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.

276
951
438

4 × 4

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

5 × 5

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

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

References

  1. ↑ William Walkington, http://carresmagiques.blogspot.fr/2012/03/from-magic-square-to-magic-torus.html
  2. ↑ Craig Knecht, http://www.knechtmagicsquare.paulscomputing.com
  3. 1 2 Harry White http://budshaw.ca/Associative.html
  4. ↑ Robert Dickter "Archived copy". Archived from the original on 2012-01-06. Retrieved 2012-02-21.
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