Maximal facet-to-facet snakes of unit cubes (Q1591757)

Let \(\mathcal C=\langle C_1,C_2,\dots,C_n\rangle\) be a finite sequence of unit cubes in the \(d\)-dimensional space. The sequence \(\mathcal C\) is called a facet-to-facet snake if \(C_i\cap C_{i+1}\) is a common facet of \(C_i\) and \(C_{i+1}\), \(1\leq i\leq n-1\), and \(\dim(C_i\cap C_j)\leq \max\{-1,d+i-j\}\), \(1\leq i<j\leq n\). Note that \(\dim(C_i\cap C_j)=-1\) if and only if \(C_i\cap C_j=\emptyset\). 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 it is proved that the minimum number of \(d\)-dimensional unit cubes which can form a maximal facet-to-facet snake is \(8d-1\) for all \(d\geq 3\).