Techniques are developed for the logical design of magnetic core circuits to produce arbitrary single-output combinational switching functions. The approach is based on the relationship of a single magnetic core circuit to a linearly separable switching function. A synthesis procedure is developed which uses a pair of logical primitives, AND with NOT and OR with NOT, which are similar to the STROKE primitive and its inverse. Procedures are developed for the synthesis of symmetric functions which require no more than the integral part of (n+3)/2 cores, approximately half the number used in previously published procedures. The synthesis of arbitrary switching circuits is treated as a linear programming problem, and a table of all four-variable circuits is presented in which no circuit requires more than three cores.