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MAGIC SQUARES HAVE LONG BEEN CONSIDERED A MATHEMATIcal recreation providing entertainment and an interesting outlet for creating mathematical knowledge. An nth-order magic square is a square array of n^sup 2^ distinct integers in which the sum of the n numbers in each row, column, and diagonal is the same. The magic lies in the fact that the numbers in each row, column, and diagonal always sum to the same number, called the magic constant. Figure 1 shows an example of a third-order magic square with a magic constant of 15.
A Brief History of Magic Squares
MAGIC SQUARES HAVE A RICH HISTORY DATING to around 2200 B.c. A Chinese myth claimed that while the Chinese Emperor Yi was walking along the Yellow River, he noticed a tortoise with a unique diagram on its shell (see fig. 2). The Emperor decided to call the unusual numerical pattern lo shu. The earliest magic square on record, however, appeared in the first-century book Da-Dai Liji.
Magic squares in China have been used in various areas of study, including astrology; divination; and the interpretation of philosophy, natural phenomena, and human behavior. Magic squares also permeated other areas of Chinese culture. For example, Chinese porcelain plates on display in museums and private collections were decorated with Arabic inscriptions and magic squares.
Magic squares most likely traveled from China to India, then to the Arab countries. From the Arab countries, magic squares journeyed to Europe, then to Japan. Magic squares in India served multiple purposes other than the dissemination of mathematical knowledge. For example, Varahamihira used a fourth-order magic square to specify recipes for making perfumes in his book on seeing into the future, Brhatsamhita (ca. 550 A.D.). The oldest dated third-order magic square in India appeared in Vrnda's medical work Siddhayoga (ca. 900 A.D.), as a means to ease childbirth.
little is known about the beginning of research on magic squares in Islamic mathematics. Treatises in the ninth and tenth centuries revealed that the mathematical properties of magic squares were already developed among what were then Islamic Arabic-speaking nations. Further, history suggests that the introduction of magic squares was entirely mathematical rather than magical. The ancient Arabic designation for magic squares, wafq ala'dad, means "harmonious disposition of the numbers."...