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According to the legend, thereafter people were able to use this pattern in a certain way to control the river and protect themselves from floods.The Lo Shu Square, as the magic square on the turtle shell is called, is the unique normal magic square of order three in which 1 is at the bottom and 2 is in the upper right corner.For the 6×6 case, there are estimated to be approximately 1.8 × 10 The moment of inertia of a magic square has been defined as the sum over all cells of the number in the cell times the squared distance from the center of the cell to the center of the square; here the unit of measurement is the width of one cell.

There are also references to the use of magic squares in astrological calculations, a practice that seems to have originated with the Arabs.

The 3×3 magic square has been a part of rituals in India since Vedic times, and still is today. There is a well-known 10th-century 4×4 magic square on display in the Parshvanath temple in Khajuraho, India. Each row, column, and diagonal, as well as each 2×2 sub-square, the corners of each 3×3 and 4×4 square, the corners of each 2×4 and 4×2 rectangle, and the offset diagonals (12 8 5 9, 1 11 16 6, 14 2 3 15 and 7 11 10 6, 12 2 5 15, 1 13 16 4) sum to 34.

Every normal magic square of order three is obtained from the Lo Shu by rotation or reflection.

The 10th-century Persian mathematician Buzjani, for example, left a manuscript that on page 33 contains a series of magic squares, filled by numbers in arithmetical progression, in such a way that the sums of each row, column and diagonal are equal.

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.

Excluding rotations and reflections, there is exactly one 3×3 magic square, exactly 880 4×4 magic squares, and exactly 275,305,224 5×5 magic squares.

For the order-3 case the moment of inertia is always 60, while for the order-4 case the moment of inertia is always 340.

In general, for the n×n case the moment of inertia is and to Islamic mathematicians possibly as early as the seventh century AD.

Normal magic squares of all sizes can be constructed except 2×2 (that is, where order n = 2).

Any magic square can be rotated and reflected to produce 8 trivially distinct squares.

According to the legend, there was at one time in ancient China a huge flood.

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