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MAGIC SQUARE

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Originally appearing in Volume V17, Page 313 of the 1911 Encyclopedia Britannica.
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MAGIC SQUARE, a square divided into. equal squares, like a chess-board, in each of which is placed one of a series of consecutive numbers from 1 up to the square of the number of cells in a side, in such a manner that the sum of the numbers in each row or column and in each diagonal is constant. From a very early period these squares engaged the attention of mathematicians, especially such as possessed a love of the marvellous, or sought to win for themselves a superstitious regard. They were then supposed to possess magical properties, and were worn, as in India at the present day, engraven in metal or stone, as amulets or talismans. According to the old astrologers, relations subsisted between these squares and the planets. In later times such squares ranked only as mathematical curiosities; till at last their mode of construction was systematically investigated. The earliest known writer on the subject was Emanuel Moscopulus, a Greek (4th or 5th century). Bernard Frenicle de Bessy constructed magic squares such that if one or more of the encircling bands of numbers be taken away the remaining central squares are still magical. Subsequently Poignard constructed squares with numbers in arithmetical pro- 300 217 232 249 8 25 40 57 72 89 104 12I 136 153 z68 z85 58 39 26 7 250 231 2,8 z49 186 167 154 135 I22 103 90 71 198 219 230 251 6 27 38 59 70 91 10Z 123 134 155 z66 z87 6o 37 28 5 252 229 220 197 z88 z65 z56 133 124 101 92 69 201 sz6 233 248 9 24 41 56 73 88 105 120 137 z5z 169 184 55 42 z3 10 247 234 215 202 183 770 z5z 138 119 zo6 87 74 303 214 335 246 II 22 43 54 75 86 107 118 139 150 171 182 53 44 21 12 245 236 213 204 z81 172 149 140 117 io8 85 76 305 212 237 344 13 20 45 53 77 84 109 116 141 148 173 180 51 46 19 14 243 238 211 sob 179 174 147 142 115 zzo 83 . 78 207 210 239 2 42 15 18 47 5o 79 82 111 114 143. 146 175 178 49 48 17 z6 241 240 309 208 177 176 145 144 113 112 81 8o 196 221 328 253 4 39 36 6z 6g 93 100 135 132 157 164 189 62 35 30 3 254 z27 222 195 190 763 158 131 336 99 94 67 194 223 226 255 3 31 34 63 66 95 98 127 130 159 16z 191 64 33 .32 1 256 225 224 193 192 163 160 129 128 97 96 65 gression, having the magical summations. The later researches of Phillipe de la Hire, recorded in the Memoires de l'Academie Royale in 1705, are interesting as giving general methods of construction. He has there collected the results of the labours of earlier pioneers; but the subject has now been fully systematized, and extended to cubes. Two interesting magical arrangements are said to have been given by Benjamin Franklin; these have been termed the " magic square of squares " and the " magic circle of circles." The first (ft. I) is a square divided into 256 squares, i.e. 16 squares along a stile, in which are placed the numbers from 1 to 256. The chief properties of this square are (1) the sum of the 16 numbers in any row or column is 2056; (2) the sum of the 8 numbers in half of any row or column is 1028, i.e. one half of 2056; (3) the sum of the numbers in two half-diagonals equals 2056; (4) the sum of the four corner numbers of the great square and the four central numbers equals Io28; (5) the sum of the numbers in any 16 cells of the large square which them-selves are disposed in a square is 2056. This square has other curious properties. The " magic circle of circles " (fig. 2) consists of eight annular rings and a central circle, each ring being divided into eight cells by radii drawn from the centre; there are therefore 65 cells. The number 12 is placed in the centre, and the consecutive numbers 13 to 75 are placed in the other cells. The properties of this figure include the following: (1) the sum of the eight numbers in any ring together with the central number 12 is 360, the number of degrees in a circle; (2) the sum of the eight numbers in any set of radial cells together with the central number is 36o; (3) the sum of the numbers in any four adjoining cells, either annular, radial, or both radial and two annular, together with half the central number, is 180. Construction of Magic Squares.—A square of 5 (fig. 3) 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 squares, 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 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, (3, y, 5, e pursue another 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 i, 2, 3, a, b, c, a, /3, y, .... are said to lie in "paths." Squares whose Roots are Odd.—Figs 4, 5, and 6 exhibit one of the earliest methods of constructing magic squares. Here the Squares whose Roots are Even..—These were constructed in various ways, similar to that of 4 in figs. 14, 15 and 16. The numbers in fig. 15 being multiplied by 4, and the squares of figs. 14 and 15 being superimposed, give fig. i6. The application of a4 20 as 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 -9 5 -a -6 4 25 18 11 7 2 7 -11 4 -: 14 20 2 17 :a -8 .3 0 3 8 5 10 13 16 21 10-4 11 .7 -10 23 9 24 6 3 6 -12 .5 a 9 19 1 8 16 22 paths, may be formed, by observing, as when the root is 4, that the series 1 to 16 may be changed into the series 15, 13, ... . 3, 1,—1,—3, .... -13,- 15, by multiplying each number by 2 a 5 e 4 b a 4 a Y c 3 7 P d 1c e a a 6 7 :I 7 3 li 3 20 24 4 12 a 8 2a 4 9 a1 17 5 13 5 a5 al 27 :6 9 18 13 14 as :8 10 14 l0 as a 23 19 6 Is 19 15 a 5 3 5 3 3 a 4 17 m 13 a 11 9 24 1a 23 3 a 4 0 3 3 0 as 14 4 20 8 4 a 3 1 a : a la 6 7 9 3 16 a 3 r, t a a I1 5 al 4 3 a 4 3 0 0 3 13 3 a 16 l4 22 FIG. 14. FIG. 15. FIG. 16. 4 s 5 3 I a 6 4 I : a 4 a 3 3 0 0 4 3 0 3 4 0 a 4 a 3'S in fig. 4 and 2'S in fig. 5 are placed in opposite diagonals to secure the two diagonal summations; then each number in fig. 5 is multiplied by 5 and added to that in the corresponding square in fig. 4, which gives the square of fig. 6. Figs. 7, 8 and g give De la Hire's method; the squares of figs. 7 and 8, being combined, give the magic square of fig. 9. C. G. Bachet arranged the numbers as in fig. ro, where there are three numbers in each of four surrounding squares; these being placed in the corresponding cells of the central square, the square of fig. 11 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 subtractedfrom each number, the summations will be zero, and the twenty-five cells will contain the series t 1, t 2, 3, .... t 12, the odd cell having o. 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 fig. 12; then, 13 beingadded in each cell, the magic square of fig. 13 is obtained. and subtracting 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 fig. 17, make zero; and, to obtain the same in the rows , and columns, we must assign such values to the p's and q's as satisfy the equations pi + p2 + al + a2 = 0, pa + p4 + a3 + a4 = o, p, + p3 — al —a3 = o, and P2 + p4 — a2 — a4 = o,a solution of which is 'readily obtained by inspection, as in fig. 18; this leads to the square, fig. 19. When the root is 8, the upper four subsidiary rows may at once be written, as in fig, 20; then, if 65 be 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 summations than in rows, columns and diagonals were investigated by A. H. Frost (Cambridge Math. _lour., 1857), and called Nasik squares, from the town in India where he resided; and he 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 s s 4 15 5 6 3 33 14 30 20 17 0 4 1 15 1a 10 3 24 16 a 4 $0 3 10 0 S 5 as 5 ao II 8 22 a 4 ao O 4 t0 3 5 15 a a 4 at 5 a0 :o 3 9 0 Is 18 :5 1 a0 as :0 O a 7 19 13 15 S 3 4 t 14 9 7 2 a, Ps -3 a, 7 at -9 6 to 13 -15 Pa P4 a4 a3 -5 8 -I I 13 -I 3 3 -a2 -P, 9 -a, P2 a 11 Is 5 -13 -a3 -P3 -P4 a4 4 :S so 3 -I s -33 -7 37 .39 9 -II 41 -13 Is :7 -2I -43 -45 47 55 -19 49 -53 -51 23 61 -a5 29 27 -57 -63 31 59 5 18 6 I 23 19 4 a 3 63 88 74 13 8 24 53 48 34 II 9 25 51 49 35 61 89 75 52 47 36 6a 87 76 12 7 a6 68 84 73 18 4 23 58 44 33 19 5 '21 59 45 31 69 85 71 57 46 32 67 86 72 17 6 22 64 83 78 14 3 z8 54 43 38 INj 7 29 55 41 39 65 8x 79 56 42 37 66 8a 77 16 2 27 a g f e d c b a d g c f b e a c e g b d f a f d b g e c a e b f e g' d a b c d e f g 0 h It It It It It consider carefully fig. 2I, 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 o, 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 differ- ent paths, will appear together in one and only one cell. Hence, if p, be placed in the cells of one of the n+1 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,, p3, . p„ in the same order, commencing from the pi in that row, the p2's, p3's and p„'s will lie each in a path similar to that of p,, and each of the n normal paths will contain one, whose sum will be p. Similarly, if qi be placed along any of the normal paths, different from that of the p's, and each row filled as above with the letters q2, q3, . . q,,, the sum of the q's along any normal path different from that of the qi will be q. The 5t cells of the square will now be found to contain all the combinations of the p's and q's; and if the q's be multiplied by n, the p's made equal to i, 2, ... It, and the q's to 0, I , 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 p's and q's, will be the constant Enq + Ep. 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 pi and q1 were placed, instead of having each of the p's and q's, some will be wanting, while some are repeated. Thus, in the case of 9, the triplets, p,p4p7, p,p1p8, p3p,;p9, and q,g4g7, g2q,q,, g3q,g9 occur, each triplet thrice, along paths whose summation should be—Ep 45 and Er 36. But if we make p„ 122, ... 129, =I, 3, 6, 5, 7, 9, 8, 2, and the r1, r2, ... r9 =o, 2, 5, 4, 3, 6, 8, 7, I, thrice each of the above sets of triplets will equal Ep and Eq respectively. If now the q'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 fig. 22 the numbers are in the nonary scale; that in the centre is the middle one of I to 92, and the sum of pair 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, is the constant Nasical summation, e.g. 72 and 32, 22, 76, 77, 26, 37, 36, 27. The numbers in fig. 22 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, 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 respectively I, 2, so, 8, 6, 14, 15, II, 4, 13, 9, 7, 3, 12, 5, and o, 1, 9, 7, 5, 13, 14, 10, 3, 12, 8, 6, 2, II, , where five times the sum of every third number and three times the sum of every fifth number makes Ep 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 the multiple of 4, the same process gives us, for the square of 4, fig. 23. Here the columns give Ep, but alternately 2q„ 2q3, and 2q2, 2q4; and the rows give Eq, but alternately 2121, 2123, and 2p2, 2p4; the diagonals giving Ep and Eq. If p„ p2, p3, p4 and q„ q2, q3, q4 be 1, 2, 4, 3, and o, 1, 3, 2, we have the Nasik square of fig. 24. A square like this is engraved in the Sanskrit character on the gate of the fort of Gwalior, in India. The squares of higher multiples of 4 are readily obtained by a similar adjustment. Nasik Cubes.—A Nasik cube is composed of n3 small equal cubes, here called cubelets, in the centres of which the natural numbers from i to n3 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. Fig. 25 shows by dots the way in which these cubes are constructed. )14g3 d39, 15294 X293 1491 1'292 p, 9z 1'4 94 75372 /'39a 15291 15193 ~,qa 734 92 15394 15 4 14 IO 5 1I 8 3 16 13 6 9 7 I2 p7q,r, are placed in the other cube-lets of the edge AO, and dispersed in the same manner as Awl. Every cubelet will then be found to contain a different combination of the p's, q's and r's. If therefore the p's are made equal to I, 2, 7, and the q's and is to o, I, 2,...6, in any order, and the q's multiplied by 7, and the r's by 72, then, as in the case of the squares, the 73 cubelets will contain the numbers from I to 73, and the Nasical summations will be E72r+-E7q + p. 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 o, 1, 2, . . . 6 in each of the other places. In all Nasik cubes, if such values are given to the letters on the central cubelet that the number is the middle one of the series I to n3, the sum of all the pairs of numbers opposite s 8 29 28 II 14 23 18 30 27 a 7 2I 20 9 16 4 5 32 25 10 15 22 19 31 26 3 6 24 17 12 13 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 same way,—and if the surrounding space be conceived thus filled with similar cubes, and a straight line of unlimited length be drawn through any two cubelet centres, one in each of any two cubes,—the numbers along that line will be found to recur in groups of seven, which (except in the three cases where the same p, q or r recur in the group) together make the Nasical summation of the cube. Further, if we take it similarly filled Nasik cubes of n, n new letters, s„ s2, ...s,,, can be so placed, one in each of the n4 cubelets of this group of n cubes, that each shall contain a different combination of the p's, q's, is and s's. This is done by placing s1 on each of the n2 cubelets o£ the first cube that 30 21 6 15 28 19 7 16 29 20 5 14 22 31 8 35 18 27 9 36 17 26 13 4 32 23 2 11 34 25 1 10 33 24 3 12 contain pl, and on the n2 cubelets oI the 2d, 3d, ... and nth cube that contain plc p3, ... p,, respectively. This process is repeated with s2, beginning with the cube at which we ended, and so on with the other s's; the n4 cubelets, after multiplying the q's, r's, and s's by is, n2, and n3 respectively, will now be filled with the numbers from i to n4, and the constant summation will be n3s + n2r + Enq + p. This process may be carried on without limit; for, if the is cubes are placed in a row with their faces resting on each other, and the corresponding faces looking the same way, is such parallelepipeds might be put side by side, and the n5 cubelets of this solid square be Nasically filled by the introduction of a new letter t; while, by introducing another letter, the no cubelets of the compound cube of n3 Nasik B C 23 10 19 14 6 II 28 5 24 17 4 22 13 2 2I 9 8 20 15 25 12 7 16 3 cubes might be filled by the numbers from i to n6, and so ad infinitum. When the root is an odd composite number the values of the three groups of letters have to be adjusted as in squares, also in cubes of an even root. A similar process enables us to place successive numbers in the cells of several equal squares in which the Nasical summations are the same in each, as in fig. 26. Among the many ingenious squares given by various writers, this article may justly close with two by L. Euler, in the Histoire de l'academie royale des sciences (Berlin, 1759). In fig. 27 the natural numbers show the path of a knight that moves within an odd square in such a manner that the sum of pairs of numbers opposite to and equidistant from the middle figure is its double. In fig. 28 the knight returns to its starting cell in a square of 6, and the difference between the pairs of numbers opposite to and equidistant from the middle point is 18. A model consisting of seven Nasik cubes, constructed by A. H. Frost, is in the SouthKensington Museum. The centres of the cubes are placed at equal distances in a straight line, the similar faces looking the same way in a plane parallel to that line. Each of the cubes has seven parallel glass plates, to which, on one side, the seven numbers in the septenary scale are fixed, and behind each, on the other side, its value in the common scale. 1201, the middle number from 1 to 74, occupies the central cubelet of the middle cube. Besides each cube having separately the same Nasical summation, this is also obtained by adding the numbers in any seven similarly situated cubelets, one in each cube. Also, the sum of all pairs of numbers, in a straight line, through the central cube of the system, equidistant from it, in whatever cubes they are, is twice 1201. (A. H. F.) Fennell's Magic Ring.—It has been noticed that the numbers of magic squares, of which the extension by repeating the rows and columns of n numbers so as to form a square of 211–1 sides yields n2 magic squares of n sides, are arranged as if they were all inscribed round a cylinder and also all inscribed on another cylinder at right angles to the first. C. A. M. Fennell explains this apparent anomaly by describing such magic squares as Mercator's projections, so to say, of " magic rings. " The surface of these magic rings is symmetrically divided into n2 quadrangular compartments or cells by n equidistant zonal circles parallel to the circular axis of the ring and by n transverse circles which divide each of the n zones between any two neighbouring zonal circles into n equal quadrangular cells, while the zonal circles divide the sections between two neighbouring transverse circles into n unequal quadrangular cells. The diagonals of cells which follow each other passing once only through each zone and section, form similar and equal closed curves passing once quite round the circular axis of the ring and once quite round the centre of the ring. The position of each number is regarded as the intersection of two diagonals of its cell. The numbers are most easily seen if the smallest circle on the surface of the ring, which circle is concentric with the axis, he one of the zonal circles. In a perfect magic ring the sum of the numbers of the cells whose diagonals form any one of the 2n diagonal curves aforesaid is in(n2 + I) with or without increment, i.e. is the same sum as that of the numbers in each zone and each transverse section. But if n be 3 or a multiple of 3, only from 2 ton of the diagonal curves carry the sum in question, so that the magic rings are imperfect; and any set of numbers which can be arranged to make a perfect magic ring or magic square can also make an imperfect magic ring, e.g. the set i to 16 if the numbers 1, 6, 11, 16 lie thus on a diagonal curve instead of in the order 1, 6, 16 I t. From a perfect magic ring of n2 cells containing one number each, n2 distinct magic squares can he read off; as the four numbers round each intersection of a zonal circle and a transverse circle constitute corner numbers of a magic square. The shape of a magic ring gives it the function of an indefinite extension in all directions of each of the aforesaid n2 magic squares. (C. A. M. F.) See F. E. A. Lucas, Recreations mat hematiques (1891–1894) ; W.W.R. Ball, Mathematical Recreations (1892) ; W. E. M. G. Ahrens, Mathematische Unterhaltungen and Spiele (1901) ; H. C. H. Schubert, Mathematische Mussestunden (19oo). A very detailed work is B. Violle, Traite compiet des carrel magiques (3 vols., 1837-1838). The theory of " path nasiks " is dealt with in a pamphlet by C. Planck (1906).
End of Article: MAGIC SQUARE
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