Dissecting d-Cubes into Smaller d-Cubes

Author: Hudelson M.

Source: Journal of Combinatorial Theory, Series A, Volume 81, Number 2, February 1998 , pp. 190-200(11)

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Abstract:

In this paper, we explore the following question: Given integers d and k, is it possible to subdivide a d-dimensional cube into k smaller d-dimensional cubes? In particular, we investigate bounds on the integer c(d) which is the smallest integer for which it is possible to subdivide the d-cube into any numberk ges c(d) smaller d-cubes. We derive specific bounds for d les 5, and furthermore, we investigate, for given k, the asymptotic behavior of c(d) for those d such that gcd(2^d-1, k^d-1)=1. Specifically, we show that if gcd (2^d-1, 3^d-1) then c(d)<6^d and that if gcd(2^d-1, k^d-1) then c(d)=O((2k)^d). Finally, we derive the general asymptotic bound c(d)=O((2d)^d-1) which improves the currently known bound of c(d)=O((2d)^d). Copyright 1998 Academic Press.