Vol. 2, No. 3, 2009

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Construction and enumeration of Franklin circles

Rebecca Garcia, Stefanie Meyer, Stacey Sanders and Amanda Seitz

Vol. 2 (2009), No. 3, 357–370
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.

Keywords
Franklin squares, Franklin circles, magic circles, enumeration, polyhedral cones, Ehrhart series
Primary: 05A15
Secondary: 15A48