2-N-Composites

Description

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.

Order 2 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

Even Order n

Order 8 Composites

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 zigzag2, and the 4x4 square is associative. The 8x8 composite is associative and zigzag2.

Pandiagonal example: .txt

The array of 2x2 squares having this aspect pattern is pandiagonal and zigzag2 2-way (left, right), and the 4x4 square is pandiagonal. The 8x8 composite is pandiagonal and zigzag2 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).

Other 2-N-Composites for Even n

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

Odd Order n

Order 6 Composites

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.

REFERENCES

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