Abstract:
Y. Itoh's problem on random integral packings of the $d$-dimensional $(4\times4)$-cube by $(2\times2)$-cubes is formulated as follows: $(2\times2)$-cubes come to the cube $K_4$ sequentially and randomly until it is possible by the following way: no $(2\times2)$-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 $(2\times2)$-cubes in a random saturated packing of the $(4\times4)$-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: $M \ge (3/2)^d$.
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