
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:
