The Menage Problem: Seating, Graphs, and Knots

We unravel the classic combinatorics puzzle: seat n couples around a circle with alternating genders and no partners beside each other. Starting with the familiar 3 couples (12 valid arrangements) and 4 couples (96), we climb to 5 couples (3,120) and 6 couples (115,200), and see why the numbers explode. Along the way we meet the OEIS sequences that codify these counts (A059375, A000179) and the menage hit polynomials (A00033) that hide the patterns via generating functions. The ride then dives into graph theory with crown graphs and Hamiltonian cycles, and even lands in knot theory through Dowker notation—revealing a surprising bridge between seating plans and alternating knot diagrams. We’ll also peek at the inclusion-exclusion backbone, variations like letting one couple sit together, and why shifting the entire table matters in the math. This episode shows how a dinner party problem opens doors to deep ideas, practical problem-solving, and unexpected connections across combinatorics, graph theory, and topology.

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