Construction and enumeration of Franklin circles

Abstract

Around 1752, Benjamin Franklin constructed a variant on the popular magic squares and what we call a magic $\left(a,r\right)$-circle. We provide a definition for magic $\left(a,r\right)$-circles, magic $a$-circles and, more specifically, Franklin magic $a$-circles. In this paper, we use techniques in computational algebraic combinatorics and enumerative geometry to construct and to count Franklin magic 8-circles. We also provide a description of its minimal Hilbert basis and determine the symmetry operations on Franklin magic 8-circles.