The ten minimum universal sets of MacMahon cubes
A collection of six-coloured MacMahon cubes is universal if eight of its cubes can be selected to build a $2\times2\times2$ model of any target variety, with matching exterior colours and no internal matching requirement. We prove that exactly ten labelled collections attain the known minimum size of twelve, even when repeated varieties are allowed, resolving the conjectures of Johnson and Roldán. Usable-cube inequalities force a symmetric multiplicity matrix with row and column sums two; corner coverage then forces two disjoint triangles in Conway's tableau. The written proof is elementary…
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