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In recreational mathematics, a magic square is an arrangement of distinct numbers (i.e. each number is used once), usually integers, in a square grid, where the numbers in each row, and in each column, and the numbers in the main and secondary diagonals, all add up to the same number. A magic square has the same number of rows as it has columns, and in conventional math notation, "''n''" stands for the number of rows (and columns) it has. Thus, a magic square always contains ''n''2 numbers, and its size (the number of rows (columns ) it has) is described as being "of order ''n''".〔"(Magic Square )" by Onkar Singh, Wolfram Demonstrations Project.〕 A magic square that contains the integers from 1 to ''n''2 is called a ''normal'' magic square. (The term "magic square" is also sometimes used to refer to any of various types of word squares.) Normal magic squares of all sizes except 2 × 2 (that is, where ''n'' = 2) can be constructed. The 1 × 1 magic square, with only one cell containing the number 1, is trivial. The smallest (and unique up to rotation and reflection) nontrivial case, 3 × 3, is shown below. Any magic square can be rotated and reflected to produce 8 trivially distinct squares. In magic square theory all of these are generally deemed equivalent and the eight such squares are said to comprise a single equivalence class.〔(The lost theorem, by Lee Sallows ) The Mathematical Intelligencer, Fall 1997, Volume 19, Issue 4, pp 51-54, Jan 09, 2009〕 The constant that is the sum of every row, column and diagonal is called the magic constant or magic sum, ''M''. Every normal magic square has a constant dependent on ''n'', calculated by the formula . For normal magic squares of order ''n'' = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS). Magic squares have a long history, dating back to 650 BC in China. At various times they have acquired magical 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. ==History== Magic squares were known to Chinese mathematicians as early as 650 BC, and explicitly given since 570 AD, and to Islamic mathematicians possibly as early as the seventh century AD. The first magic squares of order 5 and 6 appear in an encyclopedia from Baghdad ''circa'' 983, the ''Encyclopedia of the Brethren of Purity'' (''Rasa'il Ihkwan al-Safa''); simpler magic squares were known to several earlier Arab mathematicians.〔 Some of these squares were later used in conjunction with magic letters, as in (Shams Al-ma'arif), to assist Arab illusionists and magicians.〔The most famous Arabic book on magic, named "Shams Al-ma'arif ((アラビア語:كتاب شمس المعارف)), for Ahmed bin Ali Al-boni, who died about 1225 (622 AH). Reprinted in Beirut in 1985〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Magic square」の詳細全文を読む スポンサード リンク
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