Number of 2×2 square tiles in a discrete hypercube
Year: 2025
Type: article
Abstract: This note studies the problem of counting 2×2 square tiles contained in discrete hypercubes. A 2D discrete square of side n contains exactly (n-1)^2 tiles, while a 3D cube contains 3n(n-1)^2. We extend this construction to four dimensions, where a 4D hypercube can be seen as a sequence of n discrete n × n × n cubes arranged along a fourth axis. We prove that the number of 2×2 tiles in this discrete 4D object is given by the closed polynomial formula: T(n) = (n-1)^2 (3n^2 + 2n) = 3n^4 - 4n^3 - n^2 + 2n.
Author: ferraro, salvatore
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7077501032
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