Magic Square

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 i 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.

200	217	232	2 49	8	2 5	4 o	57	72	89	r04	221	136	1 53	168	185
58	39	2 6	7	250	231	218	r99	186	267	2 54	1 35	122	103	90	71
198	219	230	251	6	27	3 8	59	70	91	102	123	1 34	155	166	187
60	37	2 8	5	252	229	220	297	r88	165	156	1 33	124	101	92	69
201	226	2 33	2 4 8	9	24	4 2	5 6	73	88	105	120	1J7	152	169	184
55	42	23	1 0	2 47	234	215	202	183	270	151	238	r19	rob	87	74
203	214	2 35	246	r1	22	43	54	75	86	107	118	239	150	171	r82
53	44	22	22	2 45	236	213	204	181	172	249	140	117	108	85	76
205	212	2 37	2 44	1 3	20	45	5 2	77	84	209	r16	241	248	273	280
51	46	19	14	2 43	238	zrr	206	1 79	2 74	1 47	142	115	1r0	83	7 8
207	2 1 0	2 39	242	15	18	47	5 0	79	8z	111	114	1 4 3.	146	1 75	278
49	48	17	16	241	240	209	z03	1 77	2 7 6	1 45	1 44	113	11z	8z	80
196	22 2	228	2 53	4	29	36	61	68	9 3	zoo	125	132	1 57	164	189
62	35	3 o	3	2 54	227	222	1 95	190	163	258	132	126	99	94	67
1 94	223	226	2 55	2	3 1	34	6 3	66	95	98	1 27	x 3	1 59	z 6	192
6 4	33	32		256	225	224	1 9 3	192	i 6	r60	2 29	128	97	9 6	65

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 astro logers, 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 FIG. I.

gression, having the magical summations. The later researches of Phillipe de la Hire, recorded in the Mdmoires 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 nov.' 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 (fig. 1) is a square divided into 256 squares, i.e. 16 squares along a side, in FIG 2.

which are placed the numbers from i to 256. The chief properties of this square are (I) 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 halfdiagonals equals 2056; (4) the sum of the four corner numbers of the great square and the four central numbers equals 1028; (5) the sum of the numbers in any 16 cells of the large square which themselves 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: (I) 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 360; (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.

7		3
	8		4
13		9
	14		10
19		15
20
25

II	24	7	20	3
4	12	25	8	16
1 7	5	13	21	9
10	:8		14	22
23	6	19	2	15

	a	e	5				E
	4			5		4			5
	7			3		7
		=	Q
c					e

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. 16. The application of 6 z II 12 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. I, 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, 0, 7, 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 17 ZI :8 Z2 23 24 FIG. IO. FIG. II.

S		-6	4	a5	18	II	7
-II	4	- I	14	30		17	12
0	3	8		10	33	36	21
II	- 7	-10	23	9	24	6	3
-s	2	9	19		8	IS	22

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 FIG. 3. -9 :2 7 - 8 -3 10 -4 6 -12 FIG. 12. FIG. 13. 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 I 15 14434 0 3 3 8 4 II ' '24 5033 4 20647 9 12 3 2 16 4 9 Ta 4 0 3 25 3 3 a 5 a 84320 II I.

S 2 17 4 I 9 13533 0 s 4 21 4 3) 4 16 a 3 132400318605 5 3 4 a ' '4 2 5 3 2 4 14 ' '22 FIG. 14. FIG 15 FIG. 16. 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 3340 23 I 2 19 7 15 FIG. 4. FIG. 5. FIG. 6.

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 9 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. Io, 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. I I is formed. He also con A p2 7 14 4 a: a2 -3 II -9 P. A 6 12 15 a3 a4 7 -25 -s 13 8 Io -P, -p2 3 :3 -a1 -a2 -II 9 - I 3 16 II 5 a -p3 -P4 -a 3 -a4 S 15 -13 - 7 FIG 17 FIG. 18 FIG. 19._ and columns, we must assign such values to the p's and q's as satisfy the equations p i + P2 + al + az = 0, P3 + P4 + a3 + a4 = o, pi + p3 - al - a3 = 0, and p2 -I- p 4 - a 2 - a 4 = 0, - 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.

-1	3	5	-7	-33	35	31	-39
9	-II	-13	1 5	4 1	-43	-45	47
17	-19	-21	23	49	-5 1	-53	55
- 2 5 27		29	-3 1	-57	59	6,	-63

17 6 1 54523 3 14 15 ,:0 164324 3 12 102510 155020 8 4 II 20 22453 5 15 0 20 10 18934 25 2 I 2 5 3 15 20 :O 5 0 25 13 a I 7 3 15 19 5 5 20 ' '1 0 4 FIG. 7. FIG. 8. FIG. 9.

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 5 X13; if therefore 13 is subtracted from each number, the summations will be zero, and the twenty-five cells will contain the series t 1, = 2, 3, .... 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 being added in each cell, the magic square of fig. 13 is obtained.

FIG 20 Nasik Squares. - Squares that have many more summations than in rows, columns and diagonals were investigated by A. H. Frost (Cambridge Math. Jour., 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 consider carefully fig. 21, 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 different paths, will appear together in one and only one cell. Hence, if p, be placed in the cells of one of the n+I normal paths, each of the remaining n normal paths will contain one, and only one, of these p i 's. If now we fill each row with p 2, p 3,.. p m in the same order, commencing from the p i in that row, the p 2 's, p 3 's and p n 's will lie each in a path similar to that of p i, and each of the n normal paths will contain one, and only one, of the letters p„ p„, ...pm, whose sum will be Ep. Similarly, if q i be placed along any of the normal paths, different from that of the p's, and each row filled as above with the letters q 2, 3,. .. q,l, the sum of the q's along any normal path different from that of the q i will be Eq. The n 2 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, ... n, 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 p i and q i 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 p p 7, p 2 p 5 p 8, - 3l 8p 9, and g 144 g ?, g 9 g 5 q 3g 3 g 6 q 9 o ccur, each triplet thrice, along paths whose summation should be-EP 45 and Er 36. But if we make p i, p 2, ... p9, = 3, 6, 5, 4, 7, 9, 8, 2, and the r i, r2, ...r 9 =o, 2, 5, 4, 3, 6, 8, 7, 1, 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 9 2, 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 22. column, is the constant Nasi cal summation, e.g. 72 and 32, 22, 7 6, 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, I, 2, 3,....9, and those in the other place three triplets of 6, I, 5. For the squares of 15 the p's and q's may be respectively I, 2, Io, 8, 6, 14, 15, II, 4, 13, 9, 7, 3, 12, 5, and 0, I, 9, 7, 5, 13, 14, JO., 3, 12, 8, 6, 2, II, 4, where five times the sum of every third number and three times the sum of every fifth number makes Ep and /q; 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 1, 2q 3, and 2q3, 2q 4; and the rows give Eq, FIG. 23. but alternately 2p i, 2p3, and 2p 21 2p 4; the diagonals giving Ep and Eq. If p i, p 2, p 3, p4 and q„ q 2, q 3, q 4 be I, 2, 4, 3, and 0, I, 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 n 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. Fig. 25 shows by dots the way in which these cubes are constructed.

^ 1	3	29	28		LI	14	23	18
30	27	2	7			20	9	16
4	5	32
	z6	3	6		24	17

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, BOC, 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 BOC. Dots are now placed in the cubelets of similar paths to that on BOC in the other six sections parallel to BOC, 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 observed that the dots in sections perpendicular to BO have arranged themselves in similar paths. In this manner, p i, q i, ri 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, p 2 g 2r2, p, g 3 r 3,. p 7 g 7 r, are placed in the other cubelets of the edge AO, and dispersed in the same manner as pigiri. 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 r's to 0, I, 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 will contain the numbers from I to 7 3, and the Nasical summations will be /7 2 r+/7q -{- 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 I, 2, 3,. .. 7 in the unit place, and 0, I, 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 n 3, the sum of all the pairs of numbers opposite FIG. 26.

30	21	6	15	2 8
7	16	29	20	5	14
22	31	8	35	18	27
9	3 6	17	26	13	4
32	23	2	i	34	25
1	x0	33	2 4	3	1

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 n similarly filled Nasik cubes of n, n new letters, s i, s2,...s, , can be so placed, one in each of the n 4 cubelets of this group of n cubes, that each shall contain a different combination of the p's, q's, r's and s's. This is done by placing s, on each of the n 2 cubelets of the first cube that FIG. 27. FIG. 28.

a		f	e	d	c	b
ad			c	f		e
		e			d
a	e
a	b	c	d	e
0	h					h