The magic squares which we have so far considered contain only the
natural numbers from 1 upwards. It is possible, however, easily to
deduce from a correct magic square other squares in which a different
law controls the sequence of the numbers to be inscribed. Of the squares
obtained in this manner, we shall devote our attention here only to such
in which, although formed by the inscription of successive numbers, the
sum obtained from the addition of the rows is a determinate number which
we have fixed upon beforehand, as _the number of a year_. In such a case
we have simply to add to the numbers of the original square a determinate
number so to be calculated, that the required sum shall each time appear.
If this sum is divisible by 3, magic squares will always be obtainable
with 3 times 3 spaces which shall give this sum. In such a case we divide
the sum required by 3 and subtract 5 from the result in order to obtain
the number which we have to add to each number of the original square.
If the sum desired is even but not divisible by 4, we must then subtract
from it 34 and take one fourth of the result, to obtain the number which
in this case is to be added in each place. If, for example, we wish to
obtain the number of the year 1890 as the resulting sum of each row, we
shall have to add to each of the numbers of an ordinary magic square of
4 times 4 spaces the number 464; in other words, instead of the numbers
from 1 to 16 we have to insert in the squares the numbers from 465 to
480. As the number of the present year 1892 is divisible by 11, it must
be possible to deduce from the magic square constructed by us at the
conclusion of Section III a second magic square in which each row of 11
cells will give the number of the year 1892. To do this, we subtract from
1892 the sum of the original square, namely 671, and divide the remainder
by 11, whereby we get 111 and thus perceive that the numbers from 112 to
232 are to be inscribed in the cells of the square required. We get in
this way the preceding square, from which _one and the same sum, namely
1892, can be obtained 44 times_, first from each of the 11 horizontal
rows, secondly from each of the 11 vertical rows, thirdly from each of
the two diagonal rows, and fourthly twenty additional times from each
and every pair of any two rows that lie parallel to a diagonal, have
together 11 cells, and lie on different sides of the diagonal, as for
example, 196, 122, 158, 205, 131, 167, 214, 140, 187, 223, 149.
[Illustration: Fig. 24.
+----+----+----+----+----+----+----+----+----+----+----+
| 112| 124| 136| 148| 160| 172| 184| 196| 208| 220| 232| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 147| 159| 171| 183| 195| 207| 219| 231| 122| 123| 135| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 182| 194| 206| 218| 230| 121| 133| 134| 146| 158| 170| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 217| 229| 120| 132| 144| 145| 157| 169| 181| 193| 205| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 131| 143| 155| 156| 168| 180| 192| 204| 216| 228| 119| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 166| 167| 179| 191| 203| 215| 227| 118| 130| 142| 154| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 190| 202| 214| 226| 117| 129| 141| 153| 165| 177| 178| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 225| 116| 128| 140| 152| 164| 176| 188| 189| 201| 213| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 139| 151| 163| 175| 187| 199| 200| 212| 224| 115| 127| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 174| 186| 198| 210| 211| 223| 114| 126| 138| 150| 162| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 209| 221| 222| 113| 125| 137| 149| 161| 173| 185| 197| = 1892
+----+----+----+----+----+----+----+----+----+----+----+
1892 1892 1892 1892 1892 1892 1892 1892 1892 1892 1892
]