Associative magic square


An associative magic square is a magic square for which every pair of numbers symmetrically opposite to the center sum up to the same value.

3 × 3

The Lo Shu Square, the unique 3 × 3 normal magic square, is associative, and as such is the only 3 × 3 associative magic square. The square is shown below in Frénicle standard form. All rows, columns and diagonals sum to 15 and all pairs symmetrically opposite the center sum to 10.
276
951
438

4 × 4

There are 48 order four associative magic squares. Dürer's associative magic square is shown below.
163213
510118
96712
415141

5 × 5

There are 48,544 order 5 × 5 associative magic squares.
25225103
712111718
26132024
89151419
23162141

7 x 7

There are 1125154039419854784 order 7 x 7 associative magic squares.
2628233491540
2912338374511
32649348730
1946142536431
204324714418
3951342173821
10354116272224

Larger sizes

Estimates for larger order associative magic squares are given below.
square sizemagic squareassociative magic square
3×311
4×488048
5×5275,305,22448,544
6×6~1.77 × 10190
7×7~3.79 × 1034~1.12 × 1018
8×8~5.22 × 1054~2.52 × 1027
9×9~7.84 × 1079~7.28 × 1040
10×10~2.41 × 101100

Physical Properties

The image below shows areas completely surrounded by larger numbers with a blue background. A water retention topographical model is one example of the physical properties of magic squares. The water retention model progressed from the specific case of the magic square to a more generalized system of random levels. A quite interesting counter-intuitive finding that a random two-level system will retain more water than a random three-level system when the size of the square is greater than 51 X 51 was discovered. Percolation theory explains the counter-intuitive retention. This was reported in the Physical Review Letters in 2012 and referenced in the Nature article in 2018.

How many different properties can a square have ?

This 16 x 16 associative magic square has five additional properties.
The "Get Type" utility evaluates a square for over twenty different characteristics.