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 V zigzag, and the 4x4 square is associative. The 8x8 composite is associative and V zigzag.
Pandiagonal example: .txt
The array of 2x2 squares having this aspect pattern is pandiagonal, V zigzag 2-way (left, right), and U zigzag 2-way (left, right). The 4x4 square is pandiagonal. The 8x8 composite is pandiagonal, V zigzag 2-way (left, right), and U zigzag 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. 7,296 squares are near-associative. 3,744 squares are V zigzagA3 2-way. 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-2p SQUARES" http://magictesseract.com./large_squares