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[71] [72] For an even square, there are n/2 pairs of rows or columns that can be interchanged; thus 2 n/2 × 2 n/2 = 2 n equivalent magic squares by combining such interchanges can be obtained. For odd square, there are (n - 1)/2 pairs of rows or columns that can be interchanged; and 2 n-1 equivalent magic squares obtained by combining such ...
Magic Square Lexicon: Illustrated (co-author Harvey Heinz), HDH, 2000, 0-9687985-0-0 Through his life, Hendricks published 53 articles and papers on magic squares and cubes, 14 articles on statistics, 15 articles on meteorology, 14 miscellaneous articles and 12 books.
The number zero for n = 6 is an example of a more general phenomenon: associative magic squares do not exist for values of n that are singly even (equal to 2 modulo 4). [3] Every associative magic square of even order forms a singular matrix, but associative magic squares of odd order can be singular or nonsingular. [4]
For example the following sequence can be used to form an order 3 magic square according to the Siamese method (9 boxes): 5, 10, 15, 20, 25, 30, 35, 40, 45 (the magic sum gives 75, for all rows, columns and diagonals). The magic sum in these cases will be the sum of the arithmetic progression used divided by the order of the magic square.
Since each 2 × 2 subsquare sums to the magic constant, 4 × 4 pandiagonal magic squares are most-perfect magic squares.In addition, the two numbers at the opposite corners of any 3 × 3 square add up to half the magic constant.
In their book, Kathleen Ollerenshaw and David S. Brée give a method of construction and enumeration of all most-perfect magic squares. They also show that there is a one-to-one correspondence between reversible squares and most-perfect magic squares. For n = 36, there are about 2.7 × 10 44 essentially different most-perfect magic squares.
Conway's LUX method for magic squares is an algorithm by John Horton Conway for creating magic squares of order 4n+2, where n is a natural number. Method
This page was last edited on 18 January 2024, at 22:36 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.
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