Which regular polytopes have their vertices a subset of other regular polytopes in the same dimension (https://cp4space.hatsya.com/2020/10/01/subsumptions-of-regular-polytopes/)? We don't know! The answer is closely connected to the existence of Hadamard matrices (https://en.wikipedia.org/wiki/Hadamard_matrix), which are famously conjectured to exist in dimensions divisible by four. A solution to the Hadamard matrix existence problem would also solve the polytope problem.
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