Chapter 10 of 34 · 1320 words · ~7 min read

IV.

EVEN-NUMBERED SQUARES.

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.

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.