Completely perfect magic square

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Magic square at the Parshvanatha temple, Khajuraho.png
੧੨ ੧੪
੧੩ ੧੧
੧੬ ੧०
੧੫
7th 12 1 14th
2 13 8th 11
16 3 10 5
9 6th 15th 4th
Transcription of the above
Indian numerals
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:

  1. the order of the squares is a multiple of 4
  2. every 2 × 2 sub-square (including those that can be created by breaking the sides) gives the same sum 2 (1 + n 2 )
  3. 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.

02 11 05 16 02 11 05 16 02 11
13 08 10 03 13 08 10 03 13 08
12 01 15th 06 12 01 15th 06 12 01
07 14th 04 09 07 14th 04 09 07 14th
02 11 05 16 02 11 05 16 02 11
13 08 10 03 13 08 10 03 13 08
12 01 15th 06 12 01 15th 06 12 01
07 14th 04 09 07 14th 04 09 07 14th
02 11 05 16 02 11 05 16 02 11
13 08 10 03 13 08 10 03 13 08
Jaina square
02 11 14th 07 02 11 14th 07 02 11
13 08 01 12 13 08 01 12 13 08
03 10 15th 06 03 10 15th 06 03 10
16 05 04 09 16 05 04 09 16 05
02 11 14th 07 02 11 14th 07 02 11
13 08 01 12 13 08 01 12 13 08
03 10 15th 06 03 10 15th 06 03 10
16 05 04 09 16 05 04 09 16 05
02 11 14th 07 02 11 14th 07 02 11
13 08 01 12 13 08 01 12 13 08
05 11 14th 04 05 11 14th 04 05 11
10 08 01 15th 10 08 01 15th 10 08
03 13 12 06 03 13 12 06 03 13
16 02 07 09 16 02 07 09 16 02
05 11 14th 04 05 11 14th 04 05 11
10 08 01 15th 10 08 01 15th 10 08
03 13 12 06 03 13 12 06 03 13
16 02 07 09 16 02 07 09 16 02
05 11 14th 04 05 11 14th 04 05 11
10 08 01 15th 10 08 01 15th 10 08

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

Web links