\input zb-basic \input zb-ioport \iteman{io-port 06092026} \itemau{Billington, Elizabeth J.; Cavenagh, Nicholas J.; Khodkar, Abdollah} \itemti{Super-simple twofold 4-cycle systems.} \itemso{Bull. Inst. Comb. Appl. 63, 48-50 (2011).} \itemab Summary: There are many papers on super-simple block designs, which are $(v,k,\lambda)$-designs in which any two blocks have at most two points in common; see the handbook of combinatorial design [{\it C. J. Colbourn} (ed.) and {\it J. H. Dinitz} (ed.), CRC Press Series on Discrete Mathematics and its Applications. Boca Raton, FL: CRC Press. xviii, 753 p. (1996; Zbl 0836.00010)] for further details. But as far as we can ascertain, there are no results on the simpler (no pun intended) problem regarding the existence of super-simple cycle systems. Despite the fact that pairs of disjoint cycle systems of the same order have been constructed [the first author, J. Comb. Des. 1, No. 6, 435--452 (1993; Zbl 0885.05044)], combining two disjoint systems, even for 4-cycles, does not necessarily result in the twofold system being super-simple. Of course 3-cycle systems are also designs -- triple systems -- so the place to start is with 4-cycles. In this note we prove the following result: There exists a super-simple twofold 4-cycle system if and only if $n \equiv 0$ or $1 \pmod 4$, $n \ne 4, 5, 9$. \itemrv{~} \itemcc{} \itemut{super-simple block designs} \itemli{} \end