III.

MODERN MODES OF CONSTRUCTION OF ODD-NUMBERED SQUARES.

The reader will justly ask whether there do not exist other correct magic squares which are constructed after a different method from that just given, and whether there do not exist modes of construction which will lead to all the imaginable and possible magic squares of a definite number of cells. A general mode of construction of this character was first given for odd-numbered squares by De la Hire, and recently perfected by Professor Scheffler.

To acquaint ourselves with this general method, let us select as our example a square of 5. First we form two auxiliary squares. In the first we write the numbers from 1 to 5 five times; and in the second, five times, the following multiples of five, viz.: 0, 5, 10, 15, 20. It is clear now that by adding each of the numbers of the series from 1 to 5 with each of the numbers 0, 5, 10, 15, 20, we shall get all the 25 numerals from 1 to 25. All that additionally remains to be done therefore, is, so to inscribe the numbers that by the addition of the two numbers in any two corresponding cells each combination shall come out once and only once; and further that in each horizontal, vertical, and diagonal row in each auxiliary square each number shall once appear. Then the required sum of 65 must necessarily result in every case, because the numbers from 1 to 5 added together make 15, and the numbers 0, 5, 10, 15, 20 make 50.

We effect the required method of inscription by imagining the numbers 1, 2, 3, 4, 5 (or 0, 5, 10, 15, 20) arranged in cyclical succession, that is 1 immediately following upon 5, and, starting from any number whatsoever, by skipping each time either none or one or two or three etc. figures. Cycles are thus obtained of the first, the second, the third etc. orders; for example 3 4 5 1 2 is a cycle of the first order, 2 4 1 3 5 is a cycle of the second order, 1 5 4 3 2 is a cycle of the fourth order, etc. The only thing then to be looked out for in the two auxiliary squares is, that the same “cycle” order be horizontally preserved in all the rows, that the same also happens for the vertical rows, but that the cycle order in the horizontal and vertical rows is different. Finally we have only additionally to take care that to the same numbers of the one auxiliary square not like numbers but _different_ numbers correspond in the other auxiliary square, that is lie in similarly situated cells. The following auxiliary squares are, for example, thus possible:

[Illustration: Fig. 8.

+--+--+--+--+--+ |3 |4 |5 |1 |2 | +--+--+--+--+--+ |5 |1 |2 |3 |4 | +--+--+--+--+--+ |2 |3 |4 |5 |1 | +--+--+--+--+--+ |4 |5 |1 |2 |3 | +--+--+--+--+--+ |1 |2 |3 |4 |5 | +--+--+--+--+--+ ]

and

[Illustration: Fig. 9.

+--+--+--+--+--+ | 0|10|20| 5|15| +--+--+--+--+--+ | 5|15| 0|10|20| +--+--+--+--+--+ |10|20| 5|15| 0| +--+--+--+--+--+ |15| 0|10|20| 5| +--+--+--+--+--+ |20| 5|15| 0|10| +--+--+--+--+--+ ]

Adding in pairs the numbers which occupy similarly situated cells, we obtain the following correct magic square:

[Illustration: Fig. 10.

+--+--+--+--+--+ | 3|14|25| 6|17| +--+--+--+--+--+ |10|16| 2|13|24| +--+--+--+--+--+ |12|23| 9|20| 1| +--+--+--+--+--+ |19| 5|11|22| 8| +--+--+--+--+--+ |21| 7|18| 4|15| +--+--+--+--+--+ ]

It will be seen that we are able thus to construct a very large number of magic squares of 5 times 5 spaces by varying in every possible manner the numbers in the two auxiliary squares. Furthermore, the squares thus formed possess the additional peculiarity, that every 5 numbers which fill out two rows that are parallel to a diagonal and lie on different sides of the diagonal also give the constant sum of 65. For example: 3 and 7, 11, 20, 24; or 10, 14 and 18, 22, 1. Altogether then the sum 65 is produced out of 20 rows or pairs of rows. On this peculiarity is dependent the fact that if we imagine an unlimited number of such squares placed by the side of, above, or beneath an initial one, we shall be able to obtain as many quadratic cells as we choose, so arranged that the square composed of any 25 of these cells will form a correct magic square, as the following figure will show:

[Illustration: Fig. 11.

2|13|24|10|16| 2|13|24|10|16| 2 --+--+--+--+--+--+--+--+--+--+-- 9|20| 1|12|23| 9|20| 1|12|23| 9 --+--+--+--+--+==============+-- 11|22| 8|19| 5¦11|22| 8|19| 5¦11 --+--+--+--+--¦--+--+--+--+--¦— 18| 4|15|21| 7¦18| 4|15|21| 7¦18 --+--+--+--+--¦--+--+--+--+--¦— 25| 6|17| 3|14¦25| 6|17| 3|14¦25 --+--+--+--+--¦--+--+--+--+--¦— 2|13|24|10|16¦ 2|13|24|10|16¦ 2 --+--+========¦=====+--+--+--¦— 9|20¦ 1|12|23¦ 9|20¦ 1|12|23¦ 9 --+--¦--+--+--+=====¦========+-- 11|22¦ 8|19| 5|11|22¦ 8|19| 5|11 --+--¦--+--+--+--+--¦--+--+--+-- 18| 4¦15|21| 7|18| 4¦15|21| 7|18 --+--¦--+--+--+--+--¦--+--+--+-- 25| 6¦17| 3|14|25| 6¦17| 3|14|25 --+--¦--+--+--+--+--¦--+--+--+-- 2|13¦24|10|16| 2|13¦24|10|16| 2 --+--+==============+--+--+--+-- 9|20| 1|12|23| 9|20| 1|12|23| 9 --+--+--+--+--+--+--+--+--+--+-- 11|22| 8|19| 5|11|22| 8|19| 5|11 ]

Every square of every 25 of these numbers, as for example the two dark-bordered ones, possesses the property that the addition of the horizontal, vertical, and diagonal rows gives each the same sum, 65.

As an example of a higher number of cells we will append here a magic square of 11 times 11 spaces formed by the general method of De la Hire from the two auxiliary squares of Figs. 12 and 13. From these two auxiliary squares we obtain by the addition of the two numbers of every two similarly situated cells, the magic square, exhibited in Diagram 14, in which each row gives the same sum 671.

[Illustration: Fig. 12.

+---+---+---+---+---+---+---+---+---+---+---+ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10| 11| +---+---+---+---+---+---+---+---+---+---+---+ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10| 11| 1 | 2 | +---+---+---+---+---+---+---+---+---+---+---+ | 5 | 6 | 7 | 8 | 9 | 10| 11| 1 | 2 | 3 | 4 | +---+---+---+---+---+---+---+---+---+---+---+ | 7 | 8 | 9 | 10| 11| 1 | 2 | 3 | 4 | 5 | 6 | +---+---+---+---+---+---+---+---+---+---+---+ | 9 | 10| 11| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | +---+---+---+---+---+---+---+---+---+---+---+ | 11| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10| +---+---+---+---+---+---+---+---+---+---+---+ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10| 11| 1 | +---+---+---+---+---+---+---+---+---+---+---+ | 4 | 5 | 6 | 7 | 8 | 9 | 10| 11| 1 | 2 | 3 | +---+---+---+---+---+---+---+---+---+---+---+ | 6 | 7 | 8 | 9 | 10| 11| 1 | 2 | 3 | 4 | 5 | +---+---+---+---+---+---+---+---+---+---+---+ | 8 | 9 | 10| 11| 1 | 2 | 3 | 4 | 5 | 6 | 7 | +---+---+---+---+---+---+---+---+---+---+---+ | 10| 11| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | +---+---+---+---+---+---+---+---+---+---+---+ ]

[Illustration: Fig. 13.

+---+---+---+---+---+---+---+---+---+---+---+ | 0 | 11| 22| 33| 44| 55| 66| 77| 88| 99|110| +---+---+---+---+---+---+---+---+---+---+---+ | 33| 44| 55| 66| 77| 88| 99|110| 0 | 11| 22| +---+---+---+---+---+---+---+---+---+---+---+ | 66| 77| 88| 99|110| 0 | 11| 22| 33| 44| 55| +---+---+---+---+---+---+---+---+---+---+---+ | 99|110| 0 | 11| 22| 33| 44| 55| 66| 77| 88| +---+---+---+---+---+---+---+---+---+---+---+ | 11| 22| 33| 44| 55| 66| 77| 88| 99|110| 0 | +---+---+---+---+---+---+---+---+---+---+---+ | 44| 55| 66| 77| 88| 99|110| 0 | 11| 22| 33| +---+---+---+---+---+---+---+---+---+---+---+ | 77| 88| 99|110| 0 | 11| 22| 33| 44| 55| 66| +---+---+---+---+---+---+---+---+---+---+---+ |110| 0 | 11| 22| 33| 44| 55| 66| 77| 88| 99| +---+---+---+---+---+---+---+---+---+---+---+ | 22| 33| 44| 55| 66| 77| 88| 99|110| 0 | 11| +---+---+---+---+---+---+---+---+---+---+---+ | 55| 66| 77| 88| 99|110| 0 | 11| 22| 33| 44| +---+---+---+---+---+---+---+---+---+---+---+ | 88| 99|110| 0 | 11| 22| 33| 44| 55| 66| 77| +---+---+---+---+---+---+---+---+---+---+---+ ]

[Illustration: Fig. 14.

+---+---+---+---+---+---+---+---+---+---+---+ | 1 | 13| 25| 37| 49| 61| 73| 85| 97|109|121| +---+---+---+---+---+---+---+---+---+---+---+ | 36| 48| 60| 72| 84| 96|108|120| 11| 12| 24| +---+---+---+---+---+---+---+---+---+---+---+ | 71| 83| 95|107|119| 10| 22| 23| 35| 47| 59| +---+---+---+---+---+---+---+---+---+---+---+ |106|118| 9 | 21| 33| 34| 46| 58| 70| 82| 94| +---+---+---+---+---+---+---+---+---+---+---+ | 20| 32| 44| 45| 57| 69| 81| 93|105|117| 8 | +---+---+---+---+---+---+---+---+---+---+---+ | 55| 56| 68| 80| 92|104|116| 7 | 19| 31| 43| +---+---+---+---+---+---+---+---+---+---+---+ | 79| 91|103|115| 6 | 18| 30| 42| 54| 66| 67| +---+---+---+---+---+---+---+---+---+---+---+ |114| 5 | 17| 29| 41| 53| 65| 77| 78| 90|102| +---+---+---+---+---+---+---+---+---+---+---+ | 28| 40| 52| 64| 76| 88| 89|101|113| 4 | 16| +---+---+---+---+---+---+---+---+---+---+---+ | 63| 75| 87| 99|100|112| 3 | 15| 27| 39| 51| +---+---+---+---+---+---+---+---+---+---+---+ | 98|110|111| 2 | 14| 26| 38| 50| 62| 74| 86| +---+---+---+---+---+---+---+---+---+---+---+ ]