ISSN 2590-9770 The Art of Discrete and Applied Mathematics 3 (2020) #P2.09 https://doi.org/10.26493/2590-9770.1285.fd8 (Also available at http://adam-journal.eu) Sphere decompositions of hypercubes * Richard H. Hammack † Virginia Commonwealth University, Dept. of Mathematics, Richmond, VA 23284, USA Paul C. Kainen Georgetown University, Dept. of Mathematics and Statistics, Washington, DC 20057, USA Received 28 December 2018, accepted 28 August 2019, published online 23 August 2020 Abstract For d ≡ 1 or 3 (mod 6), the 2-skeleton of the d-dimensional hypercube is decomposed into the union of pairwise face-disjoint isomorphic 2-complexes, each a topological sphere. If d =5 n , then such a decomposition can be achieved, but with non-isomorphic spheres. Keywords: Face-disjoint union of spheres, combinatorial design, 2-skeleton of a cube. Math. Subj. Class. (2020): 57M20, 57M15, 05C45 By Euler’s theorem [9, Prop. 1.2.27], any graph (1-complex) with all vertices of even degrees is an edge-disjoint union of cycles. We say a 2-complex is even if every edge lies in a positive even number of (2-dimensional) faces. Is every even 2-complex a face-disjoint union of “2-dimensional cycles”? (A 2-complex X is a face-disjoint union of 2-complexes X 1 ,...,X n if X =  n i=1 X i and each face of X is a face of exactly one X i .) There are (at least) two natural choices for a 2-dimensional interpretation of cycle – sphere or manifold. As even complexes include surfaces like the torus, one cannot always decompose them into face-disjoint spheres. But we show below that sphere decompositions do exist in more than two-thirds of the odd-dimensional hypercubes. For d ≡ 1 or 3 (mod 6), we can decompose the 2-skeleton Q 2 d of the d-dimensional hypercube Q d into face-disjoint copies of ∂Q 3 , the boundary of a 3-cube. That is, Q 2 d is factored by ∂Q 3 . In [6], when d is odd (so the 2-skeleton is even), Q 2 d is decomposed into a face-disjoint union of tori and 3-cube boundaries. In [4] we showed that the 2-skeleton of any d- dimensional Platonic polytope is a face-disjoint union of surfaces if the 2-skeleton is even. Except for the hypercubes, all such decompositions were decompositions into spheres. (A polytope is Platonic if it is maximally symmetric. In dimension greater than four, the Platonic polytopes are just the cubes, simplexes, and hyperoctahedra.) For which odd d is the 2-skeleton of the d-cube decomposable into spheres? For which d can the decomposition be a factorization? We address these questions below. * We thank the referees for feedback that has improved the paper. † Supported by Simons Foundation Collaboration Grant for Mathematicians 523748. E-mail addresses: rhammack@vcu.edu (Richard H. Hammack), kainen@georgetown.edu (Paul C. Kainen) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/