Embedded Magic Squares

Last Updated: September 12, 2002


In W.S. Andrews's book (Magic Squares and Cubes), Harry Sayles gives the following construction for taking a magic circle and a magic square, and combining the two to get a new magic circle with embedded magic squares.

A magic square is a square grid of numbers where the rows, columns, and diagonals sum to the same value. Here is an example of a magic square:
Here we see that each row, column, and diagonals sums to 15.

To embed a magic square in a magic circle, we also need a magic circle. For this example, we will use the following simple circle:

To embed an n x n magic square in a magic circle, we first replicate each circle n times (in our example, n=3):

Now each node in the original magic circle corresponds to an n x n grid of nodes in the new set of circles. At each grid of nodes, we place a copy of our n x n magic circle, except we add to each cell of the magic square the value (k-1)n^2, where k is the value in the original magic circle corresponding to the grid in the new set of circles. In our example, we get:

A few notes are in order. First, the orientation of the magic square on each grid can be arbitrary. Second, the new magic circle will have a sum of SI+(C-I)n^3, where S is the sum of the magic square, C is the sum of the original magic circle, and I is the number of intersections on each circle of the original magic circle. In our example, the sum is 330.
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