II.

EARLY METHODS FOR THE CONSTRUCTION OF ODD-NUMBERED SQUARES.

Since early times rules have also been known for the construction of magic squares of more than 3 times 3, or 4 times 4 spaces. In the first place, it is easy to calculate the sum which in the case of any given number of cells must result from the addition of each row. We take the determinate number of cells in each side of the square which we have to fill, multiply that number by itself, add 1, again multiply the number thus obtained by the number of the cells in each side, and, finally, divide the product by 2. Thus, with 4 times 4 cells or squares, we get: 4 times 4 are 16, 16 and 1 are 17, and one half of 17 times 4 is 34. Similarly, with 5 times 5 squares, we get: 5 times 5 are 25, and 1 makes 26, and the half of 26 times 5 is 65. Analogously, for 6 times 6 squares the summation 111 is obtained, for 7 times 7 squares 175, for 8 times 8 squares 260, for 9 times 9 squares 369, for 10 times 10 squares 505, and so on. The Hindu rule for the construction of magic squares whose roots are odd, may be enunciated as follows: To start with, write 1 in the centre of the topmost row, then write 2 in the lowest space of the vertical column next adjacent to the right, and then so inscribe the remaining numbers in their natural order in the squares diagonally upwards towards the right, that on reaching the right-hand margin the inscription shall be continued from the left-hand margin in the row just above, and on reaching the upper margin shall be continued from the lower margin in the column next adjacent to the right, noting that whenever we are arrested in our progress by a square already occupied we are to fill out the square next beneath the one we have last filled. In this manner, for example, the last preceding square of 7 times 7 cells is formed, in which the reader is requested to follow the numbers in their natural sequence (Fig. 5).

[Illustration: Fig. 5.

=175 =175 \ / +---+---+---+---+---+---+---+ |30 |39 |48 | 1 |10 |19 |28 |=175 +---+---+---+---+---+---+---+ |38 |47 | 7 | 9 |18 |27 |29 |=175 +---+---+---+---+---+---+---+ |46 | 6 | 8 |17 |26 |35 |37 |=175 +---+---+---+---+---+---+---+ | 5 |14 |16 |25 |34 |36 |45 |=175 +---+---+---+---+---+---+---+ |13 |15 |24 |33 |42 |44 | 4 |=175 +---+---+---+---+---+---+---+ |21 |23 |32 |41 |43 | 3 |12 |=175 +---+---+---+---+---+---+---+ |22 |31 |40 |49 | 2 |11 |20 |=175 +---+---+---+---+---+---+---+ 175 175 175 175 175 175 175 ]

For the next further advancements of the theory of magic squares and of the methods for their construction we are indebted to the Byzantian Greek, Moschopulus, who lived in the fourteenth century; also, after Albrecht Dürer who lived about the year 1500, to the celebrated arithmetician Adam Riese, and to the mathematician Michael Stifel, which two last lived about 1550. In the seventeenth century Bachet de Méziriac, and Athanasius Kircher employed themselves on magic squares. About 1700, finally, the French mathematicians De la Hire and Sauveur made considerable contributions to the theory. In recent times mathematicians have concerned themselves much less about magic squares, as they have indeed about mathematical recreations generally. But quite recently the Brunswick mathematician Scheffler has put forth his own and other’s studies on this subject in an elegant form.

[Illustration: Fig. 6.

| 7| | | | 6| |14| | | | | | 5| |13| |21| +==+==+==+==+==+==+==+ | 4| |12| |20| |28| ---|--+--+--+--+--+--+--|--- 3| |11| |19| |27| |35 ------|--+--+--+--+--+--+--|------ 2 |10| |18| |26| |34| 42 --------|--+--+--+--+--+--+--|--------- 1 9| |17| |25| |33| |41 49 --------|--+--+--+--+--+--+--|--------- 8 |16| |24| |32| |40| 48 ------|--+--+--+--+--+--+--|------ 15| |23| |31| |39| |47 ---|--+--+--+--+--+--+--|--- |22| |30| |38| |46| +==+==+==+==+==+==+==+ |29| |37| |45| | | | | |36| |44| | | |43| ]

The best known of the various methods of constructing magic squares of an odd number of cells is the following. First write the numbers in diagonal succession as in the preceding diagram (Fig. 6). After 25 cells of the square of 49 cells which we have to fill out, have thus been occupied, transfer the six figures found outside each side of the square, without changing their configuration, into the empty cells of the side directly opposite. By this method, which we owe to Bachet de Méziriac, we obtain the following magic square of the numbers from 1 to 49:

[Illustration: Fig. 7.

+--+--+--+--+--+--+--+ | 4|29|12|37|20|45|28| +--+--+--+--+--+--+--+ |35|11|36|19|44|27| 3| +--+--+--+--+--+--+--+ |10|42|18|43|26| 2|34| +--+--+--+--+--+--+--+ |41|17|49|25| 1|33| 9| +--+--+--+--+--+--+--+ |16|48|24| 7|32| 8|40| +--+--+--+--+--+--+--+ |47|23| 6|31|14|39|15| +--+--+--+--+--+--+--+ |22| 5|30|13|38|21|46| +--+--+--+--+--+--+--+ ]