Magic Squares of an Even To construct a magic of the sixth 'order proceed according to the following rule which is, a modified form of a method due to-La-Ont As in his rule for odd squares two auxiliary squares are employed. For a square of the sixth order the first auxiliary square E is con structed as follows: First, fill the cells of the two diagonals with the numbers 1, 2, 3, 4, 5 and 6, beginning on the left-hand side. Second, fill each of the remaining cells of the first column with the same number as that already in two of them or with the complementary number, i.e., with al or a6 in any way, provided that there are the same number of these num bers in the column. Third, cells horizontally paired with those in the first column are filled with the complementary numbers. Fourth, the remaining cells in the second and third columns are filled in an analogous way to that in which the cells in the first column were filled; and then the cells horizontally paired with them are filled with the complementary numbers. Ob serve that in the case of a singly even magic square it will be necessary in constructing E to take care in the second step that in every row at least one cell which is not in a diagonal shall have its vertically paired cell filled with the same number as itself.
The second of the auxiliary squares F is constructed as follows: Rewrite square E, making another square F in which the left hand column of E (beginning with its top number) becomes the top row of the new the second column off. E becomes the -.mind row of the new square, and so on. Then instead of each of the numbers 1, 2, 3, 4. 5 and 6, substitute the corresponding number from the series 0, 6, 12, 18, 24 and 30. The result is square F. Next, if in each cell of G the sum of the numbers in the corresponding cells of squares E and F be placed, the required magic square is formed.
The following method is applicable to the construction of doubly e en magic squares only. Imagine the square to be divided into squarelets of four cells each, the four central cells com prising one; and conceive these squarelets to be of two kinds, alternating with each other. Place 1 in the left-hand upper corner cell, and proceed horizontally to the right counting a number to each cell, hut filling successively the squarelets of one kind only. When the end of one row is reached turn to the left-hand cell of the next row and again advance, filling cells of one kind as before, and so on. For the 4-square the result of this operation is seen in diagram H.
Next, begin with the right-hand lower corner cell, considering 1 as falling on it hut not writing the number, and proceed regularly to the left, row after row, filling the empty cells with the numbers belonging to them but not writing numbers in the cells already filled.
The result of the two operations is the magic square I. This is the most perfect magic square of the fourth order. Not only do the horizontal, vertical and diagonal lines of num bers sum tip 34, hut there are 33 other ways in which sets of four numbers may be selected whose sum is 34, making 48 ways in all. By the interchange of rows and columns according to the rule enunciated above, other squares may be formed, but none so perfect as this.
The above methods for the construction of magic squares are, in the writer'- opinion, the simplest of all those proposed. Limited space permits only two other methods to he noticed, which, however, are applicable to only a limited class of cases. The first relates to the con struction of composite magic squares. For ex ample a, square of 81 cells may he considered as made up of 9 smaller squares each containing 9 cells. The magic square in diagram K is built up by this The other method consists in surrounding a magic square with a border of cells, consti tuting what is termed a concentric square. In this way from the magic square of the third order can be built up squares of any odd order; and similarly even magic squares of any order may be built up from the magic square of the fourth order. Diagram L is constructed this way.
To Dr. Franklin is due the construction of diagram M which he called The Magic Circle of Circles. (See the illustration herewith). It is composed of a series of numbers from 12 to 75, inclusive, placed in eight concentric circular spaces and arranged in eight radii, with the number 12 in the centre. Luke the centre this number is c.ommon to-all the circular spaces and to all the radii. The numbers are so placed that the sum of all those in any of the circular spaces, together with the central num ber12 is 360, the number of degrees in a cir cle. The numbers in each radius together with the central number make 360. The numbers in half of any of the circular spaces taken above or below the horizontal diameter, with half the central number, make 180, the number of de grees in a semi-eircle. If any four adjoining numbers be taken, as if in a square, in the ra dial division of the circular spaces, the sum of these with half the central number is 180 There are, moreover, included five sets of other circular spaces bounded by circles which are eccentric with respect to the common centre. The centres of the circles which bound them are at A, B, C and D. The numbers in these eccentric circular spaces possess the same magic properties as the numbers in the first-mentioned circular spaces.