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 (a,r)-circle. We provide a definition for magic (a,r)-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
Mathematical Subject Classification 2000
Primary: 05A15
Secondary: 15A48
Milestones
Received: 15 March 2009
Revised: 6 June 2009
Accepted: 6 June 2009
Published: 3 October 2009

Communicated by Scott Chapman
Authors
Rebecca Garcia
Sam Houston State University
Department of Mathematics and Statistics
Box 2206
Huntsville, TX 77340
United States
http://www.shsu.edu/~mth_reg
Stefanie Meyer
Sam Houston State University
Department of Mathematics and Statistics
Box 2206
Huntsville, TX 77340
United States
Stacey Sanders
Sam Houston State University
Department of Mathematics and Statistics
Box 2206
Huntsville, TX 77340
United States
Amanda Seitz
Sam Houston State University
Department of Mathematics and Statistics
Box 2206
Huntsville, TX 77340
United States