**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.

Fig. 1.

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 progression,
having the magical summations. The later researches of
Phillipe de la Hire, recorded in the *Mémoires de l’Académie*
*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.

Fig. 2.

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
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 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: (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 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.

*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. α, β, γ, δ, ε 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 1, 2, 3, . . ., *a*, *b*, *c*, . . .,
α, β, γ, . . . are said to lie in “paths.”

Fig. 4. | Fig. 5. | Fig. 6. |

Fig. 7. | Fig. 8. | Fig. 9. |

*Squares whose Roots are Odd.*—Figs 4, 5, and 6 exhibit one of
the earliest methods of constructing magic squares. Here the
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. 10, 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 constructed
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 5×13; if therefore 13 is subtracted from
each number, the summations 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 fig. 12; then, 13 being added
in each cell, the magic square of fig. 13 is obtained.

Fig. 10. | Fig. 11. |

Fig. 12. | Fig. 13. |

Fig. 14. | Fig. 15. | Fig. 16. |

Fig. 17. | Fig. 18. | Fig. 19. |

Fig. 20. |

*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
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 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
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 *p*_{1} + *p*_{2} + *a*_{1} + *a*_{2} = 0, *p*_{3} + *p*_{4} + *a*_{3} + *a*_{4} = 0,
*p*_{1} + *p*_{3} − *a*_{1} − *a*_{3} = 0, and *p*_{2} + *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.

*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 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

Fig. 21
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*_{1} 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*_{1}’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 *p*_{1}, and each of
the *n* normal paths will contain one,
and only one, of the letters *p*_{1}, *p*_{2},. . . *p*_{n},
whose sum will be Σ*p*. Similarly, if
*q*_{1} 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}, *q*_{3}, . . . *q*_{n},
the sum of the *q*’s along any normal path different from that of
the *q*_{1} will be Σ*q*. 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 1, 2, . . . *n*, and the *q*’s
to 0, 1, 2, . . . (n − 1) 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 Σ*nq* + Σ*p*.
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*_{1} and *q*_{1} 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*_{1}*p*_{4}*p*_{7}, *p*_{2}*p*_{5}*p*_{8}, *p*_{3}*p*_{6}*p*_{9}, and *q*_{1}*q*_{4}*q*_{7}, *q*_{2}*q*_{5}*q*_{8}, *q*_{3}*q*_{6}*q*_{9} occur,
each triplet thrice, along paths whose summation should be—Σ*p* 45
and Σ*r* 36. But if we make *p*_{1}, *p*_{2}, . . . *p*_{9}, = 1, 3, 6, 5, 4, 7, 9, 8, 2, and
the *r*_{1}, *r*_{2}, . . . *r*_{9} = 0, 2, 5, 4, 3, 6, 8, 7, 1, thrice each of the above sets
of triplets will equal Σ*p* and
Σ*q* 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 1 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
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 1, 2, 10, 8, 6, 14, 15, 11, 4, 13, 9, 7, 3, 12, 5, and 0, 1, 9, 7,
5, 13, 14, 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
Σ*p* and Σ*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 Σ*p*, but
alternately 2*q*_{1}, 2*q*_{3}, and 2*q*_{2},
2*q*_{4}; and the rows give Σ*q*,
but alternately 2*p*_{1}, 2*p*_{3},
and 2*p*_{2}, 2*p*_{4}; the diagonals
giving Σ*p* and Σ*q*. If *p*_{1}, *p*_{2}, *p*_{3}, *p*_{4} and *q*_{1}, *q*_{2}, *q*_{3}, *q*_{4} be 1, 2, 4, 3,
and 0, 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.

Fig. 22. |

Fig. 23. | Fig. 24. |

*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

Fig. 25—Nasik Cube.
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.
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 O, 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*_{1}, *q*_{1}, *r*_{1}
being placed in the corner cubelet
O, 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}*q*_{2}*r*_{2}, *p*_{3}*q*_{3}*r*_{3}, . . .
*p*_{7}*q*_{7}*r*_{7} are placed in the other cubelets
of the edge AO, and dispersed
in the same manner as *p*_{1}*q*_{1}*r*_{1}.
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 1, 2,
. . . 7, and the *q*’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 will
contain the numbers from 1 to 7^{3}, and the Nasical summations will
be Σ7^{2}*r* + Σ7*q* + *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 0, 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 1 to *n*^{3}, 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 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*_{1}, *s*_{2}, . . . *s*_{n}, 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.

Fig. 26. |

Fig. 27. | Fig. 28. |

This is
done by placing *s*_{1} on each of the *n*^{2} cubelets of the first cube that
contain *p*_{1}, and on the *n*^{2} cubelets of the 2d, 3d, . . . and *n*th cube
that contain *p*_{2}, *p*_{3}, . . . *p*_{n} respectively. This process is repeated with
*s*_{2}, beginning with the cube at which we ended, and so on with the
other *s*’s; the *n*^{4} cubelets, after multiplying the *q*’s, *r*’s, and *s*’s by
*n*, *n*^{2}, and *n*^{3} respectively, will now be filled with the numbers from
1 to *n*^{4}, and the constant summation will be Σ*n*^{3}*s* + Σ*n*^{2}*r* + Σ*nq* + Σ*p*.
This process may be carried on without limit; for, if the *n* cubes are
placed in a row with their faces resting on each other, and the corresponding
faces looking the same way, *n* such parallelepipeds might be
put side by side, and the *n*^{5} cubelets of this solid square be Nasically
filled by the introduction of a new letter *t*; while, by introducing
another letter, the *n*^{6} cubelets of the compound cube of *n*^{3} Nasik cubes might be filled by the numbers from 1 to *n*^{6}, 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’académie 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 South Kensington 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 7^{4} 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 2*n* − 1 sides
yields *n*^{2} 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
*n*^{2} 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, be 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 2*n* diagonal curves aforesaid is 12*n* (*n*^{2} + 1) 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 to *n* 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 1 to 16 if the numbers
1, 6, 11, 16 lie thus on a diagonal curve instead of in the order 1, 6, 16,
11. From a perfect magic ring of *n*^{2} cells containing one number
each, *n*^{2} distinct magic squares can be 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 *n*^{2} magic squares.
(C. A. M. F.)

See F. E. A. Lucas, *Récréations mathématiques* (1891–1894); W. W. R.
Ball, *Mathematical Recreations* (1892); W. E. M. G. Ahrens, *Mathematische*
*Unterhaltungen und Spiele* (1901); H. C. H. Schubert,
*Mathematische Mussestunden* (1900). A very detailed work is B.
Violle, *Traité complet des carrés magiques* (3 vols., 1837–1838).
The theory of “path nasiks” is dealt with in a pamphlet by C.
Planck (1906).