2-N-composite magic squares can be made from an order 2 square and a magic order n square. Since a normal, (i.e., of numbers 1, 2, 3, 4), order 2 square is not magic, the magic composite is obtained by independently rotating its 2x2 sub-squares.

There are 3 normal order 2 squares arbitrarily labelled 2a, 2b, 2c below. Their 8 aspects are:

0 1 2 3 4 5 6 7 --- --- --- --- --- --- --- --- 2a: 1 3 2 1 4 2 3 4 3 1 4 3 2 4 1 2 2 4 4 3 3 1 1 2 4 2 2 1 1 3 3 4 2b: 1 4 3 1 2 3 4 2 4 1 2 4 3 2 1 3 3 2 2 4 4 1 1 3 2 3 3 1 1 4 4 2 2c: 1 4 2 1 3 2 4 3 4 1 3 4 2 3 1 2 2 3 3 4 4 1 1 2 3 2 2 1 1 4 4 3

Order 8 magic squares can be made as composites of:

. the 2x2 square 2a: 1 3 2 4 . and the order 4 magic squares . with rotation of the sub-squares to an aspect pattern, such as: 0 2 0 2 2 0 2 0 0 2 0 2 2 0 2 0

The pattern refers to the above aspect numbers.

Associative example: .txt

The array of 2x2 squares having this aspect pattern is associative and
zigzag_{2}, and the 4x4 square is associative.
The 8x8 composite is
associative and
zigzag_{2}.

Pandiagonal example: .txt

The array of 2x2 squares having this aspect pattern is pandiagonal and
zigzag_{2} 2-way (left, right), and the 4x4 square is pandiagonal.
The 8x8 composite is
pandiagonal
and zigzag_{2} 2-way (left, right).

Some other aspect patterns are:

0 2 2 0 1 1 3 3 2 1 4 3 3 2 7 0 4 5 1 2 5 6 1 0 6 6 0 4 7 7 5 5 2 0 0 2 0 6 6 4 4 7 6 5 4 7 6 5 5 0 1 2 5 0 1 6 5 0 1 6 7 7 5 5 2 0 0 2 3 7 1 5 6 0 0 2 1 5 1 3 6 3 7 0 7 2 3 4 7 2 3 4 5 5 7 7 0 2 2 0 2 4 4 6 6 2 4 0 6 0 2 4 7 2 3 0 3 2 7 4 2 2 4 0 5 5 7 7

There are **86,671,872** aspect patterns for this 2x2 square that
make an order 8 magic composite with an order 4 magic square.
So, the total number of 8x8 magic composite squares is:

86,671,872 × 880 = 76,271,247,360

Order 8 magic squares can also be made as composites of:

. the 2x2 square 2b, 1 4, or 2c, 1 4, and the order 4 magic squares 3 2 2 3 . with appropriate aspect patterns

For each of these 2x2 squares, there are **1,871,216,640** patterns that
make an order 8 magic composite with an order 4 magic square. 7,483,226 of the patterns
are the same as for the 2a square. These include the first 4 below; the last 4 below
are not patterns for the 2a square:

2a, 2b, 2c patterns 2b, 2c only patterns ------------------------------- ------------------------------ 0 0 2 2 2 4 3 1 5 1 7 5 7 7 5 5 0 0 0 0 2 1 6 5 5 5 7 7 7 7 5 5 0 0 6 6 1 6 5 0 5 2 0 7 7 7 5 5 0 0 0 0 4 2 3 1 4 2 6 6 4 2 7 5 4 4 6 6 6 1 3 4 3 6 1 4 5 1 3 7 2 4 4 2 6 0 3 1 3 0 1 2 3 6 1 0 2 2 4 4 7 3 5 7 3 7 4 6 1 5 7 3 4 2 2 4 6 1 2 3 6 1 2 3 6 1 3 4 For 2a, diagonal sums are 5; so, any aspects are good for the diagonals. For 2b, 2c: row or column sums are 5, (depending on the aspect).

With appropriate aspect patterns, magic composites can be made of the order 2 squares and magic squares of any even order n ≥ 4.

Order 6 magic squares can be made as composites of:

. a combination of the 2x2 squares: 2a and 2c 2b and 2c 2a, 2b, and 2c . and the order 3 magic square . with appropriate aspect patterns for the 2x2 sub-squares

Here, to simplify the implementation, the aspects of all three 2x2 squares, (2a, 2b, and 2c), are mapped to transformations or "aspects" of square 2a:

square, aspects 2a "aspects" --------------- ---------------- 2a, 0 .. 7 0 .. 7 2b, 0 .. 7 8 .. 15 2c, 0 .. 7 9 .. 23

Example: .txt

There are **1,740,800** aspect patterns for the 2x2 squares that
make an order 6 magic composite with the order 3 magic square.
20,992 squares are
near-associative.
57,344 patterns involve only 2x2 squares 2a and 2c;
48,128 involve only 2x2 squares 2b and 2c.

Campbell, Dwane H. and Keith A. Campbell "ORDER-2^{p} SQUARES"
http://magictesseract.com./large_squares