This paper presents methods for realizing simple threshold functions of n arguments by networks of k-input majority gates, where k≪n. An optimal network realization of the 5-argument majority function using 3-input majority gates is given, and it is then generalized by steps with realizations for the (2n-l)-argument majority function (where n = 3, 4, ...) using (2n-3)-input majority gates, and then for the (2n-1)-argument majority function using (2k-l)-input majority gates (where k≪n). In a final generalization an array network using (2k-l)-input majority gates introduced for the realization of an (m/n), ``simple,'' threshold function (where m = 1, 2, ...,n). The array network is then applied to the synthesis of arbitrary symmetric functions; in the latter synthesis a realization of ``adjustable logic'' is given where, by simple control of network connections, the same network can be made to compute any symmetric function. The specific networks for ``5 by 3's'' (5-argument majority function realized by a 3-input majority gate), ``7 by 5's'', and ``7 by 3's'' are the best known.