Simple Magic Circles

Last Updated: September 12, 2002

In W.S. Andrews's book (Magic Squares and Cubes), Harry Sayles gives a simple construction for assigning numbers to get a magic circle. This simple assignment only applies when the circles meet certain conditions:

Each pair of circles intersect in either 0 or 2 distinct places;
no more than 2 circles intersect at one point,
each circle has the same number of intersections $I$ (which because of conditions 1 and 2 has to be an even number),

In such a configuration, there will always be an even number of intersections, say 2n, where n is the number of pairs of intersecting circles. The assignment of the numbers 1...2n to the intersection points is done by first pairing the numbers i to 2n+1-i, and then going through the n pairs of intersecting circles and arbitrarily assigning one of the number pairs to the intersection pair (naturally, each pair of numbers is assigned to one and only one intersection pair).

The result is readily seen to be a magic circle, since each circle intersects I/2 other circles in a total of I locations, with a sum of (I/2)(2n+1) as its sum.

The following simple example is easily seen to be a magic circle of this type:

Other Examples of Simple Magic Circles

Note that "simple" here just refers to the arrangement of intersections. The configuration itself may look fairly complex, as seen in the following examples: