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:

8 | 1 | 6 |

3 | 5 | 7 |

4 | 9 | 2 |

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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