Combinatorial mathematics is used to develop a method for constructing "1/2" fractions of the general K^N symmetric factorial experiment. The method does not require that K be either a prime or power of a prime. An examination of the structure of orthogonal polynomials is used to prove that the proposed method of fractionation partitions the treatment combinations of the experiment into two blocks which correspond to those formed when the least likely to be significant component of the N-factor interaction is confounded. A method based on orthogonal polynomials is given for analyzing these fractions. Questions regarding the augmentation required to obtain the balanced designs necessary for the analysis are answered. "1/2" fractions are constructed by the proposed method for certain 2 ^N, 3^N, 4^N, 5^N and 6^N designs, ande their properties are compared to those formed by the theory of Bose and Kishen