There are several variations on magic circles.
One famous magic circle was
constructed by Benjamin Franklin, circa 1752, which is an arrangement of
nonnegative integers in a circular grid consisting of eight concentric
annuli and eight radial segments. Franklin's circle had many properties,
including the standard magic property: that the annular sum and the
radial sum equal the same magic number M. Franklin's magic circle is an
example of what we call an r-magic 8-circle.
Another similar type of magic circle is what we call a d-magic
n-circle, which is an arrangement of nonnegative integers in a circular
grid consisting of n concentric annuli and n diametrical segments, where
the annular sum and the diametrical sum equal the same magic number M.
In this presentation, we discuss some techniques in computational
algebraic combinatorics and enumerative geometry to construct and to
count these variations on magic circles. We provide a very nice
description of their minimal Hilbert basis, which is useful in
determining the symmetry operations on magic circles and, consequently,
in enumerating natural magic circles. Finally, we present the
enumerating functions for the Franklin magic 8-circles, the r-magic
circles, and the d-magic circles.