Magic Circles

A set of  magic circles is a numbering of the intersections of the  circles such that the sum over all intersections is the same constant for all circles. The above sets of three and four magic circles have magic constants 14 and 39 (Madachy 1979). For  circles, the constant is , for , 2, ... corresponding to 3, 14, 39, 84, 155, 258, ... (OEIS A027444).

Another type of magic circle arranges the number 1, 2, ...,  in a number of rings, which each ring containing the same number of elements and corresponding elements being connected with radial lines. One of the numbers (which is subsequently ignored) is placed at the center. In a magic circle arrangement, the rings have equal sums and this sum is also equal to the sum of elements along each diameter (excluding the central number). Three magic circles using the numbers 1 to 33 are illustrated above.

REFERENCES

Heinz, H. "Magic Circles." http://www.magic-squares.net/perimeter-3.htm#Magic%20Circles.

Hung, K. M. "Magic Circle." http://home.hkstar.com/~hkm88630/magiccl.htmlMadachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 86, 1979.

Sloane, N. J. A. Sequence A027444 in "The On-Line Encyclopedia of Integer Sequences."