertoire spread over 14 dancers. Advanced: 20 minutes plus duet spread over 9 dancers. Per-dancer, advanced dancers are on stage far longer, which is expected developmentally but should be explained. Fairness here means every dancer has meaningful stage presence, not equal minutes. The director can point to the twice-on-stage guarantee, note that beginner pieces are ensemble works where each dancer is visible throughout, and explain that advanced dancers carry longer, more demanding roles. If equity is still a concern, the director could add a combined finale where all 38 dancers appear together, which costs 3-4 minutes but strengthens the community message and satisfies the closing-number requirement simultaneously.