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The smallest (and unique up to rotation and reflection) non-trivial case of a magic square, order 3. In mathematics, especially historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same.
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 associative magic square is a magic square for which each pair of numbers symmetrically opposite to the center sum up to the same value. For an n × n square, filled with the numbers from 1 to n 2 , this common sum must equal n 2 + 1.
A magic circle can be derived from one or more magic squares by putting a number at each intersection of a circle and a spoke. Additional spokes can be added by replicating the columns of the magic square. In the example in the figure, the following 4 × 4 most-perfect magic square was copied into the upper part of the magic circle. Each number ...
A most-perfect magic square of order n is a magic square containing the numbers 1 to n 2 with two additional properties: Each 2 × 2 subsquare sums to 2 s , where s = n 2 + 1. All pairs of integers distant n /2 along a (major) diagonal sum to s .
In their magic triangles, the sum of the k-th row and the (n-k+1)-th row is same for all k. [5] (sequence A356808 in the OEIS) Its one modification uses triangular numbers instead of square numbers. (sequence A355119 in the OEIS) Another magic triangle form is magic triangles with triangular numbers with different summation. In this magic ...
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.
If we build a pandiagonal magic square with this algorithm then every square in the square will have the same sum. Therefore, many symmetric patterns of 4 n {\displaystyle 4n} cells have the same sum as any row and any column of the 4 n × 4 n {\displaystyle 4n\times 4n} square.