Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 37 (1996), No. 1, 51-65.
On-Line q-adic Covering by the Method of the n-th Segment
and its Application to On-Line Covering by Cubes
Janusz Januszewski, Marek Lassak, G\"unter Rote, Gerhard Woeginger
Abstract.
We prove that in Euclidean $d$-space every sequence of cubes with
total volume $2^d+3$ is able to cover on-line the unit cube.
The proof is based on an on-line $q$-adic method of covering the unit
segment by segments of lengths of the form $q^{-r}$, where $q\ge2$ and
$r\ge1$ are integers.
The fact that this method is $q$-adic means that every segment has to
be placed in such a way that both end-points are at points that are
multiples of the length of the segment.
MSC 1991: 52C17, 68Q20