Of magic squares having an even number of places we have hitherto
had to deal only with the square of 4. To construct squares of this
description having a higher even number of places, different and more
complicated methods must be employed than for squares of odd numbers
of places. However, in this case also, as in dealing with the square
of 4, we start with the natural sequence of the numbers and must then
find the complements of the numbers with respect to some other certain
number (as 17 in the square of 4) and also effect certain exchanges of
the numbers with one another. To form, for example, a magic square of 6
times 6 places, we inscribe in the 12 diagonal cells the numbers that in
the natural sequence of inscription fall into these places, then in the
remaining cells the complements of the numbers that belong therein with
respect to 37, and finally effect the following six exchanges, viz. of
the numbers 33 and 3, 25 and 7, 20 and 14, 18 and 13, 10 and 9, and 5 and
2. In this way the following magic square is obtained.
[Illustration: Fig. 15.
+--+--+--+--+--+--+
| 1|35|34| 3|32| 6|
+--+--+--+--+--+--+
|30| 8|28|27|11| 7|
+--+--+--+--+--+--+
|24|23|15|16|14|19|
+--+--+--+--+--+--+
|13|17|21|22|20|18|
+--+--+--+--+--+--+
|12|26| 9|10|29|25|
+--+--+--+--+--+--+
|31| 2| 4|33| 5|36|
+--+--+--+--+--+--+
]
This square may also be constructed by the method of De la Hire, from two
auxiliary squares with the numbers 1, 2, 3, 4, 5, 6 and 0, 6, 12, 18,
24, 30 respectively. In this case, however, the vertical rows of the one
square and the horizontal rows of the other must each so contain two same
numbers thrice repeated that the summation shall always remain 21 and 90
respectively. In this manner we get the magic square last given above
from the two following auxiliary squares:
[Illustration: Fig. 16.
+--+--+--+--+--+--+
| 1| 5| 4| 3| 2| 6|
+--+--+--+--+--+--+
| 6| 2| 4| 3| 5| 1|
+--+--+--+--+--+--+
| 6| 5| 3| 4| 2| 1|
+--+--+--+--+--+--+
| 1| 5| 3| 4| 2| 6|
+--+--+--+--+--+--+
| 6| 2| 3| 4| 5| 1|
+--+--+--+--+--+--+
| 1| 2| 4| 3| 5| 6|
+--+--+--+--+--+--+
]
and
[Illustration: Fig. 17.
+--+--+--+--+--+--+
| 0|30|30| 0|30| 0|
+--+--+--+--+--+--+
|24| 6|24|24| 6| 6|
+--+--+--+--+--+--+
|18|18|12|12|12|18|
+--+--+--+--+--+--+
|12|12|18|18|18|12|
+--+--+--+--+--+--+
| 6|24| 6| 6|24|24|
+--+--+--+--+--+--+
|30| 0| 0|30| 0|30|
+--+--+--+--+--+--+
]
It is to be noted in connection with this example that here also as in
the case of odd-numbered squares, it is possible so to inscribe six
times the numbers from 1 to 6 that each number shall appear once and only
once in each horizontal, vertical, and diagonal row; for example, in the
following manner:
[Illustration: Fig. 18.
+--+--+--+--+--+--+
| 1| 2| 3| 4| 5| 6|
+--+--+--+--+--+--+
| 2| 4| 6| 1| 3| 5|
+--+--+--+--+--+--+
| 3| 6| 5| 2| 1| 4|
+--+--+--+--+--+--+
| 5| 3| 1| 6| 4| 2|
+--+--+--+--+--+--+
| 6| 5| 4| 3| 2| 1|
+--+--+--+--+--+--+
| 4| 1| 2| 5| 6| 3|
+--+--+--+--+--+--+
]
But if we attempt so to insert, in a like manner, the other set of
numbers 0, 6, 12, 18, 24, 30 in a second auxiliary square, that each
number of the first auxiliary square shall stand once and once only in
a corresponding cell with each number of the second square, all the
attempts we may make to fulfil coincidently the last named condition will
result in failure. It is therefore necessary to select auxiliary squares
like the two given above. It is noteworthy, that the fulfilment of the
second condition is impossible only in the case of the square of 6, but
that in the case of the square of 4 or of the square of 8, for example,
two auxiliary squares, such as the method of De la Hire requires, are
possible. Thus, taking the square of 4 we get
[Illustration: Fig. 19.
+--+--+--+--+
| 1| 2| 3| 4|
+--+--+--+--+
| 4| 3| 2| 1|
+--+--+--+--+
| 2| 1| 4| 3|
+--+--+--+--+
| 3| 4| 1| 2|
+--+--+--+--+
]
and
[Illustration: Fig. 20.
+--+--+--+--+
| 0| 4| 8|12|
+--+--+--+--+
| 8|12| 0| 4|
+--+--+--+--+
|12| 8| 4| 0|
+--+--+--+--+
| 4| 0|12| 8|
+--+--+--+--+
]
The reader may form for himself the magic square which these give.
The existence of these two auxiliary squares furnishes a key to the
solution of a pretty problem at cards. If we replace, namely, the
numbers 1, 2, 3, 4 by the Ace, the King, the Queen, and the Knave, and
the numbers 0, 4, 8, 12 by the four suits, clubs, spades, hearts, and
diamonds, we shall at once perceive that it is possible, and must be so
necessarily, quadratically to arrange in such a manner the four Aces,
the four Kings, the Four Queens, and the four Knaves, that in each
horizontal, vertical, and diagonal row, each one of the four suits and
each one of the four denominations shall appear once and once only. The
auxiliary squares above given furnish the appended solution of this
problem:
[Illustration: Fig. 21.
+--------+--------+--------+--------+
| CLUBS | SPADES | HEARTS |DIAMONDS|
| ACE | KING | QUEEN | KNAVE |
+--------+--------+--------+--------+
| HEARTS |DIAMONDS| CLUBS | SPADES |
| KNAVE | QUEEN | KING | ACE |
+--------+--------+--------+--------+
|DIAMONDS| HEARTS | SPADES | CLUBS |
| KING | ACE | KNAVE | QUEEN |
+--------+--------+--------+--------+
| SPADES | CLUBS |DIAMONDS| HEARTS |
| QUEEN | KNAVE | ACE | KING |
+--------+--------+--------+--------+
]
To fix the solution of the problem in the memory, observe that, starting
from the several corners, each suit and each denomination must be placed
in the spots of the move of a Knight. If we fix the positions of the four
cards of any one row, there will be only two possibilities left of so
placing the other cards that the required condition of having each suit
and each denomination once and only once in each row shall be fulfilled.
Of magic squares of an even number of places we have up to this point
examined only the squares of 4 and of 6. For the sake of completeness we
append here one of 8 and one of 10 places. The mode of construction of
these squares is similar to the method above discussed for the lower even
numbers.
[Illustration: Fig. 22.
+--+--+--+--+--+--+--+--+
| 1|63|62| 4| 5|59|58| 8|
+--+--+--+--+--+--+--+--+
|56|10|11|53|52|14|15|49|
+--+--+--+--+--+--+--+--+
|48|18|19|45|44|22|23|41|
+--+--+--+--+--+--+--+--+
|25|39|38|28|29|35|34|32|
+--+--+--+--+--+--+--+--+
|33|31|30|36|37|27|26|40|
+--+--+--+--+--+--+--+--+
|24|42|43|21|20|46|47|17|
+--+--+--+--+--+--+--+--+
|16|50|51|13|12|54|55| 9|
+--+--+--+--+--+--+--+--+
|57| 7| 6|60|61| 3| 2|64|
+--+--+--+--+--+--+--+--+
]
[Illustration: Fig. 23.
+---+---+---+---+---+---+---+---+---+---+
| 1 | 99| 3 | 97| 96| 5 | 94| 8 | 92| 10|
+---+---+---+---+---+---+---+---+---+---+
| 90| 12| 88| 14| 86| 85| 17| 83| 19| 11|
+---+---+---+---+---+---+---+---+---+---+
| 80| 79| 23| 77| 25| 26| 74| 28| 22| 71|
+---+---+---+---+---+---+---+---+---+---+
| 31| 69| 68| 34| 66| 65| 37| 33| 62| 40|
+---+---+---+---+---+---+---+---+---+---+
| 60| 42| 58| 57| 45| 46| 44| 53| 49| 51|
+---+---+---+---+---+---+---+---+---+---+
| 50| 52| 43| 47| 55| 56| 54| 48| 59| 41|
+---+---+---+---+---+---+---+---+---+---+
| 61| 32| 38| 64| 36| 35| 67| 63| 39| 70|
+---+---+---+---+---+---+---+---+---+---+
| 21| 29| 73| 27| 75| 76| 24| 78| 72| 30|
+---+---+---+---+---+---+---+---+---+---+
| 20| 82| 18| 84| 15| 16| 87| 13| 89| 81|
+---+---+---+---+---+---+---+---+---+---+
| 91| 9 | 93| 4 | 6 | 95| 7 | 98| 2 |100|
+---+---+---+---+---+---+---+---+---+---+
]
The magic squares of even numbers thus constructed are not the only
possible ones. On the contrary, there are very many others possible,
which obey different laws of formation. It has been calculated, for
example, that with the square of 4 it is possible to construct 880, and
with the square of 6, _several million_, different magic squares. The
number of odd-numbered magic squares constructible by the method of De la
Hire is also very great. With the square of 7, the possible constructions
amount to 363,916,800. With the squares of higher numbers the multitude
of the possibilities increases in the same enormous ratio.