Mathematical_Recreations-Kraitchik-2e

Magic Squares 183
Figure 58. If (Figure 59) we form from this a new array by
interchanging each sum (not on a main diagonal) and the
sum symmetric to it with respect to the center, we get a
square that is magic if A + D = B + C and a + d = b + c.
If in this new array we interchange the upper right and lower
left quadrants (corner squares of order 2), there results a
square that is magic for all values of the letters.
FIGURE 59.
It is easy to verify that if we arrange the numbers in order
of size so that a < b < c < d and A < B < C < D, and if
we take A = 0, then there are just three choices of sets of
values for the letters so that the entries in the addition table
will use the numbers from 1 to 16 without repetition or omission.
These three choices yield the three tables in Figure 60.
Permuting the rows and columns of these three tables gives
3·2(4')2
us 8· = 432 nonequivalent magic squares; 48 of these
are panmagic.
3. MAGIC SERIES. There are 86 magic series of order 4.
These are of three kinds, which we shall call algebraic,
symmetric, and arithmetical.
Algebraic magic series are those formed by taking an ele-