Recent posts reminded me of this question first suggested to me by my late brother Simon (1955-2005). Given an integer n > 0, what are the finite patterns that can partition the discrete circle C_n = Z/nZ ??? A finite pattern is simply a finite subset [X] of C_n = Z/nZ *up to isometry* I.e., X and x + K are considered the same subset of X, via the corresponding equivalence relation on certain subsets of the power set P(C_n) of C_n. In this case, *translations* of A finite pattern [X] in C_n "can partition C_n" if: *there exists for some k >= 1 a finite sets of isometric copies X_j \sub C_n, j-1..., k of X in C_n, such that a) the X_j are disjoint, 1 <= j <= n and b) the union of the X_j is all of C_n. Question: --------- For each n, classify the finite patterns that can partition C_n. —Dan