MAGIC SQUARES 147

vary according to whether the number of integers used is
odd or even.

A square, not greatly different from the first one shown
above, may be built up as follows. First, it is necessary to con-
struct the framework of square, sub-squares and diagonals,
as in Fig. (i).

  

(1) (i) (i)

Next, write in the integers in their normal order from
left to right in horizontal rows, as in Fig. (ii). In the final
stage (Fig. iii), integers in sub-squares which are crossed
by diagonals change places with their diametrically opposite
integers on the same diagonals (e.g. the integers 13 and 4
change places), but the integers in the other sub-squares
remain unaltered. Fig. (iii) is then a magic square.

If the number of integers in a magic square is to be odd,
the square may be constructed in the following manner:

(a) Write the first number in the middle sub-square of the
first row.

(b) Write the second number in the following column, but
in the bottom row, thus:

 

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