Maximal Facet-to-Facet Snakes of Unit Cubes

Abstract. Let C = 〈C1, C2,..., Cn 〉 be a finite sequence of unit cubes in the d-dimensional space. The sequence C is called a facet-to-facet snake if Ci ∩ Ci+1 is a common facet of Ci and Ci+1, 1 ≤ i ≤ n−1, and dim(Ci∩Cj) ≤ max{−1, d+i−j}, 1 ≤ i < j ≤ 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 ≥ 3. 1.