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.

2	7	6
9	5	1
4	3	8

4 × 4

There are 48 order four associative magic squares. [1] Dürer's associative magic square is shown below.[2]

16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 5 × 5 There are 48,544 order 5 × 5 associative magic squares.[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 7 x 7 There are 1125154039419854784 order 7 x 7 associative magic squares.[3] 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 Larger sizes Estimates for larger order associative magic squares are given below.[4] 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 × 10^19 0 7×7 ~3.79 × 10^34 ~1.12 × 10^18 8×8 ~5.22 × 10^54 ~2.52 × 10^27 9×9 ~7.84 × 10^79 ~7.28 × 10^40 10×10 ~2.41 × 10^110 0 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.[5][6][7] How many different properties can a square have ? This 16 x 16 associative magic square has five additional properties. [8] The "Get Type" utility evaluates a square for over twenty different characteristics. [9] References https://commons.wikimedia.org/wiki/Category:Associative_magic_squares_of_order_4 "Dürer's Magic Square". "A081262 - Oeis". "Enumeration of magic squares". "A261798 - Oeis". Knecht, Craig; Walter Trump; Daniel ben-Avraham; Robert M. Ziff (2012). "Retention capacity of random surfaces". Physical Review Letters. 108 (4): 045703. arXiv:1110.6166. Bibcode:2012PhRvL.108d5703K. doi:10.1103/PhysRevLett.108.045703. PMID 22400865. https://oeis.org/A201126 OEIS A201126 "Order 16 Franklin Squares". "Downloads". External links Associative magic square, MathWorld Item on Futility Closet https://commons.wikimedia.org/wiki/Category:Associative_magic_squares_of_order_4 https://commons.wikimedia.org/wiki/Category:Magic_squares_by_order http://oeis.org/A331507/ Maximum number of ponds examples introduce the idea of the center symmetric arrangement of ponds https://www.youtube.com/watch?v=JPEB8aT7XiU/ New method of construction Marios Mamzeris File:Associative Magic Square With Lake.png