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Journal of Geometry

, 109:19 | Cite as

Decomposing cubes into smaller cubes

  • Peter Connor
  • Phillip Marmorino
Article
  • 41 Downloads

Abstract

We explore the decomposition of n-dimensional cubes into smaller n-dimensional cubes. Let c(n) be the smallest integer such that if \(k\ge c(n)\) then there is a decomposition of the n-dimensional cube into k smaller n-dimensional cubes. We prove that \(c(n)\ge 2^{n+1}-1\) for \(n\ge 3\), improving on Hadwiger’s result that \(c(n)\ge 2^n+2^{n-1}\). We also show \(c(n)\le e^2n^n\) if \(n+1\) is not prime and \(c(n)\le 1.8n^{n+1}\) if \(n+1\) is prime, improving on upper bounds proven by Erdös, Hudelson, and Meier.

Keywords

Tiling Partitions Hypercubes Cube Decomposition 

Mathematics Subject Classification

Primary 51M04 Secondary 11D45 11H31 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndiana University South BendSouth BendUSA

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