Mathematical_Recreations-Kraitchik-2e

160 Mathematical Recreations
column of the fundamental square - in any case, the one
that contains the central number of the fundamental square.
The semimagic square can then be made magic by permuting
the rows cyclically so as to bring this number, p2 ; 1 = 13,
into the main diagonal. The lower five rows of Figure
36 show the result of this operation. It will be seen also that
from this square we can get four other magic squares by per-
10 16 2 13 24
14 25 6 17 3
18 4 15 21 7
22 8 19 5 11
1 12 23 9 20
10 16 2 13 24
14 25 6 17 3
FIGURE 36.
muting cyclically both rows and columns so as to bring the
" 13" into every position along the downward main diagonal.
This shifting of rows and columns is simple enough for a
small square, but would be very laborious for a large one.
We can avoid the difficulty by ensuring at the start that the
p2 + 1
central number, --2-' of the fundamental square is on the
desired main diagonal. This can be done by properly choosing
the origin of the generating lattice. This will surely be
accomplished if the central cell (p ; 1, p ; 1) of the fundamental
square becomes the central cell of the new square.
For this we must have
+p-1t+p-1 p+ 1
a -2- -2- v - -2- modulo p,