Random packings by cubes

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 K ₄ sequentially and randomly until it is possible in the following way: no (2 × 2)-cubes overlap, and all their centers are integer points in K ₄. 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. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 187–196, 2005.