# 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.[1] All rows, columns and diagonals sum to 15 and all pairs symmetrically opposite the center sum to 10.

 2 7 6 9 5 1 4 3 8

## 4 × 4

There are, up to rotation and reflection, 48 distinct 4 × 4 associative magic squares, four of which are given here.[2]

 1 14 15 4 8 11 10 5 12 7 6 9 13 2 3 16
 1 15 14 4 8 10 11 5 12 6 7 9 13 3 2 16
 2 13 16 3 7 12 9 6 11 8 5 10 14 1 4 15
 2 16 13 3 7 9 12 6 11 5 8 10 14 4 1 15

## 5 × 5

There are 48,544 distinct 5 × 5 associative magic squares, up to rotation and reflection, such as the two below.[3]

 25 22 5 10 3 7 12 11 17 18 2 6 13 20 24 8 9 15 14 19 23 16 21 4 1
 19 6 9 16 15 5 22 23 1 14 18 2 13 24 8 12 25 3 4 21 11 10 17 20 7

## Larger sizes

There are no 6 × 6[3] (or 10 × 10) associative magic squares. Other than when there are none the precise number of associative magic squares at sizes bigger than 5 × 5 is unknown. The following table, due to Walter Trump, gives the sizes up to 10, followed by examples of 7 × 7 and 8 × 8 tables.

square size magic square associative magic square
3×3 1 1
4×4 880 48
5×5 275,305,224 48,544
6×6 ~1.77 × 1019 0
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 × 10110 0
 26 28 23 34 9 15 40 29 12 33 8 37 45 11 32 6 49 3 48 7 30 19 46 14 25 36 4 31 20 43 2 47 1 44 18 39 5 13 42 17 38 21 10 35 41 16 27 22 24
 1 32 34 63 37 60 6 27 48 49 15 18 12 21 43 54 19 14 52 45 55 42 24 9 62 35 29 4 26 7 57 40 25 8 58 39 61 36 30 3 56 41 23 10 20 13 51 46 11 22 44 53 47 50 16 17 38 59 5 28 2 31 33 64