Associative Magic Squares

Description

Every pair of cells symmetrically opposite from the center is a complementary pair,
for example, (a,a), (b,b) in the order 4 and order 5 patterns:

Construction

AssociativeSquares makes these squares. It uses the methods outlined below.

Order5Special makes all the order 5 associative squares.

Odd Order

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.

Singly-Even Order

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

Proof

This was shown by A.H. Frost (1878) and C. Planck (1919). See references.

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 = mΣ
  Summing columns:          W + Y = mΣ
  Summing 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.

Near-associative

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 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:

Doubly-Even Order

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.

Transforms

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

         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

REFERENCES

"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.

Frost, A.H. "ON THE GENERAL PROPERTIES OF NASIK SQUARES"
http://books.google.com/books?id=qxMLAAAAYAAJ&pg=PA34#v=onepage&q&f=false

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.archive.org/stream/monistquart29hegeuoft#page/306/mode/2up

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