The problem of covering an n-dimensional \( q \times q \times \cdots \times q \) torus with n-dimensional \( p \times p \times \cdots \times p \) grid graphs is studied. This is the dual problem of a packing problem concerning the capacity of a graph, which has been studied in information theory. It is related to several other problems as well, including weighted coverings, Keller’s cube-tiling problem, and the recreational problem of how to obtain zero correct predictions in the football pools. Motivated by the last problem, bounds on the minimum size of such coverings are tabulated for q = 3, p = 2, and small n.
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