Every pair of cells symmetrically opposite from the center is a
complementary pair,

for example, (**a**,**a**),
(**b**,**b**) in these order 4 and order 5 patterns:

AssociativeSquares makes these squares. It uses the methods outlined below. Associative squares can also be made with MagicSquares, the CompleteSquare utility, and MagicSquaresSODLS.

Order5Special makes all the order 5 associative squares.

These squares can be made with the Siamese method. See "Demanding the square is associative" in math behind the Siamese method.

There are 48,544 order 5 associative magic squares.

There are no associative magic squares of singly-even order.

This was shown by C. Planck (1919). See reference.

A proof similar to that of Planck can be given directly for associative
squares, (without involving pandiagonals). Let **n** be the order of the
square, **m** = **n/2**, and **Σ** the
magic sum.
Consider a **4k+2**, k≥1, square where (**a A**),
(**b B**), ... are complement pairs and **W, X, Y** are
the sums of square quarters:

Summing rows:W + X = mΣSumming columns:W + Y = mΣSumming complement pairs:X + Y = mΣ

Thus, **W = X = Y = mΣ/2 = 2m⁴ + m²/2**
which cannot be, because if **m** is odd, **m²/2** is
fractional.

There are singly-even magic squares in which only four complement pairs are not
center symmetric.
AssociativeSquares makes these squares
based on
Conway's LUX method.

To get near-associative squares of order n = 4k+2:

- create an associative magic square for the (2k+1)x(2k+1) array
- replace each value v in this array by V = 4(v-1)+1
- use a modified LUX pattern to make the cells in each 2x2 block of the 4k+2 square symmetric with their complements in the symmetric 2x2 block
- fill each 2x2 block with V, V+1, V+2, V+3 in the order given by the letter:

where, for example, **L'** is the symmetric complement of
**L**.

Using L, U, X, and their complements makes a square that is associative, except for the center 2x2, but not magic. A little adjustment in the middle row and column of the pattern makes the square magic with only four complement pairs not symmetric. The changed blocks are indicated in blue. Possible patterns for n = 6, 10, 14, 18 are:

Orders 6, 14, 22, 30, ... and 10, 18, 26, 34, ... have similar patterns, (note the different middle letter).

See "Method of constructing a magic square of doubly-even order" in Magic square.

There are 48 order 4 associative magic squares.
These are **TYPE III** in the classification by Dudeney. See
references.

Associative squares remain associative, not only under
Transform 1 and
Transform 2,
but by swapping only the described rows **or** columns.
So, from each even order associative square, the total number of resulting squares is:

**2 ^{n-2}(n/2)!^{2}**

For odd order, the number is that for order n - 1.

Order Transforms1_2All Swap Rows or Columns Only -------- ------------------ --------------------------- 4, 5 4 16 7 24 576 8, 9 192 36,864 11 1,920 3,686,400 12, 13 23,040 530,841,600

"Associative magic square" http://en.wikipedia.org/wiki/Associative_magic_square

Conway, John H. "Conway's LUX method for magic squares"

http://en.wikipedia.org/wiki/Conway's_LUX_method_for_magic_squares

Dudeney, Henry E. "Magic Square Problems"

http://www.web-books.com/Classics/Books/B0/B873/AmuseMathC14P1.htm

Dudeney, Henry E. "Magic Square Problems"

http://www.scribd.com/doc/49756911/Amusments-in-Mathematics, page 287.

Heinz, Harvey "Order 4 Magic Squares"

http://www.magic-squares.net/order4list.htm#The 12 Groups

Hospel, Ton "The math behind the Siamese method of generating magic squares" http://www.xs4all.nl/~thospel/siamese.html

Planck, C. "PANDIAGONAL MAGICS OF ORDERS 6 AND 10 WITH MINIMAL NUMBERS."

http://www.jstor.org/stable/27900742?seq=1#page_scan_tab_contents

Weisstein, Eric W. "Associative Magic Square." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AssociativeMagicSquare.html