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Magic squares are believed to exist for all orders, except for order 2. Magic squares can be classified according to their order as odd, doubly even (n divisible by four), and singly even (n even, but not divisible by four). This classification is based on the fact that entirely different techniques need to be employed to construct these ...
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]
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 = +.
An example of a 3 × 3 × 3 magic cube. In this example, no slice is a magic square. In this case, the cube is classed as a simple magic cube.. In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four ...
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.
Consequently, all 4 × 4 pandiagonal magic squares that are associative must have duplicate cells. All 4 × 4 pandiagonal magic squares using numbers 1-16 without duplicates are obtained by letting a equal 1; letting b, c, d, and e equal 1, 2, 4, and 8 in some order; and applying some translation.
A magic square is in the Frénicle standard form, named for Bernard Frénicle de Bessy, if the following two conditions hold: . the element at position [1,1] (top left corner) is the smallest of the four corner elements; and
One version of the traditional Chinese representation of the 3x3 "Lo Shu" magic square (see MagicSquare-LoShu.png). Date: 4 January 2010: Source: Own work - Made by self from scratch, following layout of PD image Luo4shu1.jpg. Author: AnonMoos: Other versions: See also Magic square Lo Shu.png: SVG development