214 MAGIC SQUAKE struction. He has there collected the results of the labours of earlier pioneers; but the subject has now been fully systematized, and extended to cubes. In order to understand the rest of this article diagram A should be carefully examined. A square of 5 has adjoining it one of the eight equal squares by which any square may be conceived to be surrounded, each of which has two sides resting on adjoining ones, while four have sides resting on the surrounded square, and four meet it only at its four angles. 1, 2, 3 are placed along the path of a knight in chess ; 4, along the same path, would fall in a cell of the outer square, and A. is placed instead in the corresponding cell of the original square ; 5 then falls within the square. a, b, c, d are placed diagonally in the square ; but e enters the outer square, and is removed thence to the same cell of the square it had left, a, /5, y, 8, e pursue another, but regular, course ; and the diagram shows how that course is recorded in the square they have twice left. Whichever of the eight surrounding squares may be entered, the corresponding cell of the central square is taken instead. The 1, 2, 3, . . . . , a, b, c, . . . . , a, /?, y, . . . . are said to lie in paths. Squares ^vhose Soots are Odd. Diagrams B, C, D exhibit one of the earliest methods of constructing magic squares. Here the 3 s in B and 2 s in C are placed in B. C. I). opposite diagonals to secure the two diagonal summations; then each number in C is multiplied by 5 and ad led to E. F. G. that in the corresponding number in B, which gives the square D. Diagrams E, F, G give M. de la Hire s method ; H. 11 24 7 20 3 4 12 25 8 16 ]7 5 13 21 9 10 18 1 14 22 23 6 If) 2 15 the squares E, F, being combined as above, give the magic square G. M. Bachet arranged the numbers as in H, where there are three numbers in each of four surrounding squares; these being placed in the corresponding cells of the central square, the square I is formed. He also con structed squares such that if one or more outer bands of numbers are removed the remaining central squares are magical. His method of forming them may be understood from a square of 5. Here each summation is 5x13; if therefore 13 is subtracted from each number, the summa- J. K. 9 12 5 o -6 1 7 -ii 4 ^ -8 -3
3 8 10 -4 11 _ 7 -10 6 -12 5 2 9 4 25 18 11 7 14 20 2 17 12 5 10 13 1G 21 23 9 24 C 3 19 1 8 15 22 tions will be zero, and the twenty-five cells will contain the series 1, 2, 3, . . . . 12, the odd cell having 0. The central square of 3 is formed with four of the twelve numbers with + and - signs and zero in the middle ; the band is filled up with the rest, as in diagram J ; then, 13 being added in each cell, the magic square K is obtained. Squares ivkose Roots are Even. These were constructed in various ways, similar to that of 4 in diagrams L, M, N. The numbers in M being multiplied by 4, and the squares L, M being superimposed, give N". The application of L. M. K 1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16 this method to squares the half of whose roots are odd requires a complicated adjustment. Squares whose half root is a multiple of 4, and in which there are summations along all the diagonal paths, may be formed, by observ ing, as when the root is 4, that the series 1 to 16 may be 0. P. Pi P2 1 2 1 -3 11 -9 P3 Pt 3 a 4 - 5 7 -15 13 -"I -<X 2 ~P Pi. -11 9 -1 3 ~ a i a 4 -P-l ~P 15 -13 5 1 changed into the series 15, 13, . . . . 3, 1, - 1, - 3, . . . . - 13, - 15, by multiplying each number by 2 and subtract ing 17 ; and, vice versa, by adding 17 to each of the latter, and dividing by 2. The diagonal summations of a square, filled as in diagram O, make zero ; and, to obtain the same E. -1 3 5 -7 -33 35 37 -31) 9 -11 -13 15 41 -43 -45 47 17 -19 -21 23 49 -51 - 53 55 -25 27 29 -31 -57 59 61 -63 in the rows and columns, we must assign such values to the p s and q s as satisfy the equations p l + p. 2 + a l + a 2 = 0,
p s + p +a 3 + 0^ = 0, 2h +^-ai- a 3 = > and P-2 + Pi