MAGIC SQUARE - a. 2 - 4 = 0, a solution of which is readily obtained by inspection, as in diagram P ; this leads to the square, dia gram Q. When the root is 8, the upper four subsidiary rows may at once be written, as in diagram R, ; then, if the square be completed, 65 added to each, and the sums halved, the square is completed. In such squares as these, the two opposite squares about the same diagonal (except that of 4) may be turned through any number of right angles, in the same direction, without altering the summations. Nasik Squares. Squares that have many more summa tions than in rows, columns, and diagonals have been investigated by the Rev. A. H. Frost (Cambridge Math. Jour., 1857), and called Nasik squares, from the town in India where he resided ; and he has extended the method to cubes, various sections of which have the same singular properties. In order to understand their construction it will be necessary to consider carefully diagram S, which shows that, when the root is a prime, and not composite, number, as 7, eight letters a, b, ... h may proceed from any, the same, cell, suppose that marked 0, each letter being repeated in the cells along different paths. These eight paths are called normal paths, their number being one more than the root. Observe here that, excepting the cells from which any two letters start, they do not occupy again the same cell, and that two letters, starting from any two different cells along different paths, will appear together in one and only one cell. Hence, if p l be placed in the cells of one of the n + normal paths, each of the remaining n normal paths will contain one, and only one, of these p^s. If now we fill each row with p< 2 , p 3 , . . . p n in the same order, commencing from the p 1 in that row, the p. 2 s, p 3 s, and p n s will lie each in a path similar to that of 2> and each of the n normal paths will contain one, and only one, of the letters p v p 2 , . . . p n , whose g sum will be 2p. Similarly, if <?! be placed along any of the normal paths, different from that of the > s, and each row filled as above with the letters q. 2 , q. A , . . . q n , the sum of the q s along any normal path dif ferent from that of the </, will be 2<?. The 2 cells of the square will now be found to contain all the combinations of the p s and q s; and, if a 7 f e d c b a d g c ~7 b e a C e ~ b d f a f d ~ 0. e c a e b f c ff d ct b c d e f f)
h h h h h h the q s be multiplied n, and the q s to by n, the p s made equal to 1, 2, 0, 1, 2, . . . n I in any order, the Nasik square of n will be obtained, and the summations along all the normal paths, except those traversed by the jp s and q s, will be the constant 2<? + 2/>. When the root is an odd composite number, as 9, 15, &c., it will be found that in some paths, different from the two along which the p^ and q l were placed, instead of having each of the j) s and q s, some will be wanting, while some are repeated. Thus, in the case of 9, the triplets p,p 4 p 7 , P 2 p 5 p s , P 3 P 6 P Q , and g^q^ q 2 q b q s , qtfjfo occur, each triplet thrice, along paths whose summation should be 2/?, 45, and 2r, 36. But if we make p v p 2 , . . . p g = 1, 3, 6, 5, 4, 7, 9, 8, 2, and the r v r 2 , . . . r 9 = 0, 2, 5, 4, 3, 6, 8, 7, 1, thrice each of the above sets of triplets will equal 2/> and 2^ respectively. If now the ^ s are multiplied by 9, and added to the p s in their several cells, we shall have a Nasik square, with a constant summation along eight of its ten normal paths. In diagram T, the numbers are in the nonary scale ; that in the centre is the middle one of 1 to 9 2 , and the sum of pairs of numbers equidistant from and opposite to the central 45 is twice 45 ; and the sum of any number and the 8 numbers 3 from it, diagonally, and in its row and column, 03 8S 74 10 8 24 53 48 34 11
25 51 49 35 61 89 75 62 47 36 62 87 76 12 ~T 26 68 84 73 18 4 23 58 44 33 1!) 5 21 59 45 31 69 85 71 57 46 32 67 86 72 17 6 22 64 83 78 14 3 28 54 43 88 15 1 29 55 41 39 65 81 79 56 42 37 66 82 77 16 2 27 215 is the constant Nasical summation, e.g., 72 and 32, 22, 76, 77, 26, 37, 36, 27. The numbers in T being kept in the nonary scale, it is not necessary to add any nine of them together in order to test the Nasical summation ; for, taking the first column, , n the figures in the place of units are seen at once to form the series, 1, 2, 3, .... 9, and those in the other place three triplets of 6, 1, 5. For the squares of 15 the p s and q s may be respect ively 1, 2, 10, 8, 6, 14, 15, 11, 4, 13,9,7,3, 12, 5,andO, 1,9,7,5, 13, H, 10,3, 12,8, 6, 2, 11, 4, where five times the sum of every third number and three times the sum of every fifth number makes 2/> and 2q ; then, if the q s are multiplied by 15, and added to the p s, the Nasik square of 15 is obtained. When the root is a multiple of 4, the same process gives us, for the square of 4, the diagram U. Here the columns U give 2/>, but alter- sately 2q v 2q. A , and ^^zi 2^ 4 ; and the rows give 2^, but alternately 2p 1 , 2/) 3 , and 2p z , 2p ; the diagonals giving 2p and 2j. If p v p y p.., p and q v q 2 , q 3 , q be 1, 2, 4, 3, and 0, 1, 3, 2, we have the Nasik square of diagram V. A square like this is engraven in the Sanskrit character on the gate of the fort of Gwalior, ia India. The squares of higher multiples of 4 are readily obtained by a similar adjustment. A Nasik cube is composed of ?z, 3 small equal cubes, here called cubelets, in the centres of which the natural numbers from 1 to n 3 are so placed that every section of the cube by planes perpendicular to an edge has the properties of a Nasik square ; also sections by planes perpendicular to a face, and passing through the cubelet centres of any path of Nasical summation in that face. Diagram W shows by dots the way in which these cubes are constructed. A dot is here placed on three faces of a cubelet at the corner, showing that this cubelet belongs to each of the faces AOB, BOG, COA, of the cube. Dots are placed on the cubelets of some path of AOB (here the knight s path), beginning from 0, also on the cubelets of a knight s path in BOG. Dots are now placed in the cubelets of similar paths to that on BOG in the other six sections parallel to BOG, starting from their dots in AOB. Forty-nine of the three hundred and forty-three cubelets will now contain a dot ; and it will be ob served that the dots in sections perpendicular to BO have arranged themselves in similar paths. In this manner, p lt q v r, being placed in the corner cubelet 0, these letters are severally placed in the cubelets of three different paths of AOB, and again along any similar paths in the seven sections perpendicular to AO, starting from the letters position in AOB. Next,^ 2 <7 2 r 2 , p^q 3 r s , . . . p 7 g 7 r 7 are placed P*l3 P 2l4 Pti Pill P9l Pll-2 7>3?3 Pl<l^ Pi.1z T r l Ptf Ptfi Pill P3<1 I ll 3 Pa1 t in the other cubelets of the edge AO, and dispersed in the same manner as Pi9i r i- Every cubelet will then be found to contain a different com bination of them s, q s, and r s. If therefore them s are made equal to 1, 2, ... 7, and the ^ s and r s to 0, 1, 2, ... 6, in any order, and the q s multiplied by 7, and the r s by 7 2 , then, as in the case of the squares, the 7 3 cubelets Mill contain the numbers from 1 to 7 3 , and the Nasical summations will be 27 2 r + 27(?+^. If 2, 4, 5 be values of r,p,q, the number for that cubelet is written 245 in the septenary scale, and if all the cubelet numbers are kept thus, the paths along which summations are found can be seen without adding, as the seven numbers would contain 1, 2, 3, ... 7 in the unit place, and 0, 1, 2, ... 6 in each of the other places. In all Nasik cubes, if such values arc given to the letters on the central cubelet that the number is the middle one of the series 1 to 11*, the sum of all the pairs of numbers opposite to and equidistant from the middle number is the double of it. Also, if around a Nasik cube the twenty-six surrounding equal cubes be placed with their cells filled
with the same numbers, and their corresponding faces looking the