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}.

Location: http://mech.math.msu.su/~fpm/eng/k05/k055/k05514h.htm

Last modified: February 26, 2006