Most-perfect magic square

A perfectly perfect magic square is a magic square with the following additional properties:

Examples

384 perfectly perfect magic squares in 1-16 representation and color coding: (16 & 1 ) - (9 & 8 ) - (5 & 12) - (3 & 14 ) - ( 2 & 15):

This 4x4 squares (any 4x4 -neck ) are partially known since the 11th or 12th century in India. Shifts ( in single steps, each also only one row or one column), by rotating, flipping, or by the free combination of these transformations can be 384 = 4! Generate x 16 squares. The transformations ( transformations ) from one square to another form a non-commutative closed group in terms of their linkage.

Properties

Published works on the properties of perfectly perfect magic squares there by Kathleen Ollerenshaw and David S. Brée and of TVPadmakumar, India.

In the 4x4 squares, there is a unique assignment of each value to its neighbors ( up, down, right and left). This " neighborhood relations " can generally expand to an algorithm with the example for squares of order 2n total of 2n! · 22n for n = 2 and n = 3 and 16.2 n! · 22n can be generated for n> 3 perfectly perfect magic squares, without applying Exhaustionsmethoden.