Random packings by cubes

Abstract

Y. Itoh’s problem on random integral packings of the d-dimensional (4 × 4)-cube by (2 × 2)-cubes is formulated as follows: (2 × 2)-cubes come to the cube K4 sequentially and randomly until it is possible in the following way: no (2 × 2)-cubes overlap, and all their centers are integer points in K4. Further, all admissible positions at every step are equiprobable. This process continues until the packing becomes saturated. Find the mean number M of (2 × 2)-cubes in a random saturated packing of the (4 × 4)-cube. This paper provides the proof of the first nontrivial exponential bound of the mean number of cubes in a saturated packing in Itoh’s problem: M ≥ (3/2)d.