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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 = +.
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.
For instance, the Lo Shu Square – the unique 3 × 3 magic square – is associative, because each pair of opposite points form a line of the square together with the center point, so the sum of the two opposite points equals the sum of a line minus the value of the center point regardless of which two opposite points are chosen. [4]
[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 ...
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.
The sums in each of the three rows, in each of the three columns, and in both diagonals, are all 15. [notes 1] Since "5" is in the center cell, the sum of any two other cells that are directly through the five from each other must be 10; e.g., opposite squares and corners add up to 10, the number of the Yellow River Map. [citation needed]
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The game is won by the player who has said three numbers whose sum is 15. [ 26 ] [ 28 ] If all the numbers are used and no one gets three numbers that add up to 15 then the game is a draw. [ 26 ] Plotting these numbers on a 3×3 magic square shows that the game exactly corresponds with tic-tac-toe, since three numbers will be arranged in a ...