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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.
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. These squares are also called associated magic squares, regular magic squares, regmagic squares, or symmetric ...
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.
[5] [6] [7] However, if all the numbers are used and no player gets exactly 15, the game is a draw. [5] [6] The game is identical to tic-tac-toe, as can be seen by reference to a 3x3 magic square: if a player has selected three numbers which can be found in a line on a magic square, they will add up to 15. If they have selected any other 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 = +.
Sallows is an expert on the theory of magic squares [1] and has invented several variations on them, including alphamagic squares [2] [3] and geomagic squares. [4] The latter invention caught the attention of mathematician Peter Cameron who has said that he believes that "an even deeper structure may lie hidden beyond geomagic squares" [5]
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A geometric magic square, often abbreviated to geomagic square, is a generalization of magic squares invented by Lee Sallows in 2001. [1] A traditional magic square is a square array of numbers (almost always positive integers ) whose sum taken in any row, any column, or in either diagonal is the same target number .