{"id":"palomar/PALOMAR-2026-11-01-000019","url":"https://github.com/jasondavies/macmahon-min-universal-sets/tree/d8856c97e43e6fa66383c81b8ac4f5bd1c285bc8","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-11-01-000019-v1.json","repo":"https://github.com/jasondavies/macmahon-min-universal-sets","tags":["math.CO","cs.LO","msc:05A15","msc:05C70","msc:68V15","msc:00A08"],"score":0,"title":"The ten minimum universal sets of MacMahon cubes","trust":"high","domain":"math.CO","source":"palomar","authors":["Jason Davies"],"reviews":0,"subject":"math.CO","summary":"A universal inventory of six-coloured MacMahon cubes can build a 2 by 2 by 2 model of every target variety, using eight distinct physical copies, without requiring internal face colours to match. We prove that the minimum inventory size is twelve and classify all ten labelled inventories attaining it, even when repetitions are allowed. They are exactly the two-triangle sets in Conway's tableau. The proof derives row and column sums of two from usability inequalities, proves symmetry, and eliminates every other degree-two inventory by corner coverage. Finite kernel-checked witnesses establish…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-11-01","reactions":{},"formalized":true}