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 number k greater than or equal to c (d) smaller d-cubes. We derive specific bounds For d less than or equal to 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)). (C) 1998 Academic Press, Inc. [References: 7]
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