Composite magic square

For multimagic square


Definition

Normal composite magic square can be constructed from two magic squares, each magic square can be used any order.
Each cell of composite magic square C is described as following by base magic squares A and B. (Each cell of B is decreased 1 from original magic square.)

Ci, j = Aceil(i/order(B)), ceil(j/order(B)) + size(A) * Bceil(i/order(A)), ceil(j/order(A))

Semantic of each function is following.
ceil(x) = Round up x
order(X) = Order of X
size(X) = The number of cell in X (=order(X)2)
@Next is the sample magic square which is used this method.


The sample from order 3 to order 9

This is only magic square to describe compsite method, not bimagic.
294
753
618
  *  
183
642
507
S=15 S=12
111813 748176 293631
161412 797775 343230
151017 787380 332835
566358 384540 202722
615957 434139 252321
605562 423744 241926
475449 Q94 657267
525048 753 706866
514653 618 696471
S=369


Multimagic theorem on composite magic square

Composite magic square is multimagic square if both base magic squares are multimagic square. Multimagic level of composite magic square depends on lower lovel of base magic square.

As the exapmle, following is calculation of top row of bimagic square. Base magic squares are A and B, composite magic square is C.

Semantic of each function is following.
magicsum(n, X) = Magic sum of n-th multimagic square X.

C1, 12 + C1, 2n + ... + C1, order(C)2

= (C1, 12 + ... + C1, order(A)2)
  + (C1, order(A)+12 + ... + C1, order(A)*22)
  + ...
  + (C1, order(A)*(order(B)-1)+12 + ... + C1, order(A)*order(B)2)

= (A1, 1 + size(A) * B1, 1)2 + ... + (A1, order(A) + size(A) * B1, 1)2

  + (A1, 1 + size(A) * B1, 2)2 + ... + (A1, order(A) + size(A) * B1, 2)2
  + ...
  + (A1, 1 + size(A) * B1, order(A))2 + ... + (A1, order(A) + size(A) * B1, order(A))2

= (A1, 12 + ... + A1, order(A)2) + 2 * (A1, 1 + ... + A1, order(A)) * size(A) * B1, 1 + order(A) * (size(A) * B1, 1)2

  + (A1, 12 + ... + A1, order(A)2) + 2 * (A1, 1 + ... + A1, order(A)) * size(A) * B1, 2 + order(A) * (size(A) * B1, 2)2
  + ...
  + (A1, 12 + ... + A1, order(A)2) + 2 * (A1, 1 + ... + A1, order(A)) * size(A) * B1, order(B) + order(A) * (size(A) * B1, order(B))2

= magicsum(2, A) + 2 * magicsum(1, A) * size(A) * B1, 1 + order(A) * size(A)2 * B1, 12
  + magicsum(2, A) + 2 * magicsum(1, A) * size(A) * B1, 2 + order(A) * size(A)2 * B1, 22
  + ...
  + magicsum(2, A) + 2 * magicsum(1, A) * size(A) * B1, order(B) + order(A) * size(A)2 * B1, order(B)2

= order(B) * magicsum(2, A)
  + 2 * magicsum(1, A) * size(A) * (B1, 1 + B1, 2 + ... + B1, order(B))
  + order(A) * size(A)2 * ( B1, 12 + B1, 22 + ... + B1, order(B)2 )

= order(B) * magicsum(2, A)
  + 2 * magicsum(1, A) * size(A) * magicsum(1, B)
  + order(A) * size(A)2 * magicsum(2, B)

This conclusion means sum of composite magic square line independent from each row and column. Diagonal sum is same (but it omit).


Composite bimagic square

I made order 64 bimagic square from two order 8 bimagic squares. This magic square size is big, please download to see it.
composite_bimagic_square.xls


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