Ducci Sequences in Higher Dimensions

A Ducci sequence is a sequence of n-tuples of integers obtained by iterating the map (a1, . . . , an) !→ (|a1 − a2|, |a2 − a3|, . . . , |an − a1|). In this paper, we generalize the concept of Ducci sequences to sequences of d-dimensional arrays, extend some of the basic results on Ducci sequences to this case, and point out some new phenomena that occur. Our main tool is the description of Ducci sequences modulo two in terms of polynomials over the field F2.