Composite Magic Squares


CompositeSquares makes these squares.


A composite square of order mn, (m times n), can be made from a square of order m and a square of order n. It contains n² order m magic squares using the numbers 1 to (mn)² The smallest are order-9 from order-3 and order-3. They are made up of 9 order-3 magic squares using the numbers 1 to 81.

The next are order-12, consisting of 16 order-3 or 9 order-4 magic squares, using the numbers 1 to 144.

Distinct Squares

Different aspects of the same squares make distinct composite squares. For example, 8 distinct order-9 magic squares can be made from the 8 aspects of the Lo Shu:

Ultramagic Transform

These composite associative order 9 squares can be converted to ultramagic by swapping rows 2 and 4, 3 and 7, and 6 and 8. This method was received from Paul Michelet.

Example: assoc.txt ultra.txt

This method is one of the associative transforms by swapping rows and/or columns. Of the 36,864 transforms of this composite associative square, these 288 are ultramagic, compact3.

Preserved Properties

Some properties of the order m and order n squares are preserved in the composite square. If the order m square and the order n square are both associative, the composite square is associative. The 3-3Composites and the 4-3Composite above are associative.

The pandiagonal and multimagic properties are also preserved in composite squares.

See associative proof, pandiagonal proof, and bimagic proof.


Each order m sub-square in the composite can also be independently rotated to a different aspect to make a very large number, 8, of order mn squares.

Composites of Orders 3, 4, 5


Water Text

Craig Knecht's idea of water writing in magic squares is now automated. Just type some text in a file and run program CompositeCalligraphy.

Example: See Pi Are Squared notes.

Dürer Anniversary

See notes.


Heinz, Harvey "Composition Magic Square"

Knecht, Craig "Knecht Magic Squares, Craig's 2014 Update" Update 2014.html.

Melencolia I.

Rouse Ball, W.W. "Other Methods For Constructing Any Magic Square"
, page 134.