The acuteness of the mathematicians has also discovered magic squares
which possess the peculiar property that if one row after another be
taken away from each side, the smaller inner squares remaining will
still be magical squares, that is to say, all their rows when added
will give the same sum. It will be sufficient to give two examples here
of such squares, (the laws for their construction being somewhat more
complicated,) of which the first has 7 times 7 and the second 8 times 8
places. The numbers within each of the dark-bordered frames form with
respect to the centre smaller squares which in their own turn are magical.
[Illustration: Fig. 25.
+--+---+---+--+--+---+---+
| 4| 5| 6 |43|39| 38| 40|
+--++================++--+
|49||15| 16|33|30| 31|| 1|
+--||--++========++--||--+
|48||37||22|27|26||13|| 2|
+--||--||--+--+--||--||--+
|47||36||29|25|21||14|| 3|
+--||--||--+--+--||--||--+
| 8||18||24|23|28||32||42|
+--||--++========++--||--+
| 9||19| 34|17|20| 35||41|
+--++================++--+
|10| 45| 44| 7|11| 12| 46|
+--+---+---+--+--+---+---+
]
[Illustration: Fig. 26.
+---+---+---+---+---+---+---+---+
| 1 | 56| 55| 11| 53| 13| 14| 57|
+---++=====================++---+
| 63||15| 47| 22| 42| 24|45|| 2 |
+---||--+==============++--||---+
| 62||49||25| 40| 34|31||16|| 3 |
+---||--||--+---+---+--||--||---+
| 4 ||48||28| 37| 35|30||17|| 61|
+---||--||--+---+---+--||--||---+
| 5 ||44||39| 26| 32|33||21|| 60|
+---||--||--+---+---+--||--||---+
| 59||19||38| 27| 29|36||46|| 6 |
+---||--++=============++--||---+
| 58||20| 18| 43| 23| 41|50|| 7 |
+---++=====================++---+
| 8 | 9 | 10| 54| 12| 52| 51| 64|
+---+---+---+---+---+---+---+---+
]
In the first of these two squares the internal square of 3 times 3 places
contains the numbers from 21 to 29 in such a manner that each row gives
when added the sum of 75. This square lies within a larger one of 5 times
5 spaces, which contains the numbers from 13 to 37 in such a manner that
each row gives the sum of 125. Finally, this last square forms part of
a square of 7 times 7 places which contains the numbers from 1 to 49 so
that each row gives the sum of 175.
In the second square the inner central square of 4 times 4 places
contains the numbers from 25 to 40 in such a manner that each row gives
the sum of 130. This square is the middle of a square of 6 times 6 places
which so contains the numbers from 15 to 50 that each row gives the sum
165. Finally, this last square is again the middle of an ordinary magic
square composed of the numbers from 1 to 64.