FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2005, VOLUME 11, NUMBER 5, PAGES 187-196

Random packings by cubes

A. P. Poyarkov

Abstract

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Y. Itoh's problem on random integral packings of the $d$-dimensional $(4\; \times \; 4)$-cube by $(2\; \times \; 2)$-cubes is formulated as follows: $(2\; \times \; 2)$-cubes come to the cube $K$₄ sequentially and randomly until it is possible by the following way: no $(2\; \times \; 2)$-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 $(2\; \times \; 2)$-cubes in a random saturated packing of the $(4\; \times \; 4)$-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 (3/2)^d.

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Last modified: February 26, 2006