A *magic square* is a by square grid filled with distinct integers such that each cell contains a different integer and the sum of the integers, called *magic number*, is equal in each row, column and diagonal. The order of a by magic square is .

The first known example of a magic square is said to have been found on the back of a turtle by Chinese Emperor Yu in 2200 B.C. (see Fig. 1) : circular dots represents the integers 1 through 9 are arranged in a by grid. Its order and magic number are and respectively.

Fig. 1

The first magic square to appear in the Western world was the one depicted in a copperplate engraving by the German artist-mathematician Albrecht Dürer, who also managed to enter the year of engraving – – in the two middle cells of the bottom row (see Fig. 2)

Fig. 2

Let be the magic number of a by magic square. Sum of all its rows (see “Little Bird and a Recursive Generator“)

.

That is,

Fig. 3

There is no by magic square. For if there is, it must be true (see Fig. 3) that

, contradicts “each cell contains a different integer”.

Shown in Fig. 4 is a square whose rows and columns add up to , but the diagonals do not, which makes the square only* semi-magic*.

Fig. 4

What’s interesting is that a chess Knight, starting its L-shaped moves from the cell marked “” can tour all squares in numerical order. The knight can reach the starting point from its finishing cell (marked”“) for a new round of tour.

*Exercise-1* Is it true that for every there exists at least one magic square?

*Exercise-2* Is there a Knight’s Tour* magic square*?

*Exercise-3* Fill in the missing numbers in the following square so that it is a magic square: