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:

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



See notes.

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.


Heinz, Harvey "Composition Magic Square"

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

"Melencolia I"

Nakazato, Ryu "Composite magic square for multimagic square"

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

"Water retention on mathematical surfaces"