Mathematical_Recreations-Kraitchik-2e

Magic Squares 143
fore we can readily change a magic square containing fractions
into one containing only integers by multiplying every
number by a suitable constant. Hence we shall admit only
positive integers.
Usually magic squares are required to be formed from the
first n2 natural numbers, which we may call the nornwl case.
We shall have so little occasion to deal with other types that
we shall ordinarily omit the specification "normal." In this
normal case the constant sum of the horizontal, vertical and
main diagonal lines is !n(n2 + 1), and is called the magic constant.
Any set of n distinct numbers from 1 to n2 whose sum
is the nth magic constant is called a magic series.
The number n of rows (and of columns) of the square is
called its order. The rows and columns will be called orthogonals.
The term diagonal will be generalized as follows: A
line from the upper left to the lower right corner of the square
cuts across the numbers forming one of the main diagonals,
consisting of just one number from each row and from each
column. H we draw lines parallel to this line through the
pth element from the bottom in the first column from the
left, and through the pth element from left in the top row.
the two lines together cross n elements, one from each row
and from each column, and these will be said to form a broken
diagonal. These are called the downward diagonals. The
upward diagonals are similarly defined. In Figure 21, the
broken upward diagonals are (6, 16, 11, 1), (9, 5, 8, 12)
and (4, 10, 13, 7). A square of order n has just n upward
and n downward diagonals.
A square that fails to be magic only because one or both
of the main diagonal sums differs from the orthogonal sums
will be called semimagic. On the other hand there are squares,
called panmagic, in which all diagonal sums are equal to all
orthogonal sums.
A magic square is called bimagic if the square formed by