Completely perfect magic square
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perfectly perfect magical square from the Parshva Jaina Temple in Khajuraho |
A perfectly perfect magic square is a magic square with the following additional properties:
- the order of the squares is a multiple of 4
- every 2 × 2 sub-square (including those that can be created by breaking the sides) gives the same sum 2 (1 + n 2 )
- for each value a , the complement 1 + n 2 - a of this value is offset diagonally by n / 2
Examples
384 perfectly perfect magic squares in 1 to 16 representation and color coding: ( 16 & 1 ) - ( 9 & 8 ) - ( 5 & 12 ) - ( 3 & 14 ) - ( 2 & 15 ):
These 4 × 4 squares (any 4 × 4 section) have been known in India since the 11th and 12th centuries . By shifting (even in single steps, also only one row or one column), by rotating, mirroring or by freely combining these conversions, 384 = 4! · 16 squares can be generated. The conversions (transformations) from one square to another form a non-commutative closed group with regard to their connection.
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properties
Published work on the properties of perfectly perfect magic squares has been published by Kathleen Ollerenshaw and David S. Brée, and TV Padmakumar, India .
With the 4 × 4 squares there is a clear assignment of each value to its neighbors (top, bottom, right and left). This “ neighborhood relation ” can generally be expanded into an algorithm with which z. B. for squares of the order in total for and or for perfectly perfect magic squares can be generated without using exhaustion methods.
literature
- Kathleen Ollerenshaw , David S. Brée: Most-perfect Pandiagonal Magic Squares: Their Construction and Enumeration. Institute of Mathematics and its Applications, Southend-on-Sea 1998, ISBN 0-905091-06-X .
- TV Padmakumar: Number Theory and Magic Squares. Sura books, India 2008, ISBN 978-81-8449-321-4 , ( surabooks.com ( Memento of February 25, 2012 in the Internet Archive )).
Web links
- Harvey Heinz: Most-perfect Magic Squares
- TVPadmakumar: work on squares of the type "Sri Rama Chakra"
- TVPadmakumar: Strongly Magic Squares . (PDF; 1.9 MB)
- The 4 × 4 Pan-Magic Squares
- Michael Dörmann: A perfect magic square . (Online Generator - Magic Square 4 × 4 with JavaScript)