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A recreational problem from nearly two centuries ago has featured prominentlyin recent times in the mathematics of designs, codes, and signal processing.The number 15 that is central to the problem coincidentally features in areasof physics, especially in today's field of quantum information, as the numberof basic operators of two quantum spins ("qubits"). This affords a 1:1correspondence that we exploit to use the well-known Pauli spin or Lie-Cliffordalgebra of those fifteen operators to provide specific constructions as posedin the recreational problem. An algorithm is set up that, working with fourbasic objects, generates alternative solutions or designs. The choice of fourbase colors or four basic chords can thus lead to color diagrams or acousticpatterns that correspond to realizations of each design. The Fano Plane offinite projective geometry involving seven points and lines and the tetrahedralthree-dimensional simplex of 15 points are key objects that feature in thisstudy.

Placing Kirkman's Schoolgirls and Quantum Spin Pairs on the Fano Plane: A Rainbow of Four Primary Colors, A Harmony of Fifteen Tones