Synchronous Counters Constructed Entirely of J-K Flip-Flops

Synchronous counters composed entirely of J-K flip-flops can be systematically designed to fit given specifications. However, known design methods do not provide useful counters for many specifications of interest. We prove that some plausible counters are impossible to construct. For example, we prove that there is no self-starting, gateless, synchronous J-K flip-flop counter with N J-K flip-flops, N ≫ 2, and a period of 2^N - 1. Known design methods do not trivially generate many desirable counters, but nonexistence theorems do not show that these counters are impossible. Searching among all possible configurations of J-K flip-flops is an attractive notion, but the number of possibilities is so large that it renders such search infeasible on current general-purpose digital computers. A special-purpose digital device was designed, implemented, and operated to perform an exhaustive search of all configurations of five or fewer J-K flip-flops. Some of the discovered counters are listed in the Appendix.