MultiMagic 2-N-Composites

Preamble

It is known that the order mn, (m times n), composite of an order m multimagic square and an order n multimagic square is multimagic.

Also, for even orders n, 2-N-composite magic squares can be made from an order 2 square and a magic order n square. See 2-N-Composites.

Description

For some bimagic squares of order n, 2-N-composite squares can be made that are bimagic.

Why Only Some

No line, (i.e., row, column, or diagonal), of a 2x2 square has the required order 2 bimagic sum of 15. The bimagic sums are:

         line numbers        bimagic sum
        ---------------      -----------
             1  2                 5
             1  3                10
             1  4                17
             2  3                13
             2  4                20
             3  4                25

So, for two 2x2 square aspects, we get twice the order 2 bimagic sum of 15 as three combinations: 5 + 25, 10 + 20, 17 + 13.

Unlike the composite magic sum, the composite bimagic sum involves interaction between the numbers of the 2x2 array and those of the nxn square. This interaction is in the form of products of the numbers.

For a 2x2 aspect line, (that has 2x2 sums not equal to 5), the aspect pattern for the 2x2 square depends on the numbers and their arrangement in the nxn square line. See example.

For some nxn square lines there will be no 2x2 aspect pattern that makes the composite bimagic sum. See no pattern.

And, if there are 2x2 aspect patterns that result in a bimagic composite sum with each line of the order n square, these patterns may still not combine to make a nxn pattern. See no combination.

2-8 BiMagic Example

Here is a composite of the order 2 square 2a {1 3} {2 4} and the first bimagic square. .txt

For this order 2 square and order 8 square, there are 1,286,784 aspect patterns for the 2x2 sub-squares that make an order 16 bimagic composite.

For 11,296 of these aspect patterns, the corresponding arrays of 2x2 squares are zigzag2. For 200,208 of the aspect patterns, the corresponding arrays of 2x2 squares are zigzag2 2-way. The square types are:

For the order 2 square 2b {1 4} {3 2}, there is a very large number of aspect patterns that make an order 16 bimagic composite. Likewise for the order 2 square 2c {1 4} {2 3}.

2-16 BiMagic Example

Here is a composite of the order 2 square 2a {1 3} {2 4} and the first order 16 bimagic square. .txt

The aspect pattern is repeated across and down to get the 16x16 pattern.
The composite is zigzag2,3 2-way (up, down).

For this order 2 square and order 16 square, there are 12,288,160 repeating 4x4 aspect patterns for the 2x2 sub-squares that make an order 32 bimagic composite. The square types are:

Pattern examples:

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

The first example is actually a repeating 2x2 pattern. There are only 16 of those:

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

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

16x16 patterns and repeating 8x8 patterns have not been investigated.

Patterns for the order 2 square 2b or 2c have not been investigated.