# 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, compact_{3}.

## 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^{n²}, 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.