Decomposition of a Cube into Nearly Equal Smaller Cubes. Issue 10 (1st December 2017) - Frankl, Peter; Meir, Amram; Pach, János
Abstract: Let d be a fixed positive integer and let ε >0 . It is shown that for every sufficiently large n ≥ n 0( d, ε), the d -dimensional unit cube can be decomposed into exactly n smaller cubes such that the ratio of the side length of the largest cube to the side length of the smallest one is at most 1 + ε. Moreover, for every n ≥ n 0, there is a decomposition with the required properties, using cubes of at most d + 2 different side lengths. If we drop the condition that the side lengths of the cubes must be roughly equal, it is sufficient to use cubes of three different sizes.