Computer and Automation Institute, Hungarian Academy of Sciences Lagymanyosi ut 11, H-1111 Budapest, HungaryDepartment of Geometry, Eötvös University, Rakoczi ut 5, H-1088 Budapest, Hungary
Abstract: Let ${\cal C}=\langle C_1,C_2,\dots,C_n\rangle$ be a finite sequence of unit cubes in the $d$-dimensional space. The sequence $\cal C$ is called a facet-to-facet snake if $C_i\cap C_{i+1}$ is a common facet of $C_i$ and $C_{i+1}$, $1\le i\le n-1$, and $\dim(C_i\cap C_j)\le\max\{-1,d+i-j\}$, $1\le i<j\le n$. A facet-to-facet snake of unit cubes is called maximal if it is not a proper subset of another facet-to-facet snake of unit cubes. In this paper we prove that the minimum number of $d$-dimensional unit cubes which can form a maximal facet-to-facet snake is $8d-1$ for all $d\ge 3$.
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