| 词条 | Associative magic square | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 释义 |
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 × 3The 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.{{cn|date=March 2019}} All rows, columns and diagonals sum to 15 and all pairs symmetrically opposite the center sum to 10.
4 × 4There are 48 order four associative magic squares. [1] Dürer's associative magic square is shown below.[2] {{Col-begin}}{{Col-break}}
5 × 5There are 48,544 order 5 × 5 associative magic squares.[3] {{Col-begin}}{{Col-break}}
7 x 7There are 1125154039419854784 order 7 x 7 associative magic squares.[4] {{Col-begin}}{{Col-break}}
Larger sizesEstimates for larger order associative magic squares are given below.[5]
Physical PropertiesThe 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.[6][7][8] How many different properties can a square have ?This 16 x 16 associative magic square has five additional properties. [9] The "Get Type" utility evaluates a square for over twenty different characteristics. [10] References1. ^https://commons.wikimedia.org/wiki/Category:Associative_magic_squares_of_order_4 2. ^http://mathworld.wolfram.com/DuerersMagicSquare.html 3. ^https://oeis.org/A081262 4. ^https://oeis.org/A081262 5. ^http://www.trump.de/magic-squares/howmany.html 6. ^https://oeis.org/A261798 7. ^{{cite journal | last = Knecht | first = Craig | authorlink = |author2=Walter Trump |author3=Daniel ben-Avraham |author4=Robert M. Ziff | title = Retention capacity of random surfaces | journal = Physical Review Letters | volume = 108 | issue = 4 | year = 2012 | pages = 045703 | url = https://arxiv.org/trackback/1110.6166 | doi = 10.1103/PhysRevLett.108.045703|arxiv = 1110.6166 |bibcode = 2012PhRvL.108d5703K }} 8. ^https://oeis.org/A201126 OEIS A201126 9. ^https://budshaw.ca/Franklin16.html 10. ^http://budshaw.ca/Download.html#gettype External links{{Commons category|Associative magic square}}
1 : Magic squares |
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