Composite Magic Squares

Construction

CompositeSquares makes these squares.

Description

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.

Rotations

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

Calligraphy

Preserved

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.

REFERENCES

Heinz, Harvey "Composition Magic Square" http://www.magic-squares.net/glossary.htm.

Knecht, Craig "Knecht Magic Squares, Craig's 2014 Update" http://www.knechtmagicsquare.paulscomputing.com/Craigs Update 2014.html.

Melencolia I.

Nakazato, Ryu "Composite magic square for multimagic square"
http://www.geocities.jp/cocotte_rn/houjin/en/composite.html
.

Rouse Ball, W.W. "Other Methods For Constructing Any Magic Square"
http://www.gutenberg.org/files/26839/26839-pdf.pdf
, page 134.