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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]
As a running example, we consider a 10×10 magic square, where we have divided the square into four quarters. The quarter A contains a magic square of numbers from 1 to 25, B a magic square of numbers from 26 to 50, C a magic square of numbers from 51 to 75, and D a magic square of numbers from 76 to 100.
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.
Start by creating a (2n+1)-by-(2n+1) square array consisting of n+1 rows of Ls, 1 row of Us, and; n-1 rows of Xs, and then exchange the U in the middle with the L above it. Each letter represents a 2x2 block of numbers in the finished square.
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.
The 3×3 magic square in different orientations forming a non-normal 6×6 magic square, from an unidentified 19th century Indian manuscript. The 3×3 magic square first appears in India in Gargasamhita by Garga, who recommends its use to pacify the nine planets (navagraha). The oldest version of this text dates from 100 CE, but the passage on ...
The Brewers' 15-8 win over the Arizona Diamondbacks moved Milwaukee's magic number to clinch the National League Central Division to three.
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order n – that is, a magic square which contains the numbers 1, 2, ..., n 2 – the magic constant is = +.