About 幻方

幻方，有時又稱魔方，由一組排放在正方形中的整數組成，其每行、每列以及兩條對角線上的數之和均相等。通常幻方由從1到N2的連續整數組成，其中N為正方形的行或列的數目。因此N階幻方有N行N列，並且所填充的數字為從1到N2。
幻方可以使用N階方陣來表示，矩陣的每行、每列以及兩條對角線的和都等於常數M2(N)，如果填充數為
圖片參考：http://upload.wikimedia.org/math/2/9/3/293fe678a070641f189afe1b30ed5a3b.png
，那麼有


圖片參考：http://upload.wikimedia.org/math/c/1/6/c1613b32735074e14013e5c9f1e5550f.png

In recreational mathematics, a magic square of order n is an arrangement of n² numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n². The term "magic square" is also sometimes used to refer to any of various types of word square.
Normal magic squares exist for all orders n ≥ 1 except n = 2, although the case n = 1 is trivial—it consists of a single cell containing the number 1. The smallest nontrivial case, shown below, is of order 3.

圖片參考：http://upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Magicsquareexample.svg/180px-Magicsquareexample.svg.png

The constant sum in every row, column and diagonal is called the magic sum or magic constant, M. The magic constant of a normal magic square depends only on n and has the value


圖片參考：http://upload.wikimedia.org/math/1/3/6/13652227c2537f1018ce4d011d792db1.png

For normal magic squares of order n = 3, 4, 5, …, the magic constants are:

15, 34, 65, 111, 175, 260, … (sequence A006003 in OEIS)

"幻方" = "Magic Square" in English:

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In recreational mathematics, a magic square of order n is an arrangement of n² numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n². The term "magic square" is also sometimes used to refer to any of various types of word square.
Normal magic squares exist for all orders n ≥ 1 except n = 2, although the case n = 1 is trivial—it consists of a single cell containing the number 1. The smallest nontrivial case, shown below, is of order 3.

The constant sum in every row, column and diagonal is called the magic sum or magic constant, M. The magic constant of a normal magic square depends only on n and has the value

The middle number can be found by

where n is the order of the square. For normal magic squares of order n = 3, 4, 5, …, the magic constants are:
15, 34, 65, 111, 175, 260, … (sequence A006003 in OEIS)

A magic square of order 4

Types of magic squares and their construction
There are many ways to construct magic squares, but the standard (and most simple) way is to follow certain configurations/formulas which generate regular patterns. Magic squares exist for all values of n, with only one exception - it is impossible to construct a magic square of order 2. Magic squares can be classified into three types: odd, doubly even (n divisible by four) and singly even (n even, but not divisible by four). Odd and doubly even magic squares are easy to generate; the construction of singly even magic squares is more difficult but several methods exist, including the LUX method for magic squares (due to John Horton Conway) and the Strachey method for magic squares. Only odd and doubly even magic squares are discussed below.
A method for constructing a magic square of odd order
Starting from the central column of the last row with the number 1, the fundamental movement for filling the squares is diagonally down and right, one step at a time. If a filled square is encountered, one moves vertically up one square instead, then continuing as before. When a move would leave the square, it is wrapped around to the first row or last column, respectively.
The same pattern can be achieved starting from the central column of the first row; In this case the fundamental movement is diagonally up and right, one step at a time, and if a filled square is encountered, one moves vertically down one square instead, then continuing as before. When a move would leave the square, it is wrapped around the last row or first column, respectively.
Similar patterns can also be obtained by starting from other squares.
Order 3
816
357
492
Order 5
17241815
23571416
46132022
101219213
11182529
Order 9
47586980112233445
57687991122334446
67788102132435456
77718203142535566
61719304152636576
16272940516264755
26283950617274415
36384960717331425
37485970812132435

A method of constructing a magic square of doubly even order
All the numbers are written in order from right to left across each row in turn, starting from the top right hand corner. Numbers are then either retained in the same place or interchanged with their diametrically opposite numbers in a certain regular pattern. In the magic square of order four, the numbers in the four central squares and one square at each corner are retained in the same place and the others are interchanged with their diametrically opposite numbers.