All pairs of cells symmetrically opposite from the center have the same sum,
for example, (**a**,**a**),
(**b**,**b**) in these order 4 and order 5 patterns:

If the square is normal magic, the pairs are complementary.

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

Order5Special makes all the order 5 associative squares.

A method attributed to
C.-Y. Jean Chan, Meera G. Mainkar, Sivaram K. Narayan, and
Jordan D. Webster

from Holger Danielsson.

A method attributed to
J. P. De Los Reyes, Ahmad Pourdarvish, Chand K. Midha, and Ashish Das

from
Holger Danielsson.

A method attributed to L. S. Frierson from
Holger Danielsson.

These include squares from the
methods of Philippe de la Hire and
Jean-Joseph Rallier des Ourmes.

A method from Bogdan Golunski.

See directory:
/mag. squares/magic squares/symmetrical magic squares/construction method I/

A technique of Marios Mamzeris.

Method V with the added feature that rows and/or columns of the initial table are randomly swapped.

The
Siamese method. See "Demanding the square is associative"
in math behind the Siamese method.

These are two squares.
Methods for these are also given by
Holger Danielsson.

See
Ḥasan Ibn al-Haytham;
Claude Gaspard Bachet de Méziriac;
Simon de la Loubère;

some
J. P. De Los Reyes, Ahmad Pourdarvish, Chand K. Midha, and Ashish Das;
Yun-er Liao, Bao-man Zhu, and Lian-fa Wu;

Harry A. Sayles, (Lozenge-Squares);
Manuel Moschopoulos (1);
Brian S. Reiner; and
Zhao Li-hua.

William Walkington
has a remarkable method that uses equations for the relative coordinates of numbers.

See link to download
MTCVS 161019.pdf.
Some of the squares from these equations are created here.

There are 48,544 order 5 associative magic squares. These are group 7 of the Complement Pair Pattern Groups.

There are 1,125,154,039,419,854,784 order 7 associative magic squares.

The number was determined by Go Cato in 2018.

See OEIS A081262.

Even order associative magic squares can be transformed to/from pandiagonal complete magic squares by this method:

- divide the square into quarters A, B, C, D
- reflect B, (rotate 180° about Y axis)
- invert C, (rotate 180° about X axis)
- reflect and invert D, (rotate 180°)

In the example below, (**e E**), (**f F**), ...
represent complement pairs.
The cells of one broken diagonal are shown in color.

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

C. Planck showed in 1919 that there are no pandiagonal magic squares of singly-even order, and observed:

The same result also follows for
associated squares, for if an associated square

of these orders existed it could be transformed into a pandiagonal by the A-D method.

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 two complement pairs are not
center symmetric.

This is the minimum, because if one pair is not center symmetric, at least one other pair
must not be.

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 pattern letter:

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

Possible patterns for n = 6, 10, 14, 18 are:

Orders 6, 14, 22, 30, ... and 10, 18, 26, 34, ... have similar patterns, (differences shown in blue).

This gives a semi-magic square with only two complement pairs not center symmetric.

Swap appropriate row(s) or column(s) to make the square magic.

A 9-block method attributed to Muhammad Al-Asfizari from
Holger Danielsson.

The order 4 pattern is the same as given for doubly even magic squares on
Wolfram.

Method I with the added feature that the patterns of smaller 4k orders are also
used in subsquares.

For example, for order 24, patterns of orders 4, 8, 12, and 24
are used.

**Method of reversions** attributed to C. Planck from
Holger Danielsson

Margossian pandiagonal Method IV converted to associative by Planck's A-D method.

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

For odd orders, associative pair numbers are either both odd or both even. Squares that are not associative can also have all odd pairs associative or all even pairs associative. These types are an idea of Craig Knecht.

There are squares that are associative but not magic.
One example of these is made by simply writing the ordered sequence 1 to n^{2}:

These are called **natural squares**. See
Planck.
See also reversible squares.

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

Danielsson, Holger "Magic Squares" https://www.magic-squares.info/en.html

Dudeney, Henry E. "Magic Square Problems"

https://archive.org/details/AmusementsInMathematicspdf

Golunski, Bogdan "NUMBER GALAXY"
http://www.number-galaxy.eu/

See directory:
/mag. squares/magic squares/symmetrical magic squares/construction method I/

Heinz, Harvey "Order 4 Magic Squares"

http://recmath.org/Magic%20Squares/order4list.htm#The%2012%20Groups

Hospel, Ton "The math behind the Siamese method of generating magic squares"

http://www.xs4all.nl/~thospel/siamese.html

Kato, Go "The number of associative magic squares of order 7"

https://oeis.org/A081262/a081262.pdf

Mamzeris, Marios "Magic Squares of odd order by Marios Mamzeris" https://www.oddmagicsquares.com/

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

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

Trump, Walter "Notes on Magic Squares" http://www.trump.de/magic-squares/index.html

Walkington, William "Magic Squares, Spheres and Tori "

https://carresmagiques.blogspot.com/2020/04/magic-torus-coordinate-and-vector.html

Weisstein, Eric W. "Associative Magic Square."
From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/AssociativeMagicSquare.html