What is "magic square"

Square \Square\ (skw[^a]r), n. [OF. esquarre, esquierre, F. ['e]querre a carpenter's square (cf. It. squadra), fr. (assumed) LL. exquadrare to make square; L. ex + quadrus a square, fr. quattuor four. See Four, and cf. Quadrant, Squad, Squire a square.]

1. (Geom.)

   1. The corner, or angle, of a figure. [Obs.]

   2. A parallelogram having four equal sides and four right angles.

2. Hence, anything which is square, or nearly so; as:

   1. A square piece or fragment.

      He bolted his food down his capacious throat in squares of three inches.
      --Sir W. Scott.

   2. A pane of glass.

   3. (Print.) A certain number of lines, forming a portion of a column, nearly square; -- used chiefly in reckoning the prices of advertisements in newspapers.

   4. (Carp.) One hundred superficial feet.

3. An area of four sides, generally with houses on each side; sometimes, a solid block of houses; also, an open place or area for public use, as at the meeting or intersection of two or more streets.

   The statue of Alexander VII. stands in the large square of the town.
   --Addison.

4. (Mech. & Joinery) An instrument having at least one right angle and two or more straight edges, used to lay out or test square work. It is of several forms, as the T square, the carpenter's square, the try-square., etc.

5. Hence, a pattern or rule. [Obs.]

6. (Arith. & Alg.) The product of a number or quantity multiplied by itself; thus, 64 is the square of 8, for 8 [times] 8 = 64; the square of a + b is a^ 2 + 2ab + b^ 2.

7. Exact proportion; justness of workmanship and conduct; regularity; rule. [Obs.]

   They of Galatia [were] much more out of square.
   --Hooker.

   I have not kept my square.
   --Shak.

8. (Mil.) A body of troops formed in a square, esp. one formed to resist a charge of cavalry; a squadron. ``The brave squares of war.''
   --Shak.

9. Fig.: The relation of harmony, or exact agreement; equality; level.

   We live not on the square with such as these.
   --Dryden.

10. (Astrol.) The position of planets distant ninety degrees from each other; a quadrate. [Obs.]

11. The act of squaring, or quarreling; a quarrel. [R.]

12. The front of a woman's dress over the bosom, usually worked or embroidered. [Obs.] --Shak. fair and square in a fair, straightforward, and honest manner; justly; as, he beat me fair and square. Geometrical square. See Quadrat, n., 2. Hollow square (Mil.), a formation of troops in the shape of a square, each side consisting of four or five ranks, and the colors, officers, horses, etc., occupying the middle. Least square, Magic square, etc. See under Least, Magic, etc. On the square, or Upon the square,

    1. in an open, fair manner; honestly, or upon honor; justly. [Obs or Colloq.]

    2. at right angles.

       On the square with, or Upon the square with, upon equality with; even with.
       --Nares.

       To be all squares, to be all settled. [Colloq.]
       --Dickens.

       To be at square, to be in a state of quarreling. [Obs.]
       --Nares.

       To break no squares, to give no offense; to make no difference. [Obs.]

       To break squares, to depart from an accustomed order.

       To see how the squares go, to see how the game proceeds; -- a phrase taken from the game of chess, the chessboard being formed with squares. [Obs.]
       --L'Estrange.

Magic \Mag"ic\, Magical \Mag"ic*al\, a. [L. magicus, Gr. ?, fr. ?: cf. F. magique. See Magi.]

1. Pertaining to the hidden wisdom supposed to be possessed by the Magi; relating to the occult powers of nature, and the producing of effects by their agency.

2. Performed by, or proceeding from, occult and superhuman agencies; done by, or seemingly done by, enchantment or sorcery; as, a magical spell. Hence: Seemingly requiring more than human power; imposing or startling in performance; producing effects which seem supernatural or very extraordinary; having extraordinary properties; as, a magic lantern; a magic square or circle.

   The painter's magic skill.
   --Cowper.

   Note: Although with certain words magic is used more than magical, -- as, magic circle, magic square, magic wand, -- we may in general say magic or magical; as, a magic or magical effect; a magic or magical influence, etc. But when the adjective is predicative, magical, and not magic, is used; as, the effect was magical.

   Magic circle, a series of concentric circles containing the numbers 12 to 75 in eight radii, and having somewhat similar properties to the magic square.

   Magic humming bird (Zo["o]l.), a Mexican humming bird ( Iache magica), having white downy thing tufts.

   Magic lantern. See Lantern.

   Magic square, numbers so disposed in parallel and equal rows in the form of a square, that each row, taken vertically, horizontally, or diagonally, shall give the same sum, the same product, or an harmonical series, according as the numbers taken are in arithmetical, geometrical, or harmonical progression.

   Magic wand, a wand used by a magician in performing feats of magic.

n. 1 (context games English) A palindromic square word arrangement, usually in the form of a magic amulet. 2 An ''n''-by-''n'' arrangement of ''n''² numbers such that the numbers in each row, in each column and along both diagonals all have the same sum.

n. a square matrix of n rows and columns; the first n-squared integers are arranged in the cells of the matrix is such a way that the sum of any row or column or diagonal is the same

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, called the " magic constant." 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 numbers, and its size (the number of rows [and columns] it has) is described as being "of order n." A magic square that contains the integers from 1 to n 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) non-trivial 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 make up a single equivalence class.

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.