Half-cycles and chaplets

For n odd, a half-cycle for Zn is a cycle [a1, a2, . . . , am] of distinct elements from Zn such that (a) m = (n − 1)/2, (b) the elements ai (i = 1, 2, . . . ,m) are all distinct, and (c) the differences ai+1 − ai (i = 1, 2, . . . ,m, with am+1 = a1) are all distinct and no two of them are the negatives of one another, modulo n. Similarly, a chaplet for Zn is now newly defined to be a cycle [a1, a2, . . . , am] of distinct units from Zn such that (a) m = φ(n)/2 where Euler’s totient function φ(n) gives the number of units in Zn, and conditions (b) and (c) are satisfied as before. Thus the relationship between half-cycles and chaplets is analogous to that between full cycles and the daisy chains defined in a previous paper. Methods of construction are given for both half-cycles and chaplets, with emphasis on methodology that is fruitful in the range 5 < n < 300. Some of the methods are adaptations of constructions for daisy chains. Most results concern robust chaplets, where either (i) the set of elements ai is identical to the set of differences ai+1 −ai or (ii) the two sets are the negatives of one another. Examples are provided liberally, to help with the understanding of a novel subject.