每行、每列、两条对角线的和都是N。
- a+b+c + d+e+f + h+i+j = 3N 三行的和
- a+e+j + c+e+h = 2N 两条对角线的和
上面两式相减得 b+d + f+i - e = N → b+i + d+f - e = N → N-e + N-e - e = N → 3e = N
In 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 order of the magic square is the number of integers along one side, and the constant sum is called the magic constant. If the array includes just the positive integers 1,2,...,n2, the magic square is said to be normal. Some authors take magic square to mean normal magic square. Magic squares that include repeated entries do not fall under this definition and are referred to as trivial.
The mathematical study of magic squares typically deals with its construction, classification, and enumeration. Although completely general methods for producing all the magic squares of all orders do not exist, historically three general techniques have been discovered: by bordering method, by making composite magic squares, and by adding two preliminary squares.
Magic squares have a long history, dating back to at least 190 BCE in China. At various times they have acquired occult [玄奥的] or mythical significance, and have appeared as symbols in works of art. In modern times they have been generalized a number of ways, including using extra or different constraints, multiplying instead of adding cells, using alternate shapes or more than two dimensions, and replacing numbers with shapes and addition with geometric operations.
The third order magic square was known to Chinese mathematicians as early as 190 BCE, and explicitly given by the first century of the common era. The first dateable instance of the fourth-order magic square occur in 587 CE in India. Specimens of magic squares of order 3 to 9 appear in an encyclopedia from Baghdad [巴格达] c. [about] 983, the Encyclopedia of the Brethren [教友] of Purity (Rasa'il Ikhwan al-Safa). By the end of 12th century, the general methods for constructing magic squares were well established.
The Brethren of Purity (Arabic: إخوان الصفا) were a secret society of Muslim philosophers in Basra [巴士拉], Iraq, in the 8th or 10th century CE.
While ancient references to the pattern of even and odd numbers in the 3×3 magic square appears in the I Ching [易经], the first unequivocal [completely clear and without any possibility of doubt]instance of this magic square appears in the chapter called Mingtang (Bright Hall) of a 1st-century book Da Dai Liji (Record of Rites by the Elder Dai 大戴礼·明堂篇), which purported [大意] to describe ancient Chinese rites of the Zhou dynasty. 西汉宣帝时的博士戴德
These numbers also occur in a possibly earlier mathematical text called Shushu jiyi [数术记遗] (Memoir on Some Traditions of Mathematical Art), said to be written in 190 BCE. This is the earliest appearance of a magic square on record; and it was mainly used for divination and astrology. The 3×3 magic square was referred to as the "Nine Halls" [九宫格] by earlier Chinese mathematicians. The identification of the 3×3 magic square to the legendary Luoshu chart [洛书] was only made in the 12th century, after which it was referred to as the Luoshu square. The oldest surviving Chinese treatise [专题论文] that displays magic squares of order larger than 3 is Yang Hui's Xugu Zhaiqi Suanfa [续古摘奇算法] (Continuation of Ancient Mathematical Methods for Elucidating [阐明] the Strange) written in 1275. The contents of Yang Hui's treatise were collected from older works, both native and foreign; and he only explains the construction of third and fourth-order magic squares, while merely passing on the finished diagrams of larger squares. He gives a magic square of order 3, two squares for each order of 4 to 8, one of order nine, and one semi-magic square of order 10. He also gives six magic circles of varying complexity.
杨辉不是刘徽。
After Yang Hui, magic squares frequently occur in Chinese mathematics such as in Ding Yidong's Dayan suoyin (c. 1300), Cheng Dawei's Suanfa tongzong (1593) [程大位的算法统宗], Fang Zhongtong's Shuduyan (1661) which contains magic circles, cubes and spheres, Zhang Chao's Xinzhai zazu (c. 1650), who published China's first magic square of order ten, and lastly Bao Qishou's Binaishanfang ji (c. 1880), who gave various three dimensional magic configurations.
惭愧，我不知道这些拼音对应的汉字。我倒是知道墨子不是法师，荆轲是个男的。
However, despite being the first to discover the magic squares and getting a head start by several centuries, the Chinese development of the magic squares are much inferior compared to the Indian, Middle Eastern, or European developments. The high point of Chinese mathematics that deals with the magic squares seems to be contained in the work of Yang Hui; but even as a collection of older methods, this work is much more primitive, lacking general methods for constructing magic squares of any order, compared to a similar collection written around the same time by the Byzantine scholar Manuel Moschopoulos. This is possibly because of the Chinese scholars' enthralment [沉迷] with the Lo Shu principle, which they tried to adapt to solve higher squares; and after Yang Hui and the fall of Yuan dynasty, their systematic purging of the foreign influences in Chinese mathematics.
六级/考研单词: mathematics, array, magic, norm, author, trivial, construct, composite, preliminary, alternate, dimension, geometry, tertiary, implicit, era, specimen, encyclopedia, elder, dynasty, legend, indigenous, mere, diagram, seldom, cube, sphere, configuration, despite, inferior, primitive, scholar
中国古代数学 | 中国是世界数学之源 | 洛书 | 九宫格 | The History of the Chinese Reminder Theorem
标签：square,magic,Chinese,幻方,Hui,squares,order From： https://www.cnblogs.com/funwithwords/p/16583643.html