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F R I E R S O N.
AN IN T R O D U C
T I O N B Y P A U L C AR U S
CH I CA G O
T HE O P E
N
CO U R T P U B LI S H I N G CO
LO N D O N
KEG A N
A G E N TS
P A U L, TRE N CH , TRUB N ER
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MPA NY
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COP YRI G H T B Y
THE O P E N COU R T P U B CO
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m agi c
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TA B LE O F CO N TE N T S
.
P AG E
tro d u cti o n B y P u l Car u
M agi c S qu res
G ner l Q u aliti e nd Ch r ct ri ti f M gi c S qu r
O dd M gi c S qu r
E v n M gi c S qu ares
C n tr u ti o n o f E en M gi S qu r
L H ir
M th d
by D
C m p u n d M agi
S qu a r
C n ntri M agi c S qu r
G n r l N o t es o n the Co nstr u ti o n f M gi S qu re
M agi Cu b e :
Ch r t eri ti
o f M gi
Cu b e
O dd M gi C u b
E v n M gi Cu b
G ener l N t o n M gi Cu b e
Th F r nk li n S qu ar
A n A naly s i
f th F r nk li n S qu r
By P u l C r u
R efl
ecti o n s o n M agi
B y P u l C ru
S qu r
Th O rd r
f F ig u r
M gi S qu r in S ymbo l
Th M gi S qu r in Chi n
Th J
ai n
S qu r
A M ath m ati c l S t u dy f M agi S qu ar
B y L S F ri r n
A N
A naly i
A S t u dy f th P si b l N u m b r f V ri ti n i n M gi S qu r
N t o n N u mb r S ri U d in th Co n tr u ti n f M gi
S qu ar es
M agi c S qu are n d P yth g r n N u mb r B y C A B r w n
M r B ro w n S qu r nd l w nu m u m By P u l C ru
n d Co mb i n ti o n
S o m Cu ri o u M gi c S qu r
N t o n Vari o u Co nstr u tiv P l n b y whi h M gi S qu r My b
Cl s i fi ed
Th M th m t ic l V a l u
f M gi S qu re
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W
I N TR O D U CTI O N
.
H E p eculiar interest of m agic squares and all la w
s numero m
m
in general lies in the fact that they possess the char m of m ys
They appear to b etray so m e hidden intelligence which by a
tery
preconceived plan produces the impression of intentional design a
phenomenon which finds its close analogu e in nature
A lthough magic s q uares have no immediate practical u s e t h e y
have always exercised a great influence upon think ing peop le It
seems to me that they contain a lesson of great value in being a
palpable instance of the sym m etry of mathematics throwing thereby
a clear light upon the order that pervades the universe wherever
w e turn in the infinitesim a l ly small interrelation s of atoms a s well
as in the immeasurable domain of the s tarry heaven s an order
wh ich although of a di ff e r ent k ind and s till more intricate i s also
traceable in the development o f organized li fe and even in the
complex domain of human action
M agic s quares are a visible instance of the intrinsic harmony
of the laws o f number and we are thrilled with j oy at beholding
this evidence which r eflects the glorious symmetry o f the cosmic
order
Pythagoras says that numbe r is the origin o f all things and
certainly the law of nu m ber is the key that unlock s the secre ts of
the universe B ut the law o f number posse ss es an immanent order
which is at first sight mysti fying but on a more intimate acquain
tance we easily understand it to be intrinsically neces s ary ; and this
l a w o f nu m ber explains the wondrous con s istency o f the law s of
nature M agic squares a r e conspicuous in s tances of the intrin s ic
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I N T ROD U C TI ON
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harmony of number and so they will serve as an interpreter o f the
cosmic order that dominates all existence
M agic squares are a mere intellectual play that illustrates the
nature of mathe m atics and incidentally the nature o f existence
dominated by mathematical regularity They illustrate the intrinsic
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harmony of mathematics as well as the intrinsic harmony of the law s
of the cosmos
In arithmetic we create a universe o f figures by the process o f
counting ; in geometry we create another universe by drawing lines
in the abstract field o f imagination laying down definite directions ;
in algebra we p roduce m agnitu des o f a still more abst r act nature ex
pres s ed by letters In al l these cases the first step producing the gen
eral conditions in which we move lays down the rule to which all
further steps are subj ect and so every o ne o f these universes is
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dominated by a consistency producing a wonderful symmetry which
“
in the cosmic world has been called by P ythagoras the harmony o f
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the s pheres
There i s no science that teaches the harmonies o f nature more
clearly than mathematics and the magic squares are like a magic
mirror which r eflects a ray o f the symmetry of the divine norm
immanent in all things in the im m easurable immensity o f the co s mos
not les s than in the m ysteriou s depths of the human m ind
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P AU L CARU S
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MA G I C S Q UA RES
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I
A G IG
s quare s are of them s elve s only
mathematic al cu ri o s
,
but they involve principles whose un folding should lead th e
mind to a higher conception of the wonderful
and order which govern the science of number s
T H E G E NERA L Q U A L ITIE S
MA G IC
A CT ER I S TI C S
OF
S
A magic square consists of
numbers arranged in
rti ca l
horizontal and
quadratic form so that the
se s q uares can
co m er diagonal column is
be made with either an odd
n num
but as odd
which
those that
squares are
squares the two cla s
be con
govern the f o
ding s
sidered under sep
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D
MAG IC
SQ
UA R E S
.
i s not only requisite that the
amount but also that the s
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MG l C
J
R
E
S
A
t
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o
a ré f ( X 3
hfi écifi
1
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Q
O
O O Q
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or 0 0 0
shown in
F ig
.
I
NU
MB E R S
covers the smallest
.
ag
and it i s al s o the only pos s ible arrangement of nine di f ferent nu mbers
relatively to each other which fulfills the required conditions It
will be s een that the sum of each of the three vertical the three
horizontal and the two corner diagonal columns in this square is
making in all eight columns having that total :also that the
15
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sum of any two opposite numb ers is
10 ,
which is twice the center
he next largest odd magic square is that o f 5 X 5 and there
are a great many di ff erent arrangements o f twenty five numbers
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T o t al s
W
15
T o ta l s
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m a yc c /
i a l; re s ults
gen
65
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wh i ch 111 show
each arrangement b eing the pro
duction of a di ff erent constr uc tive method F ig 2 illu strates w het
5
m oldest and best known arrangem entj o f this square
The sum of each of the five horizontal the five ve r tical and the
t w o corner diagonal column s is 6 5 and the su m o f any two nu m b ers
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equidistant f r om the center num b er is 2 6
or tw i ce the center number
In order to intelligently follow the rule used in the construction
of this square it may be conceived that its upper and lower edges
are b ent around backwards and united to form a horizontal cylinde r
W i th the nu m bers on the outs i de the lower li ne o f figures thus
It may also be conceived
coming next in order to the upper
that the square is bent around backwards in a direction at right
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M AGI C SQUAR E S
3
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angles to that which was last considered so that it form s a vertical
cylinder with the extreme right and left hand columns adj acent to
each other
A n unde r
standing of this simple conception will as s ist the
et ade nt to follow tha é w methods of b uilding odd magic squares
"a d“that a r e to be de s cribed
9
based
on
a
oe m ew
/
r ight or le ft hand diagonal formation
R e ferring to Fig 2 it will be seen that the square is started
b y writing unity in the center cell of the upper row the consecutive
nu m bers proceed ing diagonally therefro m in a right hand direction
U sing the conception of a horizontal cylinder 2 will be located in the
lower ro w followed by 3 in the next upp er cell to the right H ere
the formation of the y ertic al cylinder b eing co nc eiv ed the next upper
)
square will be w here 4 is written then 5 ; further progress being
he r e b lock ed by 1 which al r eady occupies the next upper cell in
‘
diagonal order
When a bl ock thus o ccurs in the regular spacing (which will
b e at every fifth num b e r in a 5 X 5 square ) the next number must
in this case be writt e
n in the cell vertically below the one last filled
so that 6 is written in the cell below 5 and the right hand diagonal
o r der is then continued in cell s occupied by 7 and 8
H ere the
horizontal cylinder i s ima gi n
ed showing the location of 9 then the
conception of the vertical cylinder will indicate the location o f 1 0
fu rther regu lar progression being here once more blocked by 6
so 1 1 is written under 1 0 and the diagonal order continued to 1 5
A m ental picture of the combination of vertical and horizontal cyl
inders will here Show that fu rther diagonal pro gress is block ed by
1 1
so 1 6 i s written under 1 5 The vertical cylinder will then indi
cate the cell in which 1 7 must be located and the horizontal cylinder
will Show the next cell diagonally upwards to the right to be o cc u
pied by 1 8 and so on until the final num b er 2 5 is reached and the
square completed
F i g 3 illustrates the development o f a 7 X 7 s q uare con s tr ucted
according to the preceding method and the s tu den t is advised to
follow the sequence of the numbers to imp ress the rule on his mem
ory A v ariation o f the last method is shown in F ig 4 illu strating
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C
MAGI C SQUARES
4
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another 7 X 7 s q uare In thi s ex ample 1 i s placed in the ne x t c ell
horizonta lly to the right o f the center cell and the consecutiv e
number s proceed diagonally upward there from a s before in a
right hand direction until a b lock occur s The ne x t number i s then
written in the second cell horizontally to the right o f the la s t cell
fi lled (instead o f the cell below as in previou s examples ) and the
upward diagonal order is resumed until the ne x t block occu rs
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T o t al s 5 65
Fig 5
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Then two cell s to the right again and regular dia gonal order
tin ned and s o on until all the cell s are filled
,
co n
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The preceding example s may be again varied by writing t h e
number s in left hand instead o f right hand diagonal sequence
making u se of the same spacing of numb erS aS before w hen b lo cks
occur in the regular sequence of construction
We no w come to a series of very intere s ting methods for
building odd magic squares which involve the use of the knight s
move in chess and it is worthy of note that the squares formed by
these methods possess curiou s characteristics in addition to tho s e
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M AGI C SQUARES
5
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previou s ly referred to To che s s players the k night s move will
require no comment but for tho s e who are not familiar with this
game it may be explained a s a move of tw o eq uat es straight for
ward in any direction and o ne squa re to either right or left
The magic square of 5 X 5 illu s trated in F ig 5 i s s tarted by
placing 1 in the center cell of the upper row and the k night s
move employed in its construction will be two cell s upward and
one cell to the right
U sing the idea of the horizontal cylinder 2 must be written
in the se c ond
from the bottom as shown and then
in the
second
from the top N ow conceiving a combination of the
horizontal and v ertical cylinder s the ne x t move will locate 4 in the
extreme lower left hand corner and then 5 in the middle ro w We
no
w find that the next move i s bloc k ed by o ne , so 6 i s written below
5 and the k night s moves are then continue d and so on until the
ast
number
2
is
written
in
the
middle
cell
of
the
lower
J
and
w
;
5
l
the square is thu s completed
In common with the odd magic squares wh ich were previou s ly
described it will be found that in this s quare the s um of each of
the fiv e horizontal the fi ve perpendicular and t h e two corner diag
onal column s is 6 5 also that the s um o f any two number s that are
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In
addition however to these characteristics it will be noted
that each spiral row of figures around t h e horizontal and vertical
cylinder s traced either right handed or left handed also amounts
to 6 5 In the vertical cylin der there are fi ve right hand and five
le ft hand spirals two of which form the two corn er diagonal col
The-same
u m n s acros s the square leaving eight new combination s
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301
Counting therefore five horizontal c ol n s fi ve vertical col
p
r
fi t“ g
eo n r i ght and le ft hand
u m n s two corner diagonal columns and sui tspiral columns there w i ll a d in all tw enty
columns
each o f w hic h will sum up to 6 5 whereas in the 5 X 5 sq uare sh o wn
7
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l
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“i g
b
,
MAGI C SQUAR E S
J
o
in
F ig
.
2
there will be found only
.
a
tw
el ve
/
column s that will amount
to that number
This meth o d of construction is subj ect to a number o f variations
Fo r example the knight s move may be upwards and to theleft
hand instead of to the right or it may be made downwards and
either to the right or left hand and also in other directions There
are in fact eight di f f erent ways in which the knight s move may
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moves are indicated by figu re 2 s in di ff erent cells of F ig 6 and
eac h o f these moves i f continued in its own dire ction varied b y
break s as before described will produce a di ff erent but
square The re m aining two possible knight s moves indi
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c a ted
by cypher s will not
,
F ig 7
.
.
may here be de s irable to e x plain another method for locating
numbers in their proper cells which some may prefer to that which
It
involve s the conception of the double cylinder This method con
sists in con s tructing part s o f au x iliary squ are s around tw o or more
Side s o f the main s quare and temporarily writing the numbers in
the cells o f these auxiliary s quare s when their regular placing car
rie s them outside the limit s of the main sq uare The temporary
location of these number s in the cells o f the au x iliary sq uare s will
then indicate into which cell s of the main s quare they mu s t be per
m anently transferred
F ig 7 shows a 5 X 5 main square with parts of th r ee auxiliary
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MAGI C SQUAR E S
squares and the main square will
,
F ig 5
.
be
7
.
built up in the same way as
.
S t arting with
in the center of the top Ha s; the first knight s
“
m ove of two cells up wa r ds and one to the right takes 2 ac r oss the
top margin o f the main square into the second cell of the se c ond
line from the bottom in one of the auxiliary squa r es so 2 must b e
trans ferred to the same relative position in the main square S tart
ing again from 2 in the main square the next move places 3 within
the main square but 4 goes out of it into the lower left hand corner
o f an auxiliary S quare from which it must be transferred to the
same location in the main square and so on throughout
The method last descri b ed and also the conception of the double
cylinders may be considered Simply as aids to the beginne r With
a little practice th
cells in
the squa r e as fast as the figu r es can be written therein
H aving thus explained
struction the general principles governing the development of fl
odd magic squares by the se method s may now be formulated
1
The ce nter cell in the square must always contain the middle
nu m ber o f the series of numbe r s used i e a number which
one
hal
f
the
sum
of
the
fi
r
st
and
last
num
b e r s of
Q
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t magic square can therefo r e be started from its
center cell but it may be started from any cell other than
the center one
With certain specific exceptions which will be re ferred to
later on odd magic squa r es may be constructed by either
right or left hand diagonal sequence or by a number of so
called knight s moves varied in all case s by periodical and
well defined departures fro m normal spacing
The directions and di m ensions o f these departures from
normal spacing or break
are governed by the relative spacing of cell s occupied by
the fi rst and last numbers of the serie s and may be deter
mined as follows
No
,
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3
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4
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1
m
M AGI C SQUARES
.
R ULE : P lace t h e fi r s t number o f the
s in any de s ired cell
number of the s erie s
excepting
the
center
one
)
(
in the cell
the cell con
between the
cell that contain s the last number o f the series and the cell
that contains the first number of the series mu st then be
repeated whenever a block occurs in the regular p ro gres
S i on
g
e
.
EX
U sing a blank sq uare o f
m
cell of the upper fin:
the middle cell in the lower
A M PLES
5 X 5
li n e,
25
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must be written therein
will
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above in the middle vertical column
or what is the same thing and ea s ier to follow one cell below 2 5
When therefore a square of 5 X 5 is c o m m enc ed w ith the fi rst
number in the middle cell o f the upper East the b reak z m o v e will
f the method of regular
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vii
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2,
x,
x,
Fig 8
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M
Fig 9
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s how s t h e
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in a 5 X 5 sq u are as above
ar ci diagonal advance
A gain u s ing a blank 5 X 5 s quare 1 may be written in the cell
immediately to the right o f the center cell bringing 2 5 into the cell
to the le ft o f the center cell The b reak z mo v es in this ca s e w ill !
therefore be two cells to the right of the last cell occupied i rresp ec
tive o f the method u sed for regular advance F ig 9 illustrates the
b reakzm o v es in the above case when a right hand upward diagonal
advance i s u s ed The positions o f these b reak mo v es in the square
advance
.
F ig 8
.
b reak mo v es
=
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MAGI C SQUARES
9
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will natu rally vary with the method of advance but the relative
spacing o f the moves themselves will remain unchanged
NOTE :The foregoing break moves were previously de s cribed in
s everal sp ecific example s (S ee F igs 1 2 3 4 and 5 ) and
the reader will now observe ho w they agree with the gen
eral rule
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O nce more u s ing a blank square o f
5 X 5
,
1
may b e written
in
the upper left hand corner and 2 5 in the lower right hand corner
I will then occupy a position four cells removed from 2 5 in a left
hand upward diagonal or what is the same thing and ea s ier to
follow the next cell in a right hand downward diagonal Thi s will
therefore be the mea
w
spacing
F ig I O shows the break moves which occu r when a
k night s move o f two cell s to the right and one cell upward s i s u s ed
for the regular advance
A s a fi nal e x ample we will write 1 in the seco nd cell from th e
left in the upper li ne o f a 5 X 5 sq uare which calls for the placing
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1
F ig
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10
F ig
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11
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o f 25 i n the s econ d sq uare from the right in the lower li m Th e
place relation between 2 5 and I may then be described by a k night s
move o f two cells to the left and one cell downwards and thi s must
be the break move whenever a block occurs in the regular spacing
The break- moves Shown in F ig 1 1 occur when an upward right
hand diagonal sequence is u sed for the regular advance
A s before stated odd magic s quare s may be commenced in
a
any cell excepting the center one and p e f eet squares may be built
mo v es
up from such commencements by a great variety of
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MAGI C SQUARES
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such as right hand diagonal sequence upward s or downward s left
hand diagonal sequence upwards or downwards or a number of
knight s moves in variou s directions There are four possible moves
from each cell in diagonal sequence and eight possible moves from
each cell b y the knight s move S ome o f these moves will pro duce
m ag ic squares b u t the r e will be found m any exceptions
which can b e Shown m ost r eadily by diagrams
F ig 1 2 is a 5 X 5 square in which the pointed arrow head s i n
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dic ate
m
rea
l
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the directions of diagonal sequence by which
G R
may be con s tructed while the blunt arrow heads Show the directions
,
12
1g
1g
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13
/
éfi 5 ;
j
o f diagonal sequence which will lead to
re s ult s F ig
1 3 illustrates the various n o rm a l knight s moves whi ch may b e
started from each cell and also indicate s with pointed an d blunt
.
.
’
squa
r
e
example it will be seen from F ig 1 2 that a m
X
5
5
cannot be b uilt by starting from either o f the four corner cells in
any direction of diagonal sequence but F ig 1 3 shows four di ff e r
ent normal knight s moves from each corner cell any of which will
produce pet -f es t squares It also sh o w s fou r other normal knight s
M
4
moves which p m duee i m perf ect sq ua res m
a o
Fo r
.
.
,
’
W
,
.
M
—
.
EXA M PL E S
OF
5 X 5
4
MAGI C SQ UARES
0
’
-
.
and 1 5 show two 5 X 5 square s each having 1 in
the upper le ft hand corner cell and 2 5 in the lower right hand
corn er cell and being constructed with di ff erent knight s moves
F igs
.
14
,
’
,
.
M AGI C SQUARES
and 2 3 illustrate three 5 X 5 square s each having
in the upper right hand corner and 2 5 in the lower left hand
F igs
I
.
.
2 1 , 22,
Fig
21
.
,
.
corner and being built up respectivel y with upwar dand downward
right hand normal k night s move s and a downward right hand
elongated k night s move
,
’
,
’
’
.
the sak e o f simplicity the s e e x ample s have been shown in
5 X 5 s quare s but the rules will naturally apply to al l size s o f odd
magic square s by u sing the appropriate numbers The explana
tions have also been given at some length becau se they cover gen
eral and comprehensiv e methods a good understanding of which
Fo r
‘
,
.
,
is clear that no s pecial significance can be attached to the
so called knight s move
per 5 6 as applied to the construction of
magic squar es it being only one o f many methods o f regular s pa
cing all o f which will produce equivalent results F o r example the
i
square
Shown
in
i
I
may
be
sa
d
to
be
built
up
by
a
suc
F
X
3
3
g
It
’
-
,
,
,
.
,
,
.
’
cession o f abbreviated k night s moves o f one cell to the right and
one cell upwards S quares illustrated in F igs 2 3 and 4 are also
constructed by thi s abbreviated knight s move but the square illu s
t rated in F ig 5 is built up by the normal knight s move
It i s equally easy to construct squares by means o f an elongated
k night s move s ay fou r cells to the right and one cell upwards
a s shown in F ig 2 4 or by a m ove consisting of two cell s to the
right and two cells downward s as shown in F ig 2 5 the latter being
.
.
,
,
’
,
’
.
.
’
,
,
.
,
,
.
,
MAG IC SQUAR E S
I3
.
e q uivalent to a right hand downward diagonal sequence wherein
alternate cells are consecutively filled
There are in fact almost innumerable combinations of moves
by which ”
odd magic squares may be constructed
.
.
T o ta l s
To t al s
Fi g
.
25
369
.
3 69
.
.
foregoing method for building od d magic squares by
he regular spacing of consecutive
s tructi on have been k nown for many years
O ne of the most interesting of these
.
- method s involve s
o lder
w
the u se of two or more pr i mary squares the sums of numbers in
,
o
z :
)
M AGI C SQUARES
14
.
similarly located cells of which constitute the correct numbers for
transfer into the corresponding cells o f the magic square that is
to b e constructed there from
This method has been ascribed primarily to D e la H ire but ha s
been more recently improved by P rof S chef fl
er
It may be si m ply illustrated by the construction of a few 5 X 5
squares as examples F igs 26 and 2 7 Show two simple primary
squares in which the num b ers 1 to 5 are so arranged that lik e num
.
.
.
.
.
bers occur once and only once in similarly placed cells in the two
squares ; also that pairs of unlike numbers are not repeated in the
same order in any si m ilarly placed cells Thus 5 occupies the ex
treme right hand cell in the lower li ne o f each square but this co m
bination does not occur in any of the other cells S o also in F ig 2 7
i
occupies
the
extreme
right
hand
cell
in
the
upper
line
and
in
F
4
g
26 this cell contains 2
N o other cell howeve r in F ig 2 7 that
c ontains 4 corresponds in position with a cell in 2 6 that contains 2
L eaving the numbe r s in F ig 26 unaltered the numbers in F ig 2 7
m u st n o w be changed to their respective hq numbe r s thus pro
duc ing the -li e s square shown in F ig 2 8
B y adding the cell num
b ers o f the m
square F i g 26 to the correspond i ng cell numbers
.
,
,
.
.
.
.
,
.
.
,
,
.
W
.
,
e
‘
.
.
”
o
f
.
,
M
”
—
d
J
.
0
.
Q
iani ln
nu m b ers
F ig
.
2
6
,
o,
2,
3
5
10,
,
4
,
,
1 5,
Fig
.
.
5
20
.
.
27
.
9 of the in square Fig 28 the magic square shown in F ig 2 9 is
fo r med which is also identical with the one previou sly given in
0
/
.
.
,
,
F ig
.
14
.
The simple and direct formation of F ig 1 4 may be thus com
pared with the D e la H ire method for arriving at the same re s ult
.
.
I
It
S
is evident that the
square shown in Fig 2 8 may b e dis
a
m
n
se
d
b
i
i
e
with
y
entally
substituting
the
num
ers
for
the
b
l
ug
p
p me
.
w
Fig
28
.
Fi g
.
29
.
ée A
/
.
num be rs gi ven in F ig 2 7 when performing the addition and by
so doin g only two primary squares are required to construct the
uare The arrangement of the numb ers 1 to 5 in the two
primary squares i s obviously open to an immense number of varia
tions each of which will result in the formation of a di ff erent bu t ( 9
magic
square
A
ny
o
f
these
squares
however
may
b
e
4
90
t
1
mo re readily constructed by the direct m ethods previous l y explained
A f ew o f these var i at i ons are g i ven as examples the In ; num
The
square F ig 3 2 is formed
b ers remaining unchanged
f r om the p rima r y square F ig 3 1 and i f the numbers in F ig 3 2
‘ /
a r e added to those in
a ny-eq uat e F ig 3 0 the magic square
Thi s square will be found identical with
F ig 33 will be produced
.
,
.
/
9
,
/
.
fl
(
/
,
,
.
,
.
M
.
.
,
.
,
.
.
that Shown in
.
F ig
.
1
5
Fig
.
.
30
F ig
.
.
31
.
i
pre
F
A
s
a
fina
l
example
the
magic
square
shown
in
g 37
X
v io u sly given in F i g I7 is made b y the addition of numbers in the
,
.
,
fl
.
9
’
‘
9
/
‘
W
W
M A GI C SQU ARES
to the numbers occupying s imilar cell s in
b
6
the
latter
eing
derived
from
h
n
t
t
e
a
s
“
3
g e
s quare
s quare
F ig
.
F ig
34
.
M
,
.
Fig 33
F ig 3 2
.
F ig
f
.
35
.
.
square shown in F ig 38 i s no w con s tructed
and
the
numbers
therein
added
34
s q uare Shown in F ig
the
If
.
.
.
m
.
ff
is
o
tained
showing
that
two
di
erent
magic
square
s
may
b
b
e
39
made from any two primary s q uares by forming a -leey sq uare from
W
,
M
—
fi
Fig 38
Fig 3 7
.
.
.
.
F i g 39
.
.
each of them in turn F ig 39 has not been given before in thi s
book but it may be directly p r oduced by an elongated knight s
.
.
’
,
9
MAGI C SQUARES
1
.
9
written in heavy figures have the correct summation The numbers
in these two colu m ns must the r efo r e b e left as they are but the loca
tion of all the othe r nu m b e r s which are written in light figu res mu s t
b e changed
A si m ple method for e ff ecting this change consists in
su b stituting for each number the complement between it and 1 7
Thus the comple m ent between 2 and 1 7 is 1 5 so 1 5 iii —s t b e written
in the place of 2 and so on throughout A ll of the light figure
,
.
,
,
,
.
.
,
,
,
.
Fi g 45
.
numbers being thus changed the result will be the pa i n t magic
squa r e Shown in F ig 42
The same relative arrangement o f figures may be attained by
leaving the light figu r e nu m b ers in their original positions as shown
,
.
in F ig 43 and changing the heavy figure numbers in the two
corner dia gonal columns to their respective complements with 1 7
.
It
,
will be seen that this is only a reversal of the order o f the figures
Fig 46
.
Fig 47
.
.
corn
er diagonal column s and the re s ulting magic s q uare
which is Shown in F ig 44 is Simply an inve r sion o f F ig 42
F ig 45 i s a geometrical diagram o f the numbers in F ig 42
in the
tw o
,
.
.
.
.
.
,
and it indicate s a regular law in their arrangement which al s o hold s
good in many larger even s q uare s a s will be s een later on
,
,
.
M AGI C SQ UAR E S
.
There are many other arrangements of sixteen number s wh ich
will fulfil the requi red conditions but the examples given will suff i ce
to illustrate the principles o f this s q uare
.
The next even magic s quare is that of
many variation s i s shown in
.
F ig 46
.
.
6 X 6
and one of its
A n analysis of this s quare
,
Fig 48
.
.
“
with the aid o f geometrical diagrams will point the way not only
to its own reconstruction but also to an easy method for building
6 X 6
0
s
6
square in which all the numbers from
F ig
.
49
.
to 36 are written in arith metical s e q uence and the twelve numbers
in the two corner diagonal columns will be found in magic square
order all other numbers requiring rearrangement L eaving there
1
,
.
,
fore the numbers in the diagonal columns unchanged the next s tep
,
will be to write in the places o f the other numbers their complements
with 3 7 making the s quare Shown in F ig 48 In this square
,
.
.
twenty four numbers (written in heavy figures ) out o f the total of
-
MAGI C SQUARES
22
.
M
the data for building a large number o f
all Showing however the sa m e general characteristics as
,
,
S,
F ig 46
.
.
A series o f these diagrams with some modification s of their
,
irregularities is given in F ig 5 1 and in order to build a variety
o f 6 X 6 magic square s there from it is only necessary to select three
.
,
F ig 5 1
.
,
P rt )
(F ir st
a
.
diagrams in the order A B and C w hi ch hav e eac h a di ff erent f o rm
o f i rregu l ari t
y and after numbering them in arithmetical sequence
from 1 to 3 6 as shown in F ig 49 copy the num b ers in di agra m m a ti c
order into the cells of a 6 X 6 square
It mu st be remembered that the cells in the corner diagonal
,
,
,
,
,
.
,
.
/ U
"
MAGI C SQUARES
23
.
g
i
columns of
en magic square s may be correctly fi lled by writing
the number s in arithmetical order according to the rule previou s ly
given so in beginning any new even square it will be found help ful
to first write the number s in these columns and they will then s erve
a s guides in the further development of the s q uare
,
,
.
F ig 5 1
.
( S ec o n d
P rt )
a
.
W
Taking for example the 6 X 6 magic square Shown in F ig 46
it will be seen from F ig 49 that it is constructed from the diagrams
marked 1 9 and 1 7 in F ig 5 1 Compari ng the first
with diagram A F ig 49 the s equence of numbers i s 1 3 5 34
in unb ro k en order ; then the diagram s how s that 33 and 3 mu s t b e
.
.
—
.
.
—
—
,
‘
.
,
,
,
.
MAGI C SQUARES
24
.
x
i
i
s
written
ne
t
in
s
tead
of
2
n
6
t
h
en
a
d
n
3
(
33 )
3
o f th i s square (s till u s ing diagram
that 5 and 2 mu s t be tran s po s ed
2 i s written in s tead o f 5 ; then 4 ; the n a s 3 and 3 3 mu s t b e tran s
m
posed 3 3 is written instead o f 3 5 in s tead o f 2 and the
fi ni shed with 3 6 D iagram B gives the development o f the s eco nd
tran s po s ed s o
,
,
,
,
m
,
W
W
W
.
TA BLE S H O I N G 1 28 C H A N GES H I C H MAY B E RU N G
TH E T
EN TY F OUR D IAG RA M S i N F I G 5 1
B
C
ON
-
.
1 7,
9
6‘
5
1
C‘
(C
(6
(C
9,
z
or
H
(l
6‘
66
U
N
8
19
or
20
(C
6
IO
1 8,
16
(c
I I
I6
12
I6
3
2
13
14
,
23
or
e
r
:
1
e
grams 2
e
,
2
I6
16
16
qu r d riv d fr
a e
1
IS
E X A M PLE S
om
qu r d riv d fr
a e
gram
1 0 , an d 1 8
.
1 28
.
S
dia
change s
16
=
Total change s
S
.
e
e
s
8
,
o
m dia
1 3 , a n d 22
.
and fi ft h
of the s q uare in the s ame manner and diagram C
the development o f the third and fourth li ne s thu s completing the
'
,
,
The annex ed table s hows 1 28 change s which may be rung on
the twenty four diagram s s hown in F igure 5 1 each combinat i on
giving a diff erent 6 X 6 sq uare and many others might be added
to the list
T h e ne x t s ize of even magic sq ua re is that of 8 X 8 and in s t ead
-
,
,
.
,
MAGI C SQUARES
.
of pre s enting one of the s e squares ready made and analyzing it
we will now use the information which has been offered by previous
e x amples in the construction of a new square o f this size
R e ferring to F ig 45 the regular geometrical diagrams of the
4 X 4 square naturally suggest that an expansion of the same may
be utilized to construct an 8 X 8 square Thi s expanded diagram
,
.
.
,
.
F ig
.
52
.
is accordingly shown in F ig 5 2 and in F ig 5 3 we have the magic
squa r e that i s produced by copying the numbers in diagrammatic
.
order
,
.
.
To tal s
Fig 5 3
.
M
9
’
260
.
.
As might be anticipated this square
,
and the ease with which it has been constructed points
to the si m plicity of the m ethod employed
The m agic squa r e shown in F ig 5 3 is howeve r only one of a
e,
.
.
,
,
MA G I C SQUAR E S
26
.
o f 8 X 8 squares all of which have the s ame general
characteristics and may be constructed with e q ual facility from
m ultitude
,
JIS 1)
Fig
.
m3
.
2
11 1
J/
02
.
55
diagram s that can be readily derived from tran s
variou s
position s
f
o
54
11
F ig
3
4
of
F ig 5 2
.
F iv e of
these variations are illu strated in
F ig
.
.
MA G I C SQUAR E S
27
.
which
also
Show
the
transpo
s
ition
s
by
which
they
are
formed
54
from the original diagrams To construct a pel éoet magic s quare
fromeither of these variations it is only necessary to make four
copie s o f the one selected annex the numbers 1 to 64 in ari th metical
,
.
,
ta l s
To
Fig
.
56
260
2
.
.
order a s before e x plained and then copy the numbers in diagram
m atic sequence into the cells of an 8 X 8 sq uare
It will be noted in the constru ction of the 4 X 4 and 8 X 8
,
.
4}
23
«a
19
4 2
1!
.
‘
F ig 5 7
sq uare s t h at only one form of diagram ha s been hitherto u s ed for
each sq uare whereas three di ff erent form s were required for the
It is possible however to u s e ei th er two three or
6 X 6 s quare
four di ff erent diagrams in the construction o f an 8 X8 sq uare as
,
.
,
,
,
,
,
MAGI C SQUARES
Fig
.
T o t al s 2
260
.
T otal s
260
.
58
.
.
Fig 5 9
.
.
Fig 60
.
.
M AGI C SQUARES
0
3
.
pand the three diagram s o f the 6 X 6 square (F ig 49 ) into five
diagra m s that are required for the construction o f a ser i es of
.
Is
Fig
sq uares
10
X
in
F ig 6 2
10
.
.
.
505
.
62
.
The s e five diagrams are Shown in
F ig 6 1 ,
.
and
we have the magic s q uare which is made by copy i ng the
“
F ig 63 (F ir st p art )
.
numbers from
10
X
1 00
in diagrammatic order into the cells of a
square
will be unnecessary to proceed further with th e construction
10
It
1
to
.
.
.
M AGI C SQUARES
1
3
.
other 1 0 X 1 0 squares fo r the reader will recognize the striking
resemblance between the diagrams of the 6 X 6 and the 1 0 X 1 0
squares especially in connection with their resp ective irregularitie s
of
,
,
,
F ig 63
.
( S eco nd
pa
rt )
.
T o ta l s 2 870
It
will al s o be s een that the same meth od s which were u sed for
M AGI C SQUARES
32
.
varying the 6 X 6 diagram s are equally applicable to the 1 0 X 1 0
diagram s s o that an almo s t infinite variety of change s may be rung
on them from which a corresponding number o f 1 0 X 1 0 s quares
may be derived each of which will be di ff erent but will re s embl e
the serie s o f 6 X 6 squares in their curiou s and characteri s tic im
perfection s
,
,
,
,
.
Fig 65
.
F ig 6 3 Show s
.
( F ir st P a rt )
.
a series of diagrams from which the
12
X
12
MAGI C SQUAR E S
Fig 65 ( S eco n d
.
Fig 66
.
.
.
P rt )
a
.
M AGI C SQUAR E S
34
.
square in F ig 64 i s derived
The geometrical de s ign of these
diagra m s is the same a s that shown in F ig 5 2 for the 8 X 8 s q uare
and it is mani fest that all the variations that were made in the 8 X 8
diagrams are also possible i n the 1 2 X 1 2 diagrams besides an
.
.
“
.
,
,
.
immense number o f additional changes which are allowed by the
increased size o f the square
.
we have a serie s of diagrams illustrating the de
v el o p m ent of the 1 4 X 1 4 magic square shown in F ig 66
These
diagrams being plainly derived from the diagram s of the 6 X 6 and
1 0 X I O squares n
o explanation o f them will be required and it is
In Fig 6 5
.
.
.
,
,
evident that the diagrammatic method may be readily applied to
the constru ction o f all sizes o f even mag i c squares
It will be
that the foregoing diagrams illustrate in a
g raphic manne r the interesting results attained by the harmoniou s
M
.
association of figures and they also clearly demonstrate the almost
infinite variety of possible combinations
,
“
.
F ig 68
.
CO N S T R
U CT I O N
Fig 69
.
.
.
EVEN M A G I C S Q UA R E S B Y
H IR E S ME T H OD
OF
D E LA
’
.
set magic
d-
po
1
.
sq uare of
4X 4
may be con s tructed as fol
the corner diagonal columns o f a 4 X 4 s q uare with t h e
number s I to 4 in arithmetical s e q uence s tarting from the
F ill
,
upper and lower left hand corner s (F ig
F ill the remaining empty cell s with the mi ss ing numbers of
the serie s 1 to 4 so that the sum o f every perpendicular and
.
2
.
c
M AGI C SQUARES
3
.
35
.
Con s truct another 4 X 4 square having all number s in the
same po s itions relatively to each other as in the la s t sq uare
but reversing the direction of all horizontal and perp endicula r
columns (F ig
F orm the a
s quare F ig 70 from F ig 69 by s ub s tituting
numbers
for
i
numbers
and
then
add
the
numbers
y
p
y
g
li oy
oquu e to simi l arly located number s i n the primary
n tht e lsquare F ig 68 The re s ult will be the per-f eet square of
4 X 4 shown i n F ig 72
,
,
M
.
4
.
{
M
/
35
fi
/
77”
.
.
,
u
g
.
.
.
.
B y making the
s q uare F ig 71 from the primary sq uare
F ig 6 8 and adding the numbers there i n to s i m i larly located number s
in the primary square F ig 69 the same magic square of 4 X 4 will
b eprod u ced but with all horizontal and perpendicular column s re
versed in direction a s s hown in F ig 73
.
.
.
,
,
.
NU
.
MB ERS
Fig 71
.
Fig 72
.
Fig 73
.
.
.
The magic sq uare of 6 X 6 s hown in F igure 46 and al s o a
large number o f variations of same may be readily constructed by
the D e la H ire method and the easiest way to e x plain the proce s s
will be to analyz e th e above mentioned square into the nece ss ary
primary and hey squares u sing the guinea-number s 1 to 6 with their
,
”i t
.
,
r e s pective
lac y
number s as follow s
numbers I
umbers 0
,
2,
3
,
6,
1 2,
,
4
5
,
6
1 8,
24,
30
,
.
.
5
M AGI C SQUARES
.
The cell s of two 6 X 6 s q uares may be re s pectively fi lled wit h
number s by analyzing the contents o f each cell in
at the left hand cell in the upper row w e
note that this cell contains 1 In order to produce this number by
,
.
mb er it
is evident that
their respective cell s
u
.
W
11
the cells the finished squares being s hown in
,
s
7
7
7
M
A nother m”3
F i g 74
.
sq uare may
b
g
by wr i t i ng i nto the var i ou s cell s of the former the
.
M
A
N
F ig 46
.
m
be derived from the
no w
(D
up
.
)
number s that correspond to the
M
sq ua re
-
n
i
p k
e
F ig 74
.
.
numbers of the latter Thi s
F ig 76
It will be seen that the
numbers in Fig 76 occupy the sam e relativ e positions to each other
as the numbers of the first primary s q uare (F ig
but the direc
he?
.
.
“
.
'
.
.
tion o f all column s i s changed from horizontal to perpendicular
and vice versa
To distinguish and identify the two primary square s which are
u s ed in these operations the first one (in this ca s e F ig 75 ) will in
future be termed the A prima ry s quare and the s econd one (in thi s
case F ig 76 ) the B primary square
It i s evident that the magic square o f 6 X 6 shown in F ig 46
may no w be reconstructed by adding the cell numbers in F ig 74
,
.
.
,
,
.
.
.
.
M AGI C SQUARES
4
.
.
The sum of every column in a 6 X 6
s t be 90
and under these conditions it follows that the sum o f every
column of a 6 X 6 magic square which is formed by the
combination o f a primary square with a leer- square mu st b e
k
1 1 1
,
With
the
necessary
changes
in
numbers
the
a
ove
rules
hold
b
5
good for all sizes o f A an d B prima ry squares and h
.
sq u a reso
f
a
t ”
a
gend a
t
We may now p roceed to show how a variety of 6 X 6 m agic
s q uares can be p roduced by di ff erent combinations of numbers in
F i g 77
.
.
F ig 78
.
.
primary and 3 9. s quares The six horizontal column s in F ig 75
show some of the combinations of numbers from 1 to 6 that can be
u s ed in 6 X 6 A primary s quares and the po s ition s of these columns
.
.
,
or rows o f figure s relatively to each other may be changed so as
to produce a vast variety of s quares which will naturally lead to
the development of a corresponding number o f 6 X 6 magic squares
.
In
order to illustrate this in a systematic manner the different
row s of figures in F ig 75 may be rearranged and identi fi ed by letters
as given in F ig 77
.
.
.
MA G I C SQUAR E S
39
.
show s the se q uence of numbers in the diagonal columns
of these 6 X 6 A primary squares and as thi s arrangement cannot
b e changed in thi s s eries the variou s horizontal columns or rows in
The small letters at the right
Fig 77 u st be selected accordingly
F ig 78
.
,
,
m
.
{
No
}
J
.
1
.
No
.
.
2
No 3
.
.
No 4
.
.
b
b
f
e
No 5
.
.
No 6
.
.
.
’
e
a
’
a
c
’
1
e
a
a
Fi g 79
.
.
of F ig 78 indicate the diffe r ent horizontal columns that may b e u s ed
fo r the respective lines in the square ; thus either a b or c column
in F i g 77 may be u sed for the first and sixth lines a e or f for the
second and fifth and c d or e for the third and fou rth lines but
neither b c or d can be u sed in the second or fifth lines and so forth
S ix diff erent combinations o f columns are given in F ig 79
from which twelve di ff erent 6 X 6 magic squares may be con
structed Tak ing column NO 1 as an example F ig 80 Shows an
,
.
,
,
,
,
,
,
,
,
,
.
,
,
,
,
.
.
F ig 80
.
.
,
F ig 8 1
.
.
A primary square made from the comb ination
.
d
and
f
F ig 8 1 is the B primary square formed b y reversing the di rection
of the horizontal and perpendicular columns of F ig 80 The key 'f j
square F ig 82 i s then made from F ig 8 1 and the 6 X 6 magi c
square in F ig 84 is the result of adding the cell numbers o f F ig 82
to the corresponding cell numbers in F ig 80
a,
c,
,
,
e,
b
,
.
.
.
.
.
.
.
.
.
t
MAGI C SQUARES
.
Th e above operation may be varied by rever s ing the horizontal
s q uare F ig 82 right and left as shown in F ig
the magic sq uare given in Fig 8 5 In th i s w av
di ff erent magic squares may be derived from each combination
.
.
.
.
.
Fig 82
.
Fig 83
.
.
.
It
will be noted that all the 6 X 6 magic squares that are co n
s tructed by these rules are s imilar in their general characteristics
to the 6 X 6 s quare s which are built up by the diagrammatic sy s tem
8 X 8 magic sq uare s may be constructed in great vari
ety by the method now under con s ideration and the diff erent co m
.
,
Fig 85
Fig 84
.
.
.
o f number s from 1 to 8 given in F ig 86 will be fo und u s e
ful for laying out a large number of A primary squares
b inatio ns
.
.
Show s the fi x ed number s in the diagonal column s of
the s e 8 X 8 A p rimary s quares and al s o de s ignate s by letter s the
speci fi c row s of figu res which may be u sed for the di ff erent hori
zo ntal column s
Thu s the row mark ed a in F ig 86 may be u s ed
F ig 87
.
,
.
.
M AGI C SQUARES
f or
41
.
the first fou rth fifth and ei ghth horizontal columns but cannot
be employed for the s econd third sixth or s eventh columns and s o
,
,
,
,
sugg
,
,
e s ts hal f a dozen combinations which will form
F ig 86
.
.
many p rimary squares and it is evident that the number of
po ss ible variations i s very large It will su ffi ce to develop the fi r s t
and third o f the series in F ig 88 as examples
as
,
.
.
.
F ig 87
.
.
i s the A primary sq uare developed from column No 1
in F ig 88 and F ig 90 is the B p rim aiz
square
made
by
rever
s
ing
y
the direction of all horizontal and perpendicular column s of Fig 89
numbers for the
numbers in F ig 90 and
S ub stituting
F ig 89
.
.
.
,
.
.
m
L
/
.
,
.
MAGI C
42
S
numb ers to the
pm
numbers in F ig 89 gives
The latter will
the p epf eet magic s quare o f 8 X 8 shown in F ig 9 1
be found ide ntical with the s q uare which may be written out directly
from diagrams in F ig 5 2
adding these
i
.
.
.
No
.
1
No
.
.
No 3
2
No 4
.
a
’
’
a a
’
a
e
b
’
a
'
b
a
'
a
e
6
dd
6
’
a
F ig 88
.
2
9
.
'
b
.
No 6
No 5
.
b
F ig
.
.
shows an A primary square produced from column
The B primary square F ig 93 being made in the
88
in F ig
regular way by reversing the direction o f the columns in
No 3
.
.
.
.
number s
Hey numbers
P ri me
F ig 89
.
1,
2,
3
0,
8,
1 6,
,
4
5
24,
2
3 ,
,
6
,
,
8
7
,
40 , 48, 5 6
Fig 90
.
.
F ig 92
.
.
.
.
.
magic square o f 8 X 8 in F ig 94 is developed from
the s e two p rimary s q uares as in the last example and it will be
.
,
found similar to the square which may be formed directly from
diagram
No
.
2
in
F ig 5 4
.
.
MA G I C SQUARES
.
Total s
Fig 9 1
.
F ig 92
.
.
.
Fig 93
.
F ig 95
.
.
.
M AGI C SQUARES
.
magic sq uare which i s con s tructed
by combining the A primary square in F ig 89 with the B primary
square in F ig 93 a fter changing the latter to a b y s quare in the
manner b e fore described
i s magic square may also be directl y
constructed from diagram N o 4 in F ig 5 4
It i s evident that an almost unlimited number of di ff er ent
8 X 8 magic s q uares may be made by the foregoing method s and
their application to the formation of other and larger sq uare s i s s o
obviou s that it will be unnecessary to present any further ex amples
Fig 95
.
shows another
8 X 8
.
.
.
.
.
.
.
,
.
CO
MP OU ND
A
M
,
5
S
M AG I C
S
Q UA R E S
.
i
n
T q
may be described as a series of small magic sq uare s ar
ranged q uadratically in magic sq uare order
The 9 X 9 square shown in F ig 96 is the smallest of thi s cla s s
e
.
.
that can be con s tructed and it consists of nine 3 X 3 sub s q uares
arrange din the same order as the numerals I to 9 inclu s ive in the
1
square
s
hown
in
F
i
The
first
sub
square
occupie
s
the
g
3 X 3
-
-
.
.
To tal s
3 69
.
F ig 96
.
.
middle s ection o f the fi r s t horizontal row o f s ub sq u are s and it
contains the num b ers 1 to 9 incl usive arranged in regular magic
1
square order being a du plicate of F ig
The s econd sub sq uare
-
,
-
.
.
.
.
MAGI C SQ UARES
46
.
s
q
uare
are
multiplied
to
a
remarkable
e
x
tent
for
whereas
in
9 X 9
the latter square (F ig 97) there are only twenty columns which
sum up to 3 69 in the compound square o f 9 X 9 there are an
immense number o f combination columns which yield this amount
This i s evident from the fact that there are eight columns in the
fi rst sub square which yield the number 1 5 ; also eight columns in
the middle sub square which yield the number 1 2 3 and eight col
and
u m n s in the last sub square which sum up to the number 2 3 1
'
,
.
,
.
-
-
—
—
—
15
23 1 z
1 23
Fig
.
6
3 9
.
Fig
1 02
.
.
1 00
F ig
.
Totals of 3 X 3 squares
Totals of 5 X 5 squares
39
65
101
.
.
.
.
T h e ne x t compound square i s that of 1 2 X 1 2 which may be
built with s ixteen s ub squares o f 3 X 3 or with nine sub squares
o f 4 X 4 the latter arrangement being s hown in F ig 98
-
-
.
.
M AGI C SQUARES
.
M AG I C S Q UA R E S
E N T RI C
.
B eginning
a s m all central magic square it i s possible to
arrange one or more p anels of numbers concentrically around it so
that afte r the addition of each panel the enlarged s q uare will still
retain magic qualifications
E i ther a 3 X 3 or a 4 X 4 magic square may be u s ed as a
nucleus and the square will obviously remain either odd or even
according to its beginning irrespective of the number o f panels
which may be successively added to it The center square will
,
.
,
,
,
.
0
Fig
.
1 03
“
2
.
F ig
.
Fig
1 04
.
Totals of 3 X 3 square
Totals o f 5 X 5 square
39
65
.
1 05
.
.
.
natu
but after one or more panel s have been added
the enlarged square will no longer retain perf ec t characteristics
becau se the peculiar features of its construction will not permit the
- opposite nu m b ers to equal the
sum o f every pair of geo metrieeHy
sum of the first and 1
every horizontal and
diagonal columns will however b e the sa m e a m ount
—
,
,
.
L
—
J
7m
7
u
g
a
v
M AGI C SQUARES
48
.
T he smalle s t concentric s q uare that can be constructed i s that
o f 5 X 5 an e x ample o f whic h i s illu s trated in F ig 99
.
,
.
The center s quare of 3 X 3 begins with 9 and continue s with
increment s o f 1 up to 1 7 the center number being I 3 in accordance
with the general rule for a 5 X 5 s q uare made with the s erie s o f
,
,
,
JX J f
Fi g
F ig
.
.
1 06
T otal s of
3 X 3
Total s of 5 X 5
Total s of 7 X 7
:
.
.
F ig
1
rc
1 08
.
sq uare
sq uare
square
yu a
.
1 10
.
75
1 25
1 75
number s 1 to 2 5 The development of the two corner diagonal
column s i s given in diagram F ig 1 0 0 the number s for these col
T he proper s e q uence of the
u mn s being indicated by s mall circle s
.
.
1
,
.
MAGI C SQUAR E S
49
.
umbers in the panels i s shown in F ig
n
T
h
e
10 1
w
r el ativ é po s itions of the nine number s in the central
3 X 3 square
ca nnot be changed but the entire square may be inverted or turned
one quarter one hal f or three quarters around s o a s to vary the
elv e
o ther/ t
.
.
,
,
,
9
X9
,
E ma ]
-
.
I XJ B
nc
3 X 3 5
l
F ig
.
Fig
.
111
2
1 15
.
.
1 14
.
.
F ig
14 0 1 6
.
Fi g
.
1 12
.
F ig
.
1 16
.
position of the numbers in it relatively to the surrounding panel
number s F ig 1 0 2 Shows a 5 X 5 concentric square in which the
panel number s occupy the same cell s a s in F ig 99 but the central
.
.
.
,
0
5
MAGI C SQ UARES
»
3 X 3
right
.
sq uare i s turned around one quarter of a revolution to the
.
S everal variation s may al s o be made in the location of the panel
number s an e x ample being given in F ig s 1 0 3 1 0 4 and 1 0 5 M any
,
.
mn d
er ?
i
,
.
,
tn
F ig
Fig
.
Fig
.
F ig
Total s o f 6 X 6
.
.
.
4X 4
1 19
1 18
1 17
T otals of
.
sq uare
sq uare
.
1 20
.
74
.
1 1 1
.
other change s in the relative po s ition s of the panel number s are
s elfevident
.
of many variations of the 7 X 7 concentric magic sq uare
i s shown in F ig 1 1 0 The 3 X 3 central square in this example i s
started with 2 1 and fi nished with 29 in order to comply with the
O ne
.
.
M AGI C SQUARES
5
.
1
general rule that 2 5
center cell in a 7 X 7 s q uare
that includes the series of iium b ers I to 49 The numbers for the
two co rner diagonal columns are indicated in their proper order
by s mall circles in F ig 1 0 6 and the arrangement of the panel num
bers l s g i ven i n F igs 1 0 7 1 0 8 and 1 09 A s a fin al example of an
3
.
.
,
.
flv
o nd
z(
‘
a
lu m
,
.
,
”J
'
F ig
1 22
.
.
F ig
Totals of 4 X
To tal s of 6 X
.
1 24
.
s q uare
sq uare
4
6
oncentric square F ig 1 1 6 s hows one of 9 X 9 it s development
being given in F igs 1 1 1 1 1 2 1 1 3 1 1 4 and 1 1 5
A ll these diagrams are simple and obvious expansion s of tho s e
s hown in Fig s 1 00 and 1 0 1 in connection with th e 5 X 5 conc entric
s q uare and they and their numerou s variations may be expanded
odd c
.
.
.
,
,
,
,
,
,
.
MAGI C SQUAR E S
2
5
.
indefinitely and u s ed for the con s truction of larger odd magic
sq uare s of this class
.
The smalle s t even concentric magic s q uare i s that o f
6
l
0
f
X
6 E
n
d
X
4
.
6 X 6
41 5 9
44
41
,
of
:
O 90
o
do
62
Fig
F ig
Fig
1 25
.
.
1 26
F ig
Totals of
Totals of
1 28
.
.
.
T otal s o f
.
4X 4
6 X 6
8 X 8
square
s quare
sq uare
1 30
1 95
2
260
.
1 29
.
.
.
.
hic h F ig 1 20 i s an examp le The development of this sq uare
may be traced in the diagrams given in F igs 1 1 7 1 1 8 and 1 1 9
w
.
.
.
The center s q uare o f
4 X 4
,
,
.
but a fter the panel is added
MA G I C SQUAR E S
54
.
have been developed in a very easy manner from successive ex p an
sions of the diagrams u sed for the 6 X 6 squa r e in F igs 1 1 7 1 1 8
and 1 1 9
The rule s gove r ning the for m ation of c o ncentric m agic squa r es
have been hitherto conside r ed somewhat di ffi cult but b y the aid o f
diagrams their const r uction in great va r iety and o f any size has
b een reduced to an operation of extreme si m plicity involving only
the necessary patience to const r uct the diag r ams and copy the n u m
.
,
,
.
'
,
,
,
bers
.
G ENE R A L N O T E S
O N T H E CO
N S T R U CT I O N
OF
M AG IC S Q UA R E S
.
The r e a r e tw o va r iab les that govern the su m m ation of magic
squares fo r m ed o f numbers that follow each other with equal in
th r oughout the series viz
The Initial o r starting number
1
The Incre m ent o r increasing nu m b er
2
When these two variab les are known the summations can be
easily determ ined o r when either o f these va r iab les and the sum
mati o n a r e known the other variab le can b e readily de r ived
The most interesting p r ob le m in this connection is the construe
c rem ents
.
,
.
.
,
.
.
,
,
,
.
,
tion o f squares with predetermined summations and this su b j ect
will therefore be fi rst considered assuming that the reader is f a
miliar
with the u sual method s o f building odd and even squares
,
,
.
a:
i
l
><
><
a squa r e of 3 X 3 is c onstructed in the u sual m anner that is
beginning with unity and proceeding with regular incre m ents o f
If
1
,
,
the total of each column will be
15
,
.
Total s
Fig
.
1 36
.
i s u sed as the initial nu m b er instead of 1 and the squa r e
i s again constru c ted with regula r incre m ents o f I the total o f each
If
2
,
column will
be
18
.
M A G I C SQUAR E S
.
Totals
Fig
.
1 37
.
is still used as the initial nu m b er and the square is on c e
I
the
are constructed with r egular increments of 2 instead o f
ta l o f each colu m n will b e 3 0
If
2
,
.
Total
Fig
.
I 38
0
3
2
.
.
therefore follows that there mu s t be initial number s the use
which with given incre m ents will entail su m m ations o f any pre
term ined a m o unt and there m ust also be increments the use o f
will likewise p r oduce p redeter
i ich with given initial numb ers
It
,
,
,
,
su m mations
These initial num b ers and increments may readily be deter m ined
a simple form of equation which will estab lish a connection b e
een them and the su m m ation nu m bers
L et :
A initial nu m b er
in ed
.
'
.
,
increment
number o f cells in
3
,
n
S
hen
,
if
o ne
.
A
-
"
t
(
A and B a r e
m ula
>r
It
,
sum m ation
I and B :
Z
If
side of square
may
be
m o re
or
-
l
)
2
5
less tha n unity the following genera l
,
u sed :
will be found convenient to substitute a constant (K) fo r
M AGI C SQUAR E S
56
.
n
i
n the above equat i on and a table of these constants
)
2
therefo r e appended for al l squares from 3 X 3 to 1 2 X 1 2
1
,
.
t
S (E
l ua res
3
X
Co ns
.
3
12
4 X 4
30
X
60
5
s
6 X 6
105
7X 7
1 68
X
8
25 2
9 X 9
6
0
3
10
X
10
495
1 1
X
1 1
12
X
12
8
K
858
When using the a b ove constants the equation will be
2
K
B
,
A rt
E X A MP LE S
1
S
.
.
What initial number is required f o r the square of 3 X
?
1
as the increment to produce 90 3 as the su m mation
Transp osing the last equation
3
,
with
,
S
BK
—
n
1 90 3
§
—
I
X
12
)
Initial
No
.
Totals
F ig
We will
no w
1 39
.
W
appl y the same equation to a square o f
which case
1 90 3
.
—
I
( X
4
0
3
)
6
4
2
I nitial
No
.
4X
4,
in
M AGI C SQUAR E S
.
Totals
F ig
A lso to a square of
1 90 3
I
5
X
X
5
.
1 40
.
,
60 )
,
2
I nitial
No
.
Totals
F ig
A nd fo r a square of
6
X
6
.
1 41
.
.
Initial N O
F ig
.
1 42
.
.
The preceding examples illustrate the const r uction of squares
built up with p r og r essive in c r e m ents o f 1 b ut the operation may be
varied by using inc r e m ents that a r e g r eate r o r less than unity
,
.
W
E XAM P L E S
hat i n iti al
'
incre m ents
of
3
,
.
nu m b e r m ust b e u sed in a squa r e of
to p r oduce a su m m ati o n of 1 90 3 ?
3
X
3
,
with
M AGI C SQUARES
8
5
.
A pplying the equat i on given on page
instead o f
56
but making 3
,
: 3
we have
1,
1 90 3
12
)
3
3
i s there fore the initial num b er and by using this in a
the
desired
results
Square
with
progressive
increments
of
X
3
3
3
are obtained
,
.
T o tal s
F ig
1 43
.
(
10
X
12
.
.
T o find the initi al number with increments of
1 90 3
1 90 3
)
10
.
Initial NO
3
.
Total s
Fig
.
1 44
.
O r to find the initial number with inc r ements of
1
1 90 3
/3 X
12
)
6 33
2
Initial
No
.
Totals
F ig
.
1 45
1 90 3
.
.
These examples being suffi cient to illu strate the rule we will
,
pass o n another step and Show how to build square s with p redeter
mined summations using any desired initial num b ers with a p r oper
,
increment
.
,
MA G I C SQUAR E S
.
What increment numb er m ust be used in a square o f 3 X 3
?
wherein 1 is the initial num b er and 1 90 3 the desired summation
R e ferring to equation o n page 5 6 and t ransposing we have
,
,
S
A
—
h
K
1 90 3
—
( X
1
increment or
B
,
3)
I nc r e m ent
1 58
12
.
S tarting therefore with unity and building up the squa re with
successive inc r ements of 1 5 8
we ob tain the desired result
.
When it is desired to start with any nu m ber larger or s m alle r
than unity the nu m b e r s in the equation can b e modified accordingly
Thus i f 4 is selected as an initial number the equation will b e
,
.
,
1 90 3
(4 X
3)
1
Increment
57
Totals
F ig
.
.
2
1 47
.
With an initial number of
I 90 3
X 3)
Inc r e m ent
Totals
F ig
.
1 48
.
.
MAGI C SQUARES
.
is thus demonstrated that any initial number may be u s ed
providing (in a square o f 3 X 3 ) it is less than one third o f the
su m m ati o n In a square o f 4 X 4 it m ust b e less than one fou r th
o f the su mm ation and so o n
It
-
-
.
.
,
To illust r ate an extre m e case we will select 6 3 4 as an initial
num b e r in a 3 X 3 square and find the increment which will result
in a summation o f 1 90 3
,
.
1 90 3
6
( 34 X 3 )
—
I ncrement
Totals
H aving
.
1 90 3
2
.
conside red the formation of magic squares with
predeter m ined su m mations by the u se of proper initial num b ers
no w
and incre m ents
,
it
only r e m ains to Show that the su m mation of any
square m ay b e found when the initial num b er and the increment
are given b y the application of the equation shown o n page 5 6 viz
,
,
,
A
n
:
K
S
+B
E XA M PL E S
F ind
.
.
the summation of a square of
number and 7 as the inc r ement
X
3
3
u sing 5 as t h e initial
.
,
(5 X 3 )
(7 X
I2
)
:
99
r
S ummation
.
Totals
F ig
.
1 50
.
What will be the summation of a square of
an initial number and
(9
X 4)
1 1
(
as an increment
11
X 30 )
99
.
366
4 X
4
?
S ummation
.
u sing 9 as
M AGI C SQUAR E S
12
.
What incre m ent must b e used in a square of 3 X 3 wherein
is the initial number and
1 2 the requi r ed su m mation ?
—
12
4
I ncrement
.
Totals
F ig
.
155
5
30
.
.
What increment must be used in a square o f
i s the initial number and 42 the summation ?
42
12
X
4
Increment
4
wherein
8
4
.
To tals
42
.
The foregoing rule s have been applied to e x ample s in sq uares
of small size only f o r the sak e o f brevity and si m plicity , but the
p rinciples explained can evidently be expanded to any extent that
may b e desired
.
N umbers following each other with uni form increment s have
been u s ed throughout this article in t h e con s truction o f magic
squares in order to illustrate their formation according to certain
,
how ever been s hown b y M r L
s erie s o f numbers u sed in the constr uction o f
every magic s q uare i s divided by the b reak m o v es into i i group s o f n
numbers per group (ii representing the number o f cells in o ne
rule s in a s impl e manner
.
It has
.
.
the square ) and that the numbers in these groups do not
n ec es s ari ly follow each other in regular orde r with equal increments
but under certain well de fi ned rules they may be arranged in a
S
ide
of
,
,
M AGI C SQUARES
m
.
63
great variet y o f irregular sequences and still produce
squares
R e fe r ring to F ig 40 as an example many di ff erent 5 X S
squares may be formed by varying the s e q uence of the fi ve groups
and also by changing the arrangement of th e n u mbers in each group
Instead of writing the five diagonal columns in Fig 40 with
the nu m b ers 1 to 2 5 in arithmetical order thus :
.
.
,
,
.
.
a
.
1
b
.
6
c
2
3
4
5
7
8
9
10
5
1 1
12
13
14
1
d
16
17
18
19
20
e
21
22
23
24
25
.
.
.
they m av be arranged in the order I) e c a d which will develop
the 5 X 5 squa r e shown in F ig 1 7
O ther variations may be made by re a rranging the con s ecutive
numbers in each group as for example thu s :
,
.
.
'
-
,
5
4
3
2
9
8
7
1 1
14
13
12
15
d
16
19
18
17
20
e
21
24
23
22
25
a
.
b
.
c
.
.
.
1
6
,
10
The foregoing may be considered as only sugge s tive o f many
ways o f grouping numbers by which pe nf eet magic squares may be
produced in great variety which however will be generally fou nd
to follow the re gular constructive rules he r etofore given providing
that these rules a r e applied to series of nu m bers arranged in similar
c o nsecutive order
,
,
.
M
a
fia
”
MA GI C CU B ES
.
H E curious and interesting characteri s tics
magic s q uares
may be developed in figures of three di m ensions constituting
of
magic cubes
C ubes o f o dd number s may be constructed by direct and co n
tinu o u s process and cubes o f even numbers may be built up by the
aid o f geometrical diagrams In each case the con s tructive meth
o ds resemble those which were previou s ly e x plained in connection
with odd and even magic s quares
.
'
,
.
.
characteri s tics of magic cubes
er running from
,
to the back
two
,
or
f
11
other and eq
l d equal the sum o f the first and l
and the center cell must contain the mid
e series
.
G
a
n
M AG I C
CU BE S/o f 0 0 0 N U
The smallest magic cube i s naturall y
3
X
3
X
3
.
MB E as
M
T he-
MA G I C
C
UB E S
.
shows one of these cubes and in columns I II and
II I F ig 1 5 8 there are given the nine di ff erent squares which it
contains In this cube there are twenty seven straight columns
two diagonal c olumns in each of the three middle squares and four
diagonal columns connecting the eight co r ners of the cu b e making
in all thirty seven columns each of which sum s up to 42 The
F ig
.
1 57
.
,
,
,
,
-
,
.
,
,
-
.
d the sum of any pair of
s
etrically
Opposite
numbers
is
m
m
y
28
Fig
or
n
3
1
e
e
g
—
i
s
In the se pe a t t his
.
1 57
.
.
de s cribing the direct method of building O dd magic squares
many fo r ms of r egular advance moves we r e explained including
“
right and left diagonal sequence and various S O called knight s
It was also shown that the order of regular advance was
moves
periodically brok en by other well defined spacings which were
“
In building odd m agic squares only one
termed b reak m o v es
form of break mo v e was employed in each square but in the con
struction o f odd magic cube s two kinds are r equired in each cube
which for distinction may be termed n and ii b reak mo v es resp ec
In magic cubes which commence with unity and proceed
tiv ely
2
with increments o f 1 the n b reak m o v es occur between each mul
2
tiple o f h and the next following numb er , which in a 3 X 3 X 3
cube bring s them between 9 and 1 0 1 8 and 1 9 and also between
the Be st and las t numbers of the series 27 and 1 The n b reak mo v es
In
,
,
’
-
,
’
.
.
,
,
'
,
‘
2
.
,
,
,
,
.
M AGI C
C
UB E S
.
a r e made b etween all othe r multiples o f n which in the above case
b r ings them between 3 and 4 6 and 7 1 2 and 1 3 1 5 and 1 6 2 1 and
22
and 24 and 2 5 With this explanation the rules for b uilding
the magi c c ube Shown in F ig 1 may no w b e form ulated and for
convenience o f ob servation and c o ns tructi o n the cu b e i s divided
horizontally into th r ee se c tions o r layers each section being sh own
sepa r ately in Colu m n 1 F ig 1 5 8
,
,
,
,
,
.
,
.
,
,
,
,
.
.
mentioned that when a m ove i s to b e continued up
from the top square it is carried around to the bottom square
It m ay b e
TH
F RO
RE E S Q U A RE S
MT OP T o B O TT O M
C OL U M
N 1
.
TH
F RO
R E E S Q U AR E S
MF R O N T T o B A C K
C OL U
MN
11
.
TH
F RO
,
RE E S Q U A RE S
ML E F T T o R IG HT
COL U M
N 1 11
.
and when a move i s to be made do w nw ard from the bottom s q uare
it i s carried ar o und to the top square the conception being similar
to that o f the horizontal cylinder used in connection with odd mag i c
,
,
squares
Commencing with 1 in the center cell o f the to p square the
cells in the three s q uares are filled with consecutive nu m bers up
to 2 7 in accordance with the following dire c tions
A dvance move O ne cell down in next square up (from la s t
.
,
.
entry )
.
MAGI C
C
UBES
67
.
cell in downwa r d r ight hand diagonal in
next squa r e down (f r o m last ent ry )
S a m e cell in next squa r e down (from last
11
b reak m o v e
ent ry )
I f it is desired to b uild this cu b e f r o m the th r ee ve r ti c al squares
fro m f r o nt to b ack o f F ig 1 5 7 as Shown in Colu m n II F ig 1 5 8
the di r e c tions will then b e as follows :co m m encing with 1 in the
m iddle c ell of the upper ro w o f nu m b e r s in the m iddle squa r e
A dvance move O ne cell up in next square up
O ne cell in downwa r d r ight hand diagonal in
i i b reak m o v e
next squa r e up
O ne
b reak m o v e
n
.
-
.
2
.
.
.
,
,
.
,
,
.
.
-
.
.
71
b rea k m o v e
2
J9
.
T
ABL E
F ig
.
I
1 59
e¢
m
.
.
the sa m e c u b e may b e const r ucted from the th r ee vertical
squares running f r om left to right side of F ig 1 5 7 as shown in
Column III F ig 1 5 8 co m mencing as in the last exa m ple with I
in the middle cell of the upp e r ro w Of nu m be r s in the middle
F inally
,
.
,
,
,
,
.
squa r e and p r oceeding as follow s
A dvance move Th r ee consecutive cells in upwa r d right hand
diagonal in sa m e sq uare (as last entry )
,
-
.
.
11
b reak m o v e
.
O ne
cell in downward right hand diagonal in
next square down
-
.
M AGI C
b reak m o v e
C
U BES
.
O ne
cell down in same square (as last entry )
F ive variations may be derived fro m this cube in th
e simpl e
way illustrated in Tab le I on the preceding page
A ssign three figu re values to the nu m b ers 1 to 27 inclusive in
te rm s o f 1 2 3 as given in Table I F ig 1 5 9 and change the
numbers in the three squares in Column I F ig 1 5 8 to their cor
re s ponding three figu re values thus producing the s quare shown in
F ig 1 60
It is evident that i f the a r rangement o f numbers in the
three squares in Column I were unknown they could be readily
p roduced from F ig 1 60 by the translation o f the three figu re values
into regular numbe r s in accordance with Table I but more than
n
?
.
.
.
-
,
,
.
,
,
.
,
,
-
,
.
.
,
-
.
,
Jy u a r o
F ig
.
1 60
.
The letters A B C in Table I i ndicat e
the normal order O f the nu m erals 1 2 3 but by changing this order
other triplets o f 3 X 3 squares can be made which will di ff er more
ell
o r le ss from the original m odels in the arrangement of their C
numbers but which will retain their general mag i c characte r istics
The changes which may be rung o n A B C are naturally S ix a s
this can be accomplished
,
,
.
,
,
,
,
.
,
,
follow s
,
,
,
,
MA G I C
0
7
C
UB E S
.
The analysis O f the numbe r s in Fig 1 5 7 and F ig 1 66 into thei r
three figu re values in te r ms of 1 2 3 as shown in F igs 1 60 and
1 6 7 makes clear the curious mathematical order of their arrange
ment which is not apparent on the face o f the regu lar numbers as
.
.
-
,
,
,
.
,
D IRE C T IO N S
F OR C
O N S T R U C T I NG T H E 3 X 3 X 3 MA G I C C U B E S HO WN
A N D F I VE VARI A T IO N S O F T H E SA ME
IN F IG
.
1
57
.
AD
A
B
C
s
.
C
.
A
A N CE
O ne c e
T
B
V
h
qu
C
B
A
.
.
.
.
A
B
A
C
.
B
c o n s ec
.
.
A
C
B
u tiv
-
a
a
e s
o
o
s
S a me
as
in
B
.
S ame
as
in
A
.
S a me
as
a
in
a re u
.
in C
.
.
.
are
ll i n
d wn
ne
xt
o
_
ll
O ne
ne
xt qu
s
l ft
to
ce
m
e
S a me
as
in
A
S ame
as
in
A
.
S am e
as
in
A
.
S ame
as
in
A
S am e
as
in
A
.
B
.
B
.
C
.
C
.
C
.
a re u p
e
a
ll t righ t
xt q u
p
ce
qu
ce
e
ar
s
e
ne
.
s
ll m u pw d
l ft h nd di g n l
in
qu r
m
.
S am e
e
o
ree
O ne
EA KMO V E S
n B R E AK
ll d w n i n n xt
sa
C
MOV E S
n2 B R
a re u p
ce
.
MOV E S
C
B
A
.
.
.
O ne
s
A
C
B
ce
qu
ll
up
ne
xt
are u p
S am e
as
in C
.
S am e
as
in
B
.
S ame
in
A
.
in
as
.
.
.
A
C
B
.
.
.
B
A
C
‘
.
.
B
.
B
.
C
.
B
.
C
.
.
.
they appear in the various cells of the cubes F o r example it may
b e seen that in eve r y sub sq u a re in F igs 1 60 and 1 6 7 (corresponding
to horizontal columns in the cubes ) the nu m bers I 2 3 are each
repeated three times A lso in every horizontal and perpendicular
,
.
.
,
.
,
MA G I C
C
UB E S
C
1
7
.
column there i s the sam e triple repetition F urthermore all the
diagonal columns in the cubes which su m up to 42 if followed int o
their analyses in F igs 1 60 and 1 6 7 will also b e found to carry si m i
la r r epetitions A brie f study of these figures will also dis c l o se
other cu r ious m athe m atical qualities pe r taining to thei r intrinsic
sym m etrical a rr ange m ent
The next Odd m agic cube in order is 5 X 5 X 5 and F ig 1 68
Shows one o f its m any possible va r iations F o r c o nvenien c e it i s
divided into five horizontal sections o r layers fo r m ing five 5 X 5
squares f r om the top to the b otto m of the cu b e
C om m encing with 1 in the first cell of the m iddle ho r i zontal
.
,
,
.
.
.
.
,
,
.
,
.
F ig
.
1 67
.
colu m n in the thi r d square this cu b e m ay b e const r ucted b y filling
in the va r iou s c ells with consecutive nu m b e r s up to 1 2 5 in ac c o r d
ance with the f o llowing di r ecti o ns
,
A dvance
cell up in next squa r e down
Two cells to the left and o ne cell down (knight s
i i b reak m o v e
m ove ) in sa m e squa r e as the last ent r y
n b reak m o v e
O n e c ell to right in sa m e squa r e as last ent r y
m o ves
.
O ne
.
’
.
2
.
M
W
.
.
esfo eta l ! exhi b its so m e inte r esting
f his cube
E xa m ining fi r st the five ho r i zo ntal squa r es f r o m
qualification s
the to p to the b ottom o f the cu b e as shown in F ig 1 68 there are
o a gh
'
.
.
,
M AGI C
2
7
.
1
3 5
corner diagonal columns summing up to
sub diagonal columns summing up to
columns having the same summation
10
0
4
DIRE C TIO N S
UBES
s traigh t columns summing up to
0
5
Total
C
.
1
3 5
.
-
1
3 5
.
1 00
.
FO R C
O N S T R U C T I NG T HE 3 X 3 X 3 M
A G I C C U B E SHO WN
A N D F I VE VARI A T IO N S O F T HE SAME
IN F I G
1 66
.
.
AD
V
MOVE S
AN CE
MOV E S
n B RE AK
n
2
MO V E S
B RE A K
ll i n d wnw d
i gh t h nd di go
n l i n n xt
u
q
d o wn
O ne c e
A
B
C
s
T
B
.
C
.
A
ll t l ft i n n xt
O ne c e
qu
are u p
s
C
.
.
B
A
.
.
A
.
O ne
.
s
B
.
A
C
B
.
.
A
C
B
ar
a
a
sa
C
.
.
.
a
c o nsec u
ree
ce
qu
ll
o
e
e
a
i
a re
e s
up
m
ne
xt
a re u
p
S ame
as
in
B
S a me
as
in
A
S ame
as
in C
ll i n u pw d
i gh t h n d di g
n n xt qu
n l
O ne
-
r
a
.
.
C
B
A
.
.
,
A
.
a
a
a
a
e
s
S a me
as
in
A
S a me
as
in
A
S ame
as
in
A
.
S a me
as
i n As
B
.
S am e
as
in
A
B
o
a re
s
e
i
are
.
B
.
C
.
.
B
.
C
.
up
ll i n d wnw d
l ft h n d di g n l
n n xt qu r
p
O ne c e
e
-
i
.
ar
ar
ce
ar
o
a
a
e
a
o
a e u
s
S ame
as
in C
C
.
S ame
as
in
B
B
.
S a me
as
in
A
A
.
B
.
.
C
.
.
B
.
A
the five vertical squares fro m front
there are :
In
-
r
e
e
tiv
d l ft
c ll i n p w
u
b nd d i g n l
n
m
qu
h
e
.
o
o
C
.
to
.
B
.
.
.
C
C
C
.
.
.
back of this cube
MAGI C C UBES
.
straight colu m ns summing up to
6 corner diagonal columns su m ming up to
20 sub diagonal columns summing up to
T otal 76 columns having the same summation
1
3 5
0
5
1
3 5
1
3 5
-
.
.
the fi ve vertical s quares fro m right to left of cube there are
as in the la s t case 76 columns W hi } all sum up to 3 1 5 In the com
g
?
”
y
l
i
l
h
l
t
a
f
o
r
o
e
cube
there
are
é
e
e
u
d
a
n
a
m
h
m
o
s
p
g
i k
In
,
.
,
T O P S Q U A RE
BO
.
F ig
A table similar to
F ig
values for the nu m bers in
changing the nu m be r s in
squa r e Similar to F ig 1 60
tions o f F ig 1 68 may be
be O b tained with less work
.
.
.
.
1 68
TTO MS Q U ARE
.
r
e
fi
u
may
be
laid
out
giving
three
g
59
1
b
cubes
from
1
to
2
and
y
5
5 X 5 X 5
F ig 1 68 to these three figu re value s a
will be p r oduced from which five varia
derived S imilar results however can
b y means o f a table of key numbers con
-
1
,
-
,
.
.
.
,
,
.
structed as shown in F ig 1 69 (Tab le I I )
The th r ee figu re values of cell nu m b ers in 5 X 5
cubes are found from this table as follows :
-
,
.
X
magic
5
MAGI C
f
C
UB E S
.
W
S elect the my num b e r which is nea r est to the cell nu m b er
-
it in
b el o w
1
.
2
.
3
v a lu e
.
,
b ut
Then write down
M
The section num b er in which t he Izzy nu m b er is found
” 4 C
TH
u m b er over the
E
The di ff erence b etween the
number and the cell numb er
-
-
.
—
M
,
—
o
t
.
t
Three figures will thu s b e determined which will rep r esent the
required three figu re value o f the cell num b er
E x a mp l es
The first number in the first ro w o f the upp er
sq uare in F ig 1 68 is 6 7 The nearest Iggy nu m ber to this and
m b er 4 and the
below it in value is 6 5 in section 3 under the
-
-
.
.
M
-
.
.
T
AB LE
F ig
11
1 69
.
‘
“
.
.
erence between the lu g:number and the cell num b er is 2 The
three nu m ber value o f 6 7 i s there fore 3 4 2 A gain the fourth
numbe r i n the sa m e row is 1 0 The nearest Es nu m ber b u t b el o w i t
i n v a l ue i s 5 in section 1 under the pr ime number 2 and the di ff er
ence between the h ey nu m ber and the cell nu m ber I S 5 The three
fig u re value o f 1 0 is there fore I 2 5 B y these simple operations
the th r ee figu re values of all the cell numbers in the 5 X 5 X 5
cu b e in F ig 1 68 may be quickly deter m ined and by the system of
transposition previou sly explained five variations o f this cube may
be constructed
di ff
-
.
-
M
.
.
.
,
t-
.
,
—
.
M
‘
.
.
r
/
7
.
-
-
.
,
,
.
MA G I C C U B E S
75
.
The shorter method of building these 5 X 5 X 5 cubes by the
direct process of filling the di ff erent cells in regular order with
consecutive numbers may however be considered by some to b e
preferab le to the more roundab out way
S
ee
directions
in
the
(
following table )
,
,
.
.
DIR E C T IO N S
FOR C
O N S T R U C T I N G T H E 5 X 5 X 5 MA G I C C U B E S HO WN
A N D F I VE VARIAT IO N S O F T H E SAME
I N F IG
.
1 68
.
AD
A
.
B
.
C
V
O ne
.
s
qu
MOVE S
A N CE
ce
ll
a re
up
in
ne
d wn
n B RE A K
xt
Tw
.
C
.
A
o ce
o ne
o
s
qu
ll
.
C
.
B
.
A
B
A
.
.
.
B
A
C
to
d wn
o
a re a s
o ce
l ft
e
a nd
O ne
in
sa m e
sa
l t
as
ty
en r
a
s
.
S a me
as
in
B
S ame
as
in
A
S am e
as
in C
,
.
.
C
.
B
.
A
.
A
C
B
a re
.
S am e
as
in C
.
S ame
as
in
B
S am e
as
in
A
.
ll
me s
ty
to
qu
.
ri ght
in
l t
a re a s
as
en r
S a me
as
in
A
S ame
as
in
A
.
S a me
as
in
A
.
S a me
as
in
A
.
S a me
as
in
A
a
o
.
A
.
B
.
C
.
o
.
.
ce
ar
a
e
.
s
e
C
s
ll i n u pw d
l ft h n d di g n l
i n n xt q u
d wn
Tw
B
MOV E S
MOV E S
2
72 B RE A K
.
.
C
.
.
B
.
B
A
C
.
B
.
B
.
.
.
C
C
.
.
B C
.
.
.
is anothe r example of a 5 X 5 X 5 m agic cube which
is commenced in the upper left hand co r ner of the to p square and
finished in the lower right hand co r ner of the bott o m square the
F ig
.
1 70
-
,
'
-
,
M AGI C
6
7
C
UBES
.
middle number o f the series (63 )appearing in the center cell of the
cu b e according to rule
O dd magic cubes may be commenced in various cells other
than those Shown in the preceding pages and they may be built
up with an almost infinite number of variations It would however
be onl y s uperfluous and tiresome to ampli fy the subj ect furthe r as
the examples already submitted cover é the important points of
con s truction and m ay readily be applied to further e x tensions
.
,
.
,
,
,
.
,
3
T OP S Q U A R E
.
BO
.
Fig
A ny sizes o f
.
1 70
TT OMS Q U ARE
.
.
magic cubes larger than 5 X 5 X 5 may be
constructed by the directions which govern the formation of 3 X 3
X 3 and 5 X 5 X 5 cu b es
O dd
.
M AG IC
M agic cubes o f even numbers
CU BE S
O
F EV E N N U
m ay
MB E R S
be built by the aid of geo
metric diagra m s similar to those illustrated in the preceding chap
te r which describes the construction of even magic squares
,
,
.
.
MAGI C
8
7
C
U BE S
.
num b ers which are diametrically op posite to each other and equi
“
distant fro m the center of the cube also equal s 6 5 o r if
1
,
.
A nother feature
this cu b e is that the sum O f the fou r num
b e r s in each of the forty eight sub squares o f 2 X 2 is 1 3 0
“
It has been shown in the chapte r o n M a gi c S quares that the
of
-
-
.
”
S ec ti o n II
S ec
ti
on
.
I II
.
F ig
.
1
71
F ig
.
.
Totals
2
1 30
1
72
.
.
square o f 4 X 4 could be formed by writing the number s 1 to 1 6
in a r ithmetical order then leaving the numbers in the tw o corner
diagonals unchanged but changing all the other numbers to their
2
1
It will be noted in the magic cube
com plements with 1 7 o r 71
of 4 X 4 X 4 given in F ig 1 71 that in the first and last of the
,
,
.
,
.
,
MA G I C
C
UB E S
79
.
fou r sections (I and I V ) this r ule also holds good I n the two
m iddle section s ( I I and III )the r ule is r eve r sed ; the nu m b e r s in the
and all
1
t w o c o r ne r diagonals b eing co m ple m ents with 6 5 or 71
the other numb e r s in a r ithmetical o r de r
F ig 1 73 Show s fou r squa r es o r sections o f a cu b e with the
numbe r s 1 to 64 w r itten in a r ithmetical order Those num b e r s
that occupy corresponding cells in F ig 1 71 a r e enclosed within
circles I f all the other nu mbe r s in F ig 1 73 are changed to their
co m plements with 6 5 the total arrangement o f numbe r s will then
b e the same as in F ig 1 71
In his inte r est i ng and instructive chapter entitled R eflections
”
on M agic S qua r es
D r P aul Ca r us gives a novel and ingeniou s
“
analysis o f even squares in di ff erent orders of numb e r ing these
o r de r s b eing te r med respectively 0 r o i and r i It i s shown that
the two magic squares of 4 X 4 (in the chapter referred to ) con
.
3
,
.
,
.
.
.
.
.
,
.
.
<
.
”
,
,
,
F ig
sist onl y o f
n
s
l
m
t
e
e
p
m agic
of
o
0
1 73
.
.
.
num b ers being in fact the c o m
I This r ule also Ob tains in the
num b ers with 71
of
1
given
in
The
fou
r
sections
i
1
F
g
7
4 X 4 X 4
and
ro
nu m b ers ;
ro
2
.
cu b e o f
this cu b e may in fact be filled out by writing the 0 numbe r s in a r ith
m etical order in the cells o f the two c o r ner diagonal colu m ns of
sections I and IV and in all the cells o f sections II and II I ex
c ep tin g those of the two co r ner diagonal colu m ns and then writing
the r o num b e r s al so in arithmetical order in the remaining empty
.
.
,
,
,
,
,
,
the fou r sections
Fig 1 71 m ay b e considered as typical of all magic cubes of
and
their
ultiples
o f this class but a g r eat m any va r ia
m
4X 4 X 4
tions may be e ff ected b y si m ple transpositions F o r exa m ple F ig
cells
of
.
.
,
,
.
l
>‘
S ee p
.
1 13
ff
.
,
.
MAGI C
is a
C
U BES
.
cu b e which is constructed by writing the f o
nu m bers that are contained in the 2 X 2 sub squares (F ig 1 71 )
a straight line and there are many other possible t ra n sp o sitio
1 74
4
X
4X 4
-
.
,
which will change the relative order o f the numbers without
stroying the magic characteristic s of the cube
,
d
.
S ec ti o n II
S ec
ti o n
III
.
.
F ig
.
1
F ig
74
.
.
Totals
2
1 30
.
75
.
.
The arrangement o f the numbers in
gram m atic order shown in F ig 1 75
.
1
F ig
.
1 74
follows the di
MA G I C C UB E S
81
.
The 8 X 8 X 8 m agic cube follows next in order F ig 1 76
Shows this cu b e divided for c o nvenience into eight hori zontal layers
1 77 gives the diagram m atic order o f the n u m
o r sections and F ig
b e r s in the first and eighth sections the inter m ediate sections b eing
built fro m Si m ilar diag r ams numb e r ed in arithmetical order
.
.
,
,
.
,
,
.
,
S ecti o n I
S ect i o n I I I
.
.
S ect i o n I I
S ecti o n IV
.
.
will b e seen f r om these diag r a m s that the 8 X 8 X 8 magic
cube is simply an expansion of the 4 X 4 X 4 cube j ust as the
In like
8 X 8 magic squa r e is an expansion o f the 4 X 4 square
manner all the d iagrams which were given for di ff e r ent arrange
m ents of 8 X 8 m agic squa r es m ay also be employed in the con
struction o f 8 X 8 X 8 mag i c cub es
It
,
.
.
MA G I C
82
A n examination of
F ig
C
UBES
.
1 76
will show that like the 4 X 4 X 4
cu b e in F ig 1 71 it i s built up of o and ro nu m b ers exclusively In
sections I IV V and V II I the cells in the corner diagonal columns
and in ce r tain other cells which are placed in definite geometrical
relation s thereto contain 0 num b ers while all the other cells co n
.
,
.
.
,
,
,
,
,
,
,
S ecti o n
V
S ecti o n V II
.
S ec ti o n V I
.
( S eco n d
F ig
.
S ecti o nV I II
.
P rt )
a
1 76
.
.
.
tain ro numbers In sections I I II I V I and V II the relative
positions of the o and r o numbers are reve r sed
B y noting the sym m etrical disposition o f the s e two orders o f
numbers in the di f ferent sections the cube may be readily con
structed without the aid O f any geometrical diagrams F ig 1 78
shows section s I and I I o f F ig 1 76 filled with o and re symbols
.
,
,
,
,
.
,
“
.
.
.
MAGI C
C
UBES
83
.
without r egard to numerical value s and the relative sym m etrical
arrangement O f the two orders is therein plainly illustrated This
,
.
« our
g i a
45
7
J?
»
F ig
.
“
‘
o
”
n
u mb
ers
1 77
.
O
.
F ig
.
to
n
u mb
ers
.
1 78
.
clear and lucid a nalysis fo r which we a re indebted to D r Caru s
reduces the formation of a rather co m pli c ated numerical structure
to an Ope ration of the ut m ost Simplicity
.
,
.
,
M A GI C
C
UBE S
.
y
}?
r
this cube there are 1 92 st r aight columns and 4 diagonal s
h
ee lum ns (which unite the eight corner s o f the cube ) each o f which
sums up to 2 0 5 2 also 3 84 hal f columns and the same number of
2 X 2 sub squares each o f which has the summation o f 1 0 2 6
It
will also be seen that the sum of any two number s whi c h are lo
c a ted in cells diametrically opposite to each other and e q uidistant
3
from the center o f the cube i s 5 1 3 or n
I
In
V
ea
Q
/
,
-
.
,
,
.
e 1 2 X 1 2 x 1 2 cu
and all larger ones that are formed with multip les o f 4 will naturally
resemble the 8 X 8 X 8 cube and will be e q ually easy to construct
I
"
7
A
0
a
"
3
0
1
.
"
A
a
5
s
,
f
A
f
:
.
.
i
"
I
o
.
G EN E R A L
N OT
ES
M A G IC CU BE S
ON
W
.
M agic cubes may be constructed having any desired summa
tions by using suitable initial numbers with given increments or
b y applying proper incre m ents to given initial numbers
l
l
.
’
s
l
><
><
,
><
The formula for determining the summations o f magic cubes
is si m ilar to that which was given in connection with magic squares
and m ay be expressed as follows
L et :
initial number
increment
num b e r o f cells in each colu m n
su m mation ;
,
,
then
1
and B
1
n
2
If
(
11
3
I
S
)
z
.
A and B are more or less than unity the following general
,
fo r mula may be employed
An
g
+e
(
fi
g
—
To shorten the above equation
as
,
I
)
Z
=
(ri
3
S
.
I
) may be express ed
a constant (K) for each Siz e o f cube as follo w s
MAGI C
S ec
ti
on
I
S ec
(To p )
.
ti
on
II
C
UBES
S ec
.
F ig
To
t al
1 80
.
s
2
.
ti
on
I II
.
S ec
ti
on
IV
B
tt
o
o m)
(
.
.
70 4
.
What initial number must be u sed with increments
produce su m m ations of 1 90 6 in a 3 X 3 X 3 cube ?
E xp r essing equation (3 ) in figure values
1 90 6
To p S ect
ion
.
(1 0 X
39 )
M iddl e
S ec ti o n
F ig
To
t l
a s
1 81
.
2
0
5 5
.
B o tto m
S ecti o n
.
.
1 906
.
What initial number is required f o r the cube Of 5
with 4 as i ncrement nu m ber to produce summations of
X
5
X
5
,
,
1 90 6
(4
X
1
0
3
)
5
The preceding Simple examples will be su f ficient to illustrate
the formul ae given and m ay suggest o the r problems to those w ho
a r e interested in the su b j ect
,
.
It
will be noted that the magic cubes which have been descri b ed
in this chapter are all in the sa m e general class as the m agic squa r es
which for m ed the su b j ect o f the previou s chapter
.
There are howeve r m any classes o f m agic squa r es and c o r
r esponding c u b es which di ff e r f r o m these in the general a rrange
,
,
f
MA G I
87
nu m b ers and in va r i ous
featu r es while r etaining the
c o m m o n cha r a c te r isti c Of having si m ila r c o lu m n values
A n ex
“
a m ple O f this di ff erentiation is seen in the inte r esting Jaina squa r e
m ent
o
f
,
.
S ec ti o n I (To p )
.
S ect i o n I II
.
S ec ti o n II
ti o n
S ec
.
IV
.
desc r i b ed by D r Ca r us in his R efle c tions on M a gic S quares
S quares o f this class can r eadily be expanded into cu b es which will
natu r ally c a r ry with the m the peculiar featu r es of the squares
.
.
.
M AGI C
88
C
U BES
A nother class i s illustrated in the
.
F ranklin S q u ar
i
these can also be expanded into cubes constructed
general p r inciples
The su b j ect o f magic squares and cubes is indeed
on
t
.
i n ex l
’
and may be indefinitely extended The p hil O S Ophi cal sig
o f these studies has been so ably set forth by D r Carus
writer considers it unnecessary to add anything in this c o r
but he trusts tha t the present endeavor to popularize the :
.
.
esting problem s may s ome time lead
to
u s e ful re s ult s
.
C H
A P T E R
T H E F R A N KL I N S Q U A RE S
.
H E following lette r with m agic squares of 8 X 8 and 1 6 X 1 6
“
i s copied f r om L ette r s and pape r s on Philosophical su b j ects
by B enj amin F ranklin LL D F R
a work which was p rinted
in L ond o n E ngland in 1 769
,
,
D EA R
.
.
.
.
,
.
,
F ROM
B EN JA M I N
F RA N K LI N E S Q
To
P E T ER COLLI N SON
ES Q
AT
.
.
OF
P H I LAD ELP H IA
L ON DO N
,
.
S IR
A ccording to your request I no w send you the arithmetical
cu riosity of which thi s is the history
B eing one day in the country at the house of our common
friend the late learned M r L ogan he showed me a folio F rench
boo k fi lled with magic s quares wrote i f I forget not by one M r
F renic l e in which he s aid the author had discovered great ingenuity
and dexterity in the management of number s ; and though several
.
,
.
,
,
.
,
,
other foreigners had distinguished themselves in the same way he
did not recollect that any one E nglishman had done anything o f the
k ind remarkable
I said it was perhaps a ma rk of the good sense of our mathe
matic ian s that they would not spend their time in thing s that were
merely difhc iles mi gae incapable o f any useful application
He
answered that many of the arithmetical or mathematical q ue s tion s
publicly proposed in E ngland were equally trifling and u s ele s s
P erhaps the considering and answering s uch question s I replied
may not be altogether useless i f it produces by practice an habitual
,
.
.
,
.
,
,
TH E
0
9
F
RA N K L I N SQ U AR E S
.
readiness and exactness in mathematical disquisitions which readi
ness may on many occasion s b e of real use In the sa m e w ay
says he may the making o f these squares be of u s e I then con
fessed to him that in my younger days having once s ome leisure
I
I
which
still
think
might
have
employed
more
usefully
had
I
(
)
amu sed mysel f in mak ing these kind o f magic squa r es and at
length had acquired such a knack at it that I could fill the cells of
any magic s q uare o f reasonable Siz e with a series of number s as
fa s t a s I could write them disposed in such a manne r that the sums
o f every row horizontal perpendicular or diagonal should be
equal ; but not being satisfied with these which I looked on as com
,
,
.
,
.
,
,
,
,
,
,
,
,
,
F ig
.
1 83
.
and easy thing s I had imposed on myself more di f ficult tasks
and s ucceeded in mak ing other magic squares with a variety of
mon
.
,
prop erties and much more curiou s H e then Showed me several
in the s ame book o f an uncommon and more curiou s kind ; but as
I thought none of them equal to some I remembered to have made
he desired me to let him see them ; and accordingly the next tim e
I vi s ited him I carried him a square o f 8 which I found among my
old paper s and which I will no w give you with an account o f its
prop erties (see F ig
The prope r ties a r e
.
,
,
,
,
.
That every straight row (horizontal o r vertical ) o f 8 num
ber s added together mak e s 260 and hal f of each row hal f of 260
2
That the bent ro w o f 8 numbers ascending and descending
1
.
,
.
,
,
.
T H E F RA N KLI N SQ U AR E S
9
.
1
diagonally viz from 1 6 ascending to 1 0 and from 2 3 descending to
1 7 and eve r y one o f its parallel bent rows of 8 numbers mak e 2
60 etc
etc A nd la s tly the fou r corner number s with the fou r middle num b ers
,
.
,
,
.
.
F ig
F ig
.
.
1 84
.
1 85
.
mak e 2 60 S O this magical square seems perfect in its kind but
these are not all its properties there are 5 other curious ones which
at s ome time I will explain to you
.
,
M L ogan then showed me an old arithmetical book in quarto
.
r
.
,
,
T H E F RA N K LI N SQUARES
2
9
.
wrote I think by one S tifeliu s which contained a sq uare of 1 6
which he s aid he should imagine to be a work of great labour ; but
if I forget not it had only the common propertie s o f mak ing th e
s ame s um viz 20 5 6 in every ro w horizontal vertical and diagon al
,
,
,
,
.
,
,
,
.
willing to be outdone by M r S ti feliu s even in the s ize o f my
sq uare I went home and made that evening the followi ng m agi cal
square o f 1 6 (see Fig 1 84) which besides having all the prope rtie s
N ot
.
,
,
,
.
of the foregoing sq uare o f 8 i
in al l the
s ame row s and diagonal s h ad t h i s added t h at a four sq u are h ol e
being cut in a piece of paper o f such a s ize a s to tak e in and s h ow
th rough it j u s t 1 6 o f the little sq uare s when laid on th e grea
ter
,
e
.
.
,
it would make
,
20 5 6
-
,
,
3 a! 1
t
0
F ig
.
1 86
71 9 0
l ei /"J
.
s quare the s um of the 1 6 number s so appearing through the h ol e
wherever it wa s placed on the greater sq uare s hould lik ewi s e make
Thi s I s ent to ou r friend the nex t morning w h o after s ome
20 5 6
day s s ent it back in a letter with the s e word s
:
,
,
.
“
,
.
I
return to thee t h y a s toni s hing
or mo s t stup endou s piece
of the ma gi cal s q uare in which
—
but the compliment i s too ex trava gant and therefore for hi s s ak e
,
N or i s it nece ss ary
mak e no q ue s tion but yo u will readily allow t h e sq uare o f 1 6
a s well as my own
for I
,
I
ought not to rep eat it
.
,
‘
TH E
F
RAN KLI N SQUAR E S
.
The dotted lines in this square indicate four bent diagonal col
u m us each o f which has a total O f 3 4 ; three o f these columns being
intact W ithin the squa r e and o ne b eing b roken F ou r b ent dia gonal
c olu m n s m a y b e for m ed fro m each o f the four Sides o f the square
b u t o nly twelve o f these sixteen columns have the p r oper totals
A dding to these the eight st r aight columns w e find that this square
“
c o ntains twe n
ty columns with summations o f 3 4 The 4 X 4 Jaina
squa r e c ontain s Sixteen columns which sum up to 3 4 while the
o r di n a r y 4 X 4 m agic square contains only t en:
The 8 X 8 F ranklin square (F ig 1 83 ) contain s forty eight
colu m n s which su m up to 2 60 viz eight h orizontal eight p erp en
dic u l ar Sixteen bent horizontal diagonals and sixteen bent p erp en
,
.
,
.
,
”
.
-
.
‘
.
,
,
,
,
,
S ec
S ec fio n 3
F ig
.
diagonals wherea s the
,
ti
on 2
.
.
1 88
M
g
.
X
o rdinary
A
8 m agic
um n s o f
sq uare con
the sa m e summation
other cha r acteristics mentioned by F ranklin
in his letter concerning his 8 X 8 magic s quare it may be stated that
the su m o f the nu m bers in any 2 X 2 sub square contained therein
tains
0
.
-
is
1 30 ,
and that
the
sum of any four nu m b ers that are arranged
a lc o
s ym
equidistant from the cente r o f the square also e q uals I
In regard to his 1 6 X 1 6 square F r anklin states in his letter
that the sum o f the numbers in any 4 X 4 sub square contained
there i n is 2 0 5 6 The sub division m ay indeed be ca r ried still further
f o r it will be Ob served that the su m Of the nu m b ers in any 2 X 2
m etrically
,
-
-
.
,
T H E F RAN KLI N SQUAR E S
[
95
.
u are is 5 1 4 and there are also othe r cu r ious features which a
study will disclose
,
.
3
gfi
t
fil
n
z
e
f
rn
j
Ja e /5
0
n
en
a
Se c /Eo n 5ej
.
t 11
3
}
1
£
!
e
ff
o
n
Av e rt :
’
J
F ig
“
3?
.
P
fg
1
.
1 89
.
he F ranklin S qua r es possess a unique and peculiar s y mmetry
arrange m ent of their nu m b ers which i s not clea r ly o bse r vable
eir faces b u t which is brought o ut very strikingly in thei r
,
96
TH E
F
RAN K LI N SQUAR E S
.
geometrical diagrams as given in F igs 1 86 and 1 8 7 which illustrate
r espectively the diagrams of the 4 X 4 and 8 X 8 squa r es
M agic cubes may be readily constructed by expanding these
diagra m s and writing in the appropriate numbe r s
The cube of 4 X 4 X 4 and its diagram are given as exa m ples
in F igs 1 88 and 1 89 and it will be observed that the cu r iou s char
acteri sti c s o f the square are carried into the cube
.
,
.
.
~
,
.
.
A N A N A LY S I S
In The L i
V
ol
(
8
I
.
pp
,
.
fe and Ti mes o f
25 5 - 2
X 8 the other
,
O F TH E F R
A N KL I N
S
Q UA R E S
.
Ja m es P arton
there i s an account o f tw o magic squares one
1 6 which are given here in F igs 1 1 and 1 2
9
9
B enj a m i n F ra n kli n b y
,
,
,
16
X
.
,
26 0
26 0
PROPER T IES
OF F
.
RAN K LI N S
’
F ig
.
1 90
.
8X 8
SQUA RE
.
.
M r P arton ex plains the 8 X 8 square as follow s
“
Thi s s q uare a s e x plained by its contriver contain s astonishing
propertie s :every s traight row (horizontal or vertical ) added to
.
,
,
TH E F
RA N K LI N SQUAR E S
1 0 28
PROPER T IES
.
20 5 6
OF F
RA N KLI N S
’
F ig
.
1 93
.
I 6X 16
—
1 28
SQUARE
.
T H E F RAN KLI N SQUAR E S
99
.
ther make s 260 and each hal f row hal f 260 The bent ro w of
1
i
ht
numbers
ascending
and
descending
diagonally
from
6
v
z
{
and fro m 2 3 descending to 1 7 and every o ne of
sending to 1 0
parallel b ent r ows of eight numbers makes 2 60 A lso the bent
.
,
.
,
,
,
.
20 5 6
PROP E R T I E S
,
OF
F
RAN KLI N S
Fig
’
.
I6
,
+ 1 28
X 1 6 SQUARE
.
1 93
from 5 2 descending to 5 4 and from 43 a s cending to 45 and
ry one of its parallel bent row s of
b
num
ers
makes
2 60
eight
so the b ent row from 4
to
43 descending to the left and from
5
,
,
“
,
,
,
.
,
TH E
1 00
F
RA N K LI N SQUARES
.
to 1 7 de s cending to the right and every one o f it s parallel bent
row s o f eight numbers makes 2 60 A lso the b ent ro w from 5 2
to 5 4 descending to the right and from 1 0 to 1 6 descending to
23
,
,
,
.
,
,
,
,
the left and every one o f its parallel bent rows of eight numbers
mak es 260 A lso the parallel bent rows next to the above men
ti o ned which are shortened to three numbers ascending and three
descending etc as from 5 3 to 4 ascending and from 29 to 44
,
,
-
.
,
,
.
,
,
descending make with the two corner numbers 260 A lso the two
numbers 1 4 6 1 ascending and 36 1 9 descending with the lower
fou r numbers Situated like them viz 5 0 1 descending and 3 2 47
ascending ma k es 2 60 A nd la s tly the four corner numbers with
the fou r middle number s mak e 2 60
,
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,
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,
“
B ut even these are not all the properties o f this marvelous
sq uare Its contriver declared that it has five other curiou s one s
‘
’
.
,
which he does not explain ; but which the ingeniou s reader may
di s cover if he can
.
These remark able characteristics which M r P arton enu merate s
are illustrated graphically in the accompanying diagrams in which the
relative position O f the cells containing the numbers which mak e up
the number 260 is indicated by the relation of the small hollow
squares (F i g
.
,
.
square is constructed upon the s ame principle
a s the smaller and M r P arton continues
“
N or was this the most wonderful of F rank lin s magical
sq uares H e made one of sixteen cells in each ro w which beside s
p ossessing the properties o f the square s given above (the amount
however added being always
had also this most remark
able peculiarity :a square hole being cut in a p iece of paper o f such
a Size as to take in and Show thro ugh it j ust Sixteen o f the little
s quares when laid o n the greater square the sum o f sixteen num
bers so appearing through the hole wherever it wa s placed on the
F rank l i n
’
s 16 X 1 6
.
,
’
,
.
,
,
,
,
,
,
greater s quare should likewise make
The additional peculiarity which M r P arton notes o f the 1 6 X
1 6 s quare is no more remarkable than the corresponding fact which
i s true o f the smaller square that the sum of the numbers in any
,
.
,
T H E F RAN KL I N SQUAR E S
.
combination of its cells yields 1 30
The properties of the
larger square are also graphically represented here (F ig
A clue to the construction o f these squares may b e found as
follows
We write down the numbers in numerical order and call the
cell s a fter the p recedent of the chess board with two sets o f symbol s
“
letters and numbers We call this the plan of construction (F ig
2X 2
.
.
—
,
,
.
I 94
.
)
B e fore we construct the general s cheme o f F ranklin s square
we will build up another magic square a little le s s comple x in prin
’
,
c ip l e,
which will be p reparatory work for more complicated squares
We will simply intermix the ordinary s erie s of numbers accordin g
to a defi nite rule alternately reversing the letters so that the odd
rows are i n alphabetical order and the even ones rever s ed In order
to distribute the numbers in a regular fashion 5 0 that no combina
tion o f letter and number would occu r twice we start with 1 in the
upper left hand corner and pas s consecutively downward s alter
nating between the first and second cells in the successive row s
thence a s cending by the same method o f simple alternation from 1
in the lower le ft hand corne r We have no w the key to a scheme
for the distribution o f number s in an 8 X 8 magic square It is the
first step in the construction of the F ranklin 8 X 8 magic square and
“
we call it the k ey to the scheme o f simple alternation (F ig 1 95 )
.
.
,
-
,
,
-
.
.
,
.
goe s without s aying that the e ff ect would be the same i f we
begin in the same way in the right hand corners Only we mu s t
b ew are of a di s tribution that would occasion repetition s
T0 c omplete the s cheme we have to repeat the letters alternatel y
invert i ng their order row after row and the first two given fi gures
mu s t be repeated throughout every ro w as they are started T he
top and bottom rows will read I 8 ; 1 8 ; 1 8 ; 1 8 The second
row from the top and also from the bottom will be 7 2 ; 7 2 ; 7 2
6
ro w
6
The
third
from
the
top
and
bottom
will
be
2
;
;
3
3
7
I
n
6
and
the
two
center
r
ows
6
:
;
;
;
;
4
4
4
4
5
5
5
5
3
3
every line the sum of two con s ecutive figu res yields 9 This is the
second step yielding the completed scheme o f Simple alternation
It
—
.
'
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,
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,
,
,
,
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,
,
,
,
,
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,
,
,
,
,
,
.
,
,
.
i
F
( g
,
.
TH E F
RA N KLI N SQUARES
1 03
.
The squa r e is no w produced by su b stituting for the letter and
figure combinations the corresponding figu r es according to the con
s ec u ti v e arrange m ent in the plan of const r uction (F ig
Trying the results we find that all ho r izontal rows su m up to
while the ve rtical rows are alte r nately 260
and
260
4
The diagonal from the upp er right to the lower left corner yields
a sum of
while the other diagonal from the left upper
2
corner descending to the right lower corner makes 260
The
3
upper halves o f the two diagonals yield 260 and also the sum o f
the lower halves and the sum total of both diagonals is accordingly
2
The
sum
of
the
t
o
left
hand
hal
f
diagonals
re
2
0
or
2
6
0
w
X
5
16
and the su m of the two hal f diagonal s to the
sul t s in 2 60
right hand side makes
The sum of the fou r central cell s
plu s the four extre m e corner cells yields also 260
Considering the fact that the figures 1 to 8 o f our scheme run
up and down in alternate succession we naturally have an arrange
ment of figures in which sets of two belong together Thi s binate
peculiarity is evidenced in the result j u st stated that the rows yield
sums which are the same with an alternate addition and subtraction
of an equal amount S O we h ave a symmetry which is a s toni s hing
and might be deemed magical i f it were not a matter o f intrin s ic
nece ss ity
We represent these p eculiarities in the adj oined diagrams (F ig
1 98 ) which however b y no means exhaust all the possi b ilities
,
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,
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,
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,
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,
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,
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,
We must bear in mind that these magic squares are to be re
garded a s continuou s ; that is to say they are as i f their Oppo s ite
sides in either direction passed over into one another a s i f they
were j oined both ways in the shape o f a cylinder In other words
whe n we cro ss the boundary o f the square on the right hand the fi rst
ro w
of cell s outside to the right has to be regarded a s identical
with the fi rst row of cells on the left ; and in the same way the
uppe rmo s t or fi rst horizontal row of cells correspond s to the fi r s t
row of cell s belo w the bottom row This remarkable p rope rty of
the sq uare will bring out some additio nal p ecul iarities which mathe
maticians may ea s ily derive according to general principle s ; espe
c ially what wa s stated of the sum of the lower and upper hal f
,
.
,
.
'
TH E
104
26 0
1
6
26 0
F
260
.
1 98
.
4
2
PROPER T I ES
OF
.
26 0
16
2 60 —
F ig
RA N K LI N SQUARES
8X 8
X
26 0
SQUARE
BY
S I M PLE
20 5 6
F ig
.
1 99
.
PROPER T IES
OF
16X 16
SQUARE
BY
SI M PLE AL T ERNA I
’
1 06
TH E
Fig
.
KEY
200
T O T H E S C H EM E
Fi g
.
S
rs t
t ep s
.
St p
A L TE R N AT IO N W I TH
202
S C H E ME O F
BI N ATE
.
OF
RA N KLI N SQUARES
i S
A LT ER NA T ION
F
.
F
ec o n d
e
.
T RA N S P O S I I I O N
’
‘
.
Fig
.
20 4
.
S CHE ME
OF F
.
W
F ig
.
20 1
.
I T H B I NA T E T RA N SPOSI T ION
.
T hi d S p
S Q UAR E C O N S T R U C TE D BY ALTE R N A
T ION W I TH BINATE T RAN SPO SI T IO N
Fi g
RAN KLIN S
'
BX S
.
20 3
.
S Q UAR E
r
te
.
T H E F RAN KLI N SQUARES
F i g 205
.
.
C
.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
ONS E C U T I VE ARRA NGEMENT
OF
S CH E ME
OF
F ig
.
206
.
KE Y T O T HE
N U MB E RS
IN
A
16
X 16
SIMPLE AL TE RNAT ION
S Q UAR E
.
1 08
TH E F
Fi g
Fi g
.
208
.
16
X
16
.
207
.
RA N KLI N SQUARES
S CH E ME
OF
SI MP L E
AL
.
TE RNAT IO N
.
MA G I C S Q UAR E C ONS T RU CTED BY SIMPL E ALTE RNAT IO N
.
T H E F RAN K LI N SQUARES
1 10
Fi g
.
21 1
.
.
S Q U AR E C ONS T RU C TE D B Y A L TE R N AT ION W I T H Q U ATE R N ATE T RANS
POSI T IO N
.
Fi g
.
21 2
.
S C HE ME
OF F
RA N KLI N S
’
16
X
16
S Q UAR E
.
TH E
F
RA N KLI N SQUAR E S
.
diagonal s represents regularities which counterbalance one another
on the right and the left hand Side In order to O ff set these results
we have to s hi ft the figures of our scheme
We tak e the diagram which forms the key to the scheme of ou r
dist r i b ution b y simple alternation (F ig
and cutting it in the
middle tu r n the lower hal f upside down giving the first two rows as
seen in F ig 20 0 in which the heavy lines indicate the cut ting Cutting
then the upper hal f in two (i e in binate sections ) and transposing
the second quarter to the bottom we have the key to the entire ar
rangement of figures ; in which the alternation starts as in the
scheme for Simple alternation but Skips the fou r center row s passing
from 2 in the second cell of the second ro w to 3 in the first cell of
’
the seventh and from 4 in the second cell of the eighth pa ss ing to
in
h
first
cell
and
thence
up
ward
in
similar
alternation
again
t
e
5
s
passing over the four central rows to the second and ending with 8
in the second cell of the first row Then the same alternation i s pro
It is obvious that this can not start
du ced in the four center rows
in the first cell as that would duplicate the first row s o we start with
1 in the second cell passing down uninterruptedly to 4 and ascending
as be fore f r om 5 to 8
A closer exa m ination will Show that the row s are binate which
mean s in set s of two The four inner numbers 3 4 5 6 and the
two outer sets O f two numbers each 1 2 and 7 8 are brought to
gether thus imparting to the whole s quare a binate character (F ig
-
.
.
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,
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,
,
,
,
,
,
.
.
,
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,
,
.
,
,
,
,
,
,
,
.
We are no w provided with a key to build up a m a gic s q uare
after the pattern of F ran k lin We have Simply to complete it in
the same way as our last square repeating the letters with their
order alternately reversed a s before and repeating the fi gure s in
each line
When we insert their figure values we have a square which is
not the same as F ranklin s but posses s es in principle the s ame
qualities (F ig
To make ou r 8 X 8 square of binate transposition into the
F ranklin square we mu st first take its obverse square ; that i s to
say we preserve exactly the same order but holding the p aper
.
,
.
’
,
.
,
TH E
1 12
F
RA N KLI N SQUAR E S
.
with the fi gures toward the light we read them o ff from the obverse
side and then tak e the mirror picture of the result holding the
mirror on either horizontal side S o far we have still our sq uare
with the peculiarities of o ur scheme but which lack s one o f the
incidental characteristics of F rank lin s square
We must notice
that he ma k es four cells in both horizontal and vertical direction s
s um up to 1 3 0 whic h property i s necessarily limited only to two
s ets o f fou r cell s in each row If we write down the sum of
we will fi nd that th e middle s et
,
,
.
,
’
.
.
.
i s e q ual to the rest consisting o f the s um o f two e x treme s
In this way we cut o ut in our s cheme (F i g
and
the
rows rep resented by the letters C D E F in either order and ac
c o rdingly w e can s h i ft either o f t h e two fi r s t o r two la s t vertical
,
.
,
,
,
rows to the oth er s ide F ran k lin did the former thu s beginning
hi s square with G 4 in the left uppe r corner as in F ig 20 4 We have
indicated this divis i on by heavier lines in both sche m es
.
,
'
.
.
.
The greater s q uare of F ran k lin which i s 1 6 X I 6 is made after
the same fashion and the adj oined diagram s (F igs 20 5 2 1 2 ) will
su f ficiently explain its construction
,
,
-
.
,
.
We do not k now the method employed by F ranklin ; we pos
se s s only t h e re s ult but it i s not probable th at h e derived hi s sq uare
according to the s cheme employed h ere
O ur 1 6 X 1 6 S q uare i s not ex actly th e s ame a s the s q uare o f
O ur method gives the
F ranklin but it belong s to t h e s ame cla ss
,
.
.
,
k ey to the con s truction and it i s understood t h at the s ystem here
represented will allow u s to con s truct many m
ore s q uares by s impl y
pu shing t h e sq uare beyond its limits into the Opposite row which
b y thi s move ha s to be tran s ferred
There i s the s ame relation between F ranklin s 1 6 X I 6 s q uare
and ou r sq uare con s tructed by alternation with quaternate tran s
p osition that e x i s t s between the corre s ponding 8 X 8 s quare s
,
.
’
.
,
P
.
C
.
R E F LE CT ION S ON M AGI C SQUAR E S
.
shape o f a cylinder This cannot b e done at once with both its two
opposite vertical and its tw o Opposite horiz o ntal sides b u t the pro
cess is easily represented in the plane b y havin g the magic square
extended o n all its sides and on passing its li m its on one Side we
mu st t r eat the extension as i f we had ente r ed into the magic square
.
,
o
,
on the side Opposite to where we left it If we now transfer the
figu res to their r espective places in the inside square they are shoved
over in a way which by a regular transposition will counteract their
regular increase o f counting and so equalize the sums of entire rows
The case is so m ewhat more co m plicated with even magic
squares and a suggestion which I propose to O ff e r here pertains
to thei r form ation M r A ndrews be g ins t hei r discussion b y stating
.
,
.
,
,
.
.
The s m allest magic square o f even num b e r s i s o f course 4 X 4 ;
and he points o ut that i f we write the figures in their regular o r der
in a 4 X 4 square those standing on the diagonal lines can remain
,
,
,
in their places while the r est are to be r eve r sed so as to r eplace
every figure by its comple m entary to 1 7 (i e 2 by 1 5 3 by 1 4 5 b y
1 2 9 by 8 ) the num b e i 1 7 b eing the sum O f the highest and lowest
2
numbe r s of the magic square (i e n
I ) It i s b y this reversal
o f figures that the inequalities of the natural order are equalized
again so as to make the sum o f each ro w equal to 3 4 which is o ne
fourth o f the sum total o f all figures the general formula being
,
.
.
,
,
,
,
.
.
.
,
,
,
,
n
n
We will no w try to find o ut more about the relation which the
magic square a r rangement b ears to the normal sequence o f figures
F o r each corner there are two ways one horizontal and o ne
vertical in which figu r es can be written in the no r m al sequence
.
,
,
accordingly there are altogether eight possible arrangements from
which we select one as fundamental and rega i d all others as me re
variations produced by inverting and reversing the order
,
‘
,
,
.
RE F L E CT IO N S ON MAGI C SQUARES
1 15
.
As the fundamental arrangement we choose the ordinary way
o f writing f r om the left to the right p r oceeding in parallel lines
“
downwa r d We call this the ordinary order o r 0 Its reverse
proceed s f r om the lower right hand co r ne r toward the left and
line b y line upward thu s b eginning the se r ies where the ordina r y
a r rangement ends and ending where it started We call this order
“
the reve r sed ordinary o r si m ply ro
A nothe r o r der is produced b y following the H ebrew and A rabic
mode of writing :we b egin in the upper r ight hand corner p r oceed
,
”
.
.
-
,
,
.
,
”
.
,
-
,
ing to the left and then continue in the same way line by line
downwa r d This the inverse direction to the ordinary way we
call b riefly i
The reverse o r de r of i starting in the lower le ft corner p r o
and line b y line upward we call ri F urther
c eedin g to the right
on we Shall have occasion to represent these four orders b y the fol
i b y 4 ; ri b y
lowin g symbols :0 b y
ro by
,
,
.
,
.
,
,
.
,
4
ORDER
O RD ER
ORD ER
0
ORD ER
ri
F ig
.
214
1.
ro
RE F LE CT ION S
1 16
W
i ll
Fi g
.
21 5 .
EVE N S Q UARE S
qu r
u i t
u t u lly i
These s a es, 4 X 4
an d
b e s ffi c en t o
0 an d ro are m
a
nte
T
it
an d
MAGI C SQUARES
ON
it s m
I N MUL T IPL E S
u lti p l
writ o ut the
rc h angeab l e
e
.
es,
i t
tw o 4
X 4
OF F
of
c o ns s
s
.
OUR
.
*
o an d r o o
qu r
a es ,
t
WW
rders
Ch S hO
ly
on
I1 0
,
R E F LE C T IO N S
1 18
ON
MA G I C SQUAR E S
.
hold the diagonal lines The 1 2 r e go parallel with one of
these diagonals and stand in such position s that i f the whole magic
square were diagonally turned up on itself they would exactly cover
the 6 i and 6 ri figures A nd again t h e 6 i and 6 ri also hold toward
each other places in the same way corresponding to one another ;
i f the magic square we r e turned upon itsel f around the othe r diag
onal each r i figu re would c ove r o n e o f the i o r der
12
0
.
,
,
,
.
‘
,
.
Fi g
.
21 6
6X6
.
EVEN S Q UAR E S
.
we compare the magic squares with the san d covered glass
plates which Chladni u sed and think of every cell as equally filled
with t he fou r figures that would fall u pon it according to the normal
sequence o f 0 re i and ri and further i f we compare their change
into a magic square to a musical note ha r m onizing whole row s into
If
-
,
,
,
,
equal sums w e w Ou l d find (i f by so m e magic process the di ff erent
values o f the several fi gu res would m echanically b e turned up so
,
0
as to be evenly balanced in rows ) that they would present geo met
ri c al ly harmoniou s designs as m uch as the Chladni acousti c figu r es
The progressive trans formations o f o r o i and ri b y m irro r i ng
,
,
,
,
.
,
are not unlik e the air waves of notes in which 0 represents the crest
o f the wave r e the trough i and ri the nodes
In placing the mirror at right angles progressively from o to
i from i to r o fro m ro to ri and from ri to 0 we return to the
beginning thu s com pleting a whole sweep o f the
The re
,
.
,
,
,
,
,
S ee
di gr
a
am o n
pa
g
e 1 15
.
,
R E F L E C T IO N S ON MAGI C SQUARE S
F i g 21 7
.
b ma
l tt r
rk t h p l
The
e
s
e
e
0
ac e
i ndi cat
wh r
e e
es
.
wh r
HLA D NI FI G U R E S ‘
C
e e
the b o w
.
.
the
s
s
tri k
u rf
es
ac e
the
t u ch ed with
gl p l t e In
is
o
as s
a
.
fi ng r ; whil
th f u r u p p r
a
e
e
o
e
e
RE F LE CT ION S ON M AGI C SQUARES
1 20
.
verse o f w h ich i s
repre s ent s one hal f turn i and ri the fi r s t and
third q uarter in the whole circuit and it is natural there fore that
0
ro
-
,
,
,
,
a symmetry producing wave Should p roduce a Similar e ff ect in the
magic s q uare to that o f a note upon the sand of a C hladni glas s
plate
-
.
M AG I C
S
Q UA R E S
IN S
YM B OL S
.
The diagrams which a r e O ff ered here in F ig 2 1 8 are the be s t
evidence of their resemblance to the Chladni figures both exhibiting
in their formation the e ff ect o f the law of symmetry The most
.
,
.
,
!
“O
O E O OOE B OO B OO
OO!
C O B OO O OOO B B O
O B OO OO B O
o e e o o e e oo a e o
8X8
.
32
0 an
d
32
ro
IO
.
S Q U AR E S
Co nstr
Co nst r
u t d fr
c e
OF
MU L T IPL E S
utd
c e
8X 8
om
al l
on
a
a
a
s
e
as
in
an
of
o
the
.
o
.
OF F
0
OU R
a nd ro
and
stene
72
ro
.
.
.
rd ers
218
72
IO
,
0,
ro ,
i
,
an d ri
.
.
d i n th c ent r whil
ex c entri c p o si t i n i n d i c t d b y
b een
fa
ly
S Q UARE S
F ig
di gr m th p l at e h
it h s b een h eld tigh t
X
e
e
,
o
o
,
a e
e
l w r o nes
the wh i t do t
in the
o
e
e
.
R E F L E CT ION S O N M A G I C SQUAR E S
1 22
here p r opose to indicate the o r der o f
ri
by
o
.
ro
i by
by
4
4
,
by
M A G I C S Q UA R E
T HE
IN C
H IN A
.
the introduction to the Chou edition of the Yi h Ki ng we
fi nd some arithmetical diagrams and among them the L o h S hu the
In
,
-
,
scroll o f the river L oh which is a mathematical square from 1 to 9
so written that all the odd numbers are expressed by white dot s
,
i
.
e
.
,
,
,
yang symbol s the emblem o f heaven while the even numbers
,
,
W
T H E S CROLL OF LO H
(A
Fig
.
21 9
.
T
MA P
TH E
.
OF H o
rdi ng t T s a i Yu ng ti ng )
O ARI T HM ET I CAL D E SI GN S O F AN C I ENT
cco
o
‘
a
*
.
—
.
CH
I NA
.
are in black dots i e yin symbols the emblem o f earth The in
v enti o n o f the scroll is attributed to F u h H i
the mythical founder
o f Chine s e civilization who according to C hinese report s lived 285 8
273 8 B
C B ut it goes without saying that we have to deal here
with a recon s truction o f an ancient document and not with t h e
document it s el f The scroll o f L oh i s shown in F ig 2 1 9
The first unequivocal appearance o f the L oh S hu in the form of
,
.
.
,
,
.
-
,
,
.
.
,
.
.
.
-
a magic sq uare i s in the latter part of the posterior Chou dynasty
r
ill u t t
rly d
irit
l g h r
ti
wh
t i
gr u
rth r
tiv ly
g tiv
di r
it t d
s an
The m ap o f Ho p o p e
be
o es no t b e o n
e e b ut w e l et
c a se i t
o f Lo h w a s
e p s to
m es
s ra e t he s p
o f t he
en the sc o
c o m p o se
in C na
o p s o f o dd an d e en
The m ap o f Ho c o n a n s fiv e
If the o m e a re re
esp ec
e
es the n m b e s o f
ea en a n d ea
o p
ff e en ce o f eac
as n e a
e t he
a
e
a
e
as p o s
e a n d the
*
u
figu r
h l
d
,
u
hi
.
r
g rd d
itiv
will u n i f rm l y y i ld
o
e
h v
5
or
l tt r
5
.
,
.
,
r ll
f r
v
r
h gr u
RE F LE CT ION S ON MAGI C SQ UARES
1 23
.
3
A
or
the
beginning
of
the
S
outhern
S
ung
dynasty
D
)
(95
A
The
L
oh
S
hu
is
incorporated
in
the
writing
s
1
1
1
2
(
7 3 33
of Ts ai Yii an Ting who lived from 1 1 3 5 1 1 98 A D (cf M ayers
1 - 1 1 26
.
.
-
-
.
‘
-
—
’
Chines e R eader s
M
.
anu al,
.
.
,
I , 75 4a ),
but similar arithmetical dia
grams are traceable a s recon s truction s of primitive documents amon g
scholar s that lived under the reign of S ung H wei Tsung which
lasted from 1 1 0 1 1 1 2 5 A D (S ee M ayers C R
p
The Yi h Ki ng is un q ue s tionably very ancient and the symbo l s
-
M
-
.
.
.
,
.
.
,
,
.
yang and yin as Em blems o f heaven and earth are inseparable from
its contents They existed at the time o f Con fuciu s (5 5 1 479 B
for he wrote several chapters which are called appendices to the
Yi h Ki ng and in them he s ays (III I I X 49 5 0
S B E XV I
-
.
.
-
,
,
.
.
,
,
.
.
,
,
P 3 65 )
.
“
To heaven belongs 1 ; to earth 2 ; to heaven 3 ; to earth 4 ;
to heaven 5 ; to ea rth 6 ; to heaven 7 ; to earth 8 ; to heaven 9 ;
to earth 1 0
,
,
,
,
,
,
,
,
,
.
“
The number s belonging to heaven are five and those belonging
to earth are five The numbers of these t w o series correspond to
each other and each one has another that may be considered its
mate The heavenly numbers amount to 2 5 and the earthly to 30
The number s o f heaven and earth together amount to 5 5 It is
by these that the change s and trans formations are eff ected and the
Spiritlike agencie s kept in movement
Thi s pa ss age wa s written about 5 00 B C and i s approximatel y
Simultaneou s with the philosophy of P ythagora s in the O ccident
who declare s number to be the e s sence o f all things
O ne thing is sure that the magic s q uare among the Chinese
cannot have been derived from E urope It is highly probable how
ever that both countries received s ugge s tions and a general impulse
from India and perhaps ultimately from B abylonia B ut the devel
,
.
,
.
.
,
.
,
”
.
.
.
,
.
,
,
.
,
.
of the yang and yin s ymbol s in their numerical and occult
signi fi cance can be traced back in C hina to a hoary anti q uity s o as
to render it typically C hine s e and thu s it s eems s trange that the
same idea o f the odd numbers a s bel onging to heaven and the even
ones to earth a pp ears in ancient G reece
I owe the following communication to a personal letter from
o p m ent
,
.
RE F LE CT ION S ON M AGI C SQUARES
1 24
P rofess o r D avid
York
E ugene
.
S mith of the Teachers College of N ew
’
There is a L atin aphorism probably as Old as P ythagora s
D eus i mpari b us n u m eris ga u det V irgil paraphrases this as follow s
N u mero deus i mpare gau det (E c l viii
In the edition I have
,
,
.
.
.
,
at han d there i s a footnote which gives the ancient idea
nature of odd and even numbers saying
*
of
.
the
,
.
i mp ar
[
,
n on
p o test
m o rtalis q uia div i di p o test ; li cet V arro di ca t P ytho
p a tare i mpare numera m habere finem p arem esse infini tu m
ar nu merus
p
or
g eos
,
m
a curi ou s idea which
m ul taru m qu e
causa
i mmo rta lis qui a div i di i nteger
nu m erus
di ctum
es t,
p eri
su
,
I
r eru
have not s een elsewhere ] ;
m i mp ares
dii i mp ari i nf eri
,
nu m ero s s erv ari
p ari
i deo m edendi
:nam
,
p
u t su ra
a
t
n
u
d
e
g
.
“
T here are several reference s among the later commentators
to the fact that the O dd numbers are ma s culine divine heavenly
while the even Ones were feminine mortal earthly but I cannot j u s t
at thi s writing place my hands upon them
“
A S to the magic square P rofe s sor F uj i s awa at t h e I nter
national C ongre ss o f M athematicians at P aris in 1 90 0 made the
as s ertion that the mathematics derived at an early time from the
C hine s e (independent o f their own native mathematics which wa s
o f a somewhat more s cienti fi c character ) included the s tudy o f
the s e sq uare s going as far a s th e fi r s t 400 number s H e did not
however give th e date s of the s e contribution s i f indeed they are
”
k nown
,
,
,
,
,
,
.
,
,
,
,
,
.
,
,
,
.
A s to other magic s q uare s P rofes s or S mith write s in anot h er
,
letter :
“
The magic s quare i s found in a work by A braham ben
E z ra
in t h e eleventh century It is also found in A r
abic work s o f the
twel fth century In 1 90 4 P ro fes s or S chilli ng contributed to the
M athematical S ociety O f G ottingen the fact that P rofessor Kielhorn
had found a Jaina in s cri ption o f the twel fth or thirteenth century
.
.
,
M
P V i rgilii
aro ni s
Op era Ic um integ ri s co mm enta ri i s l S ery u , Phi
P a nc rati s
l argyri i, P ieri i A c cedunt S c alig eri et L indenb ro gi i
x
r
i
c
I
I
c
L
n
d
a
e
o
oc
I
e
o
a
a sv i c i us l
T
m
o
a
,
|
|
|
|
*
.
M
,
u
,
.
,
R E F LE CT I ON S
M AGI C SQ UAR E S
ON
.
ward or forward upward or downward in horizontal vertical or
2
slanting lines always yield the same sum viz 3 4 which i s 2 (n + 1 )
and s o does any small square o f 2 X 2 cell s S ince we can not bend
the square upon itsel f at once in two direction s we mak e the re s ult
visi b le in F ig 2 2 1 by extending the square in each direction by
hal f its own Size
Wherever 4 X 4 cells are taken out from this extended sq uar e
,
,
,
,
.
,
,
,
.
,
.
,
.
,
w e
Of
Shall find them satis fying all the condition s o f thi s peculiar k ind
magic squares
.
T he construction of this ancient Jaina equilibrium sq uare
-
qu i re s another method than
w e
\
F ig
M
re
have s uggested for M r A ndrew s
’
.
.
22 1
.
squares and the following considerations will a ff ord us
the key a s shown in F ig 22 2
F irst we write the numbers down into the cells of the sq uare
in th eir consecutive order and call the four rows in one direction
A B C D ; in the other direction 1 2 3 4 O ur aim i s to re
di s tribute them S O as to have no two numbers o f the s ame deno mi
nation in the same row In other words each row mu st contain
,
.
,
,
.
,
,
,
,
.
one and only one
of
.
,
each o f the four letters and al s o one and only
,
each o f the four fi gure s
We s tart in the left upper corner and write down in the fi r s t
horizontal row the letters A B C and D in their ordinary s ucce s
one
of
.
,
,
,
,
sion and in the s econd horizontal row the same letters in t h ei r
,
,
RE F LE C T ION S ON MAGI C SQUARES
1 27
.
inverted order We do the same with the numbers in the fi r s t and
second vertical row s A ll that remains to be done is to fill out the
rest in such a way as not to repeat either a letter or a number In
the first row there are still missing for C and D the number s 2 and 3
.
.
.
,
of which 2 mu s t belong to C for C3 appear s already in the s econd
ro w and 3 i s left for D
In the second row there are mi ss ing I and 4 o f w h ich 1 mu s t
belong to B becau s e we have B 4 in the fi r s t row
In the fir s t vertical row the letters B and C are mi s sing of
which B must belong to 3 leaving C to 4
,
.
,
,
.
,
.
,
2
l
In Co nsec
The
P rf t d
e
4
3
u tiv
e
O
rd r
e
di tri b u ti o n
Re
ec e
s
The S
.
F
.
Fig
.
222
t art
for
a
i gu re V al u es
Re
of
di tri b u ti o n
s
the
.
S
qu r
a e
.
.
the second vertical row A and D are missing for 1 and 2
AI and D 2 exist s o A must go to 2 and D to 1
In the s ame simple fashion all the column s are fi lled out and
then the cell na mes replaced by their fi gu re value s which yields
the same kind of magic s quare as the one communicated by P rof
S mith with the s e di ff erences only that ours start s in the left
corner with number 1 and the vertical row s are e x changed with
the horizontal ones It is scarcely neces s ary to point out th e beauti
ful symmetry in the distribution of the fi gure s which become s full y
apparent when we consider their cell names B oth the letter s A
In
.
.
,
,
,
,
.
,
,
.
.
,
,
RE F LE CT ION S
1 28
MAGI C SQUARES
ON
.
and the fi gu re s 1 2 3 4 are harmoniou s ly di s tributed
over the whole sq uare so a s to leave to each small sq uare it s di s
tinct individuality as appears from F ig 223
B C
,
,
D
,
,
,
,
,
,
,
.
,
Fig
.
223
.
.
Th e center s q uare in each ca s e e x hibits a cro ss relation th u s
,
a s imilar way each one of the four group s of four cell s in
each of the corner s possesse s an arrangement of it s own which i s
In
symmetrically di ff erent from the ot h er s
.
A M A T H EMA T I CAL S T UDY
2h =
OF
M AGI C SQUARES
.
b
+g
b+ d
2n
=
2c
=
d+ m
2a
=
m +g
will be s een t h at t h e fi r s t term s o f the s e e q uation s are the
q uantities which occur in the four corner cell s and there fore that
the q uantity in each corner cell i s a mean between the two q u an
titie s in the two oppo s ite cells that are located in the middle of
the out s ide row s It i s therefore evident th at t h e lea s t q uantity in
It
,
.
the magic sq uare mu s t occupy a middle cell in
o ne
of t h e four
out s ide row s and that it cannot occupy a corner cell
S ince the middle cell o f an out s ide ro w mu s t be occupied by t he
lea s t q uantity and s ince any o f the s e cell s may be m ade t h e mid dle
cell o f t h e upper row by rotating the square we may con s ider t h i s
cell to be s o occupied
.
,
,
,
.
H aving thu s located t h e lea s t q uantity in:the ; sqna re it i s plain
,
that the ne x t h igher q uantity mu s t be placed in o ne o f the lower
corner cell s and s ince a simple reflection in a mirror would rever s e
the po s ition o f the lower corner cell s it follow s t h at the s econd
s mallest quantity may occupy ei ther o f the s e corner cell s N e x t we
may write more e q uations as follows :
_
,
,
.
a
e
n
d+
e
+g
h+
e
+
S
=
5
=
S
d + h=
S
a
+
n
+g+
c
c z
or
s
ummation
(
)
S
there fore
e
3
: S
e z
S /s
H ence the q uantity in the central cell is an arithmetical mean
between any two q uantitie s with which it form s a straight ro w or
column
.
A MA T H E M A T I CAL S T UDY
OF
M A G I C SQUAR E S
With these facts in view a magic square may
as Shown in F ig 2 2 5
.
L et
no w
1
.
be
1
3
constructed
.
representing the least quantity be placed in the middle
upper cell and x
in
the
lower
right
hand
corner
cell
being
y
y
the increment over x
S ince x y is the mean b etween x and the quantity in the left
hand central cell this cell must evidently contain x
2y
N ow writing x + v in the lower le ft hand corn er cell (con
sidering v as the increment over 2 ) it follows that the central righ t
hand cell must contain x
2v
x
,
,
-
,
,
.
,
.
-
,
.
N ext as the quantity in the central cell in the square i s a mean
,
2y and x
between x
2 v it must be filled with x
I
t
v
y
no w follows that the lower central cell mu st contain x
2v
2y
and t h e upper left hand corner cell x l zv l y and fi nally the
upper right h and corner cell must contain x
v + 2y thu s com
i
i
s
l
i
t
n
the
sq
uare
which
nece
s
sarily
must
magic
qualificat
on
e
a
i
y
b
e
p
g
with any conceivable values which may be assigned to x v and y
f
We may no w p ro geed to giye values to x v and y which will
produce a 3 X 3 m ag ic Sq uare contai ning the numbers 1 to 9 in
1
c l u siv e in arithmetical pro ression
E
vidently
x
must
e
q
ual
and
g
,
.
,
-
— —
—
,
-
,
,
,
.
,
,
-
e
,
.
,
there must be a number 2 either v or y must equal 1 al s o ;
A ss uming y 1 i f v
1 or 2 duplicate number s would re
‘
f
o
”
?
7
C
i f
s ult therefore v mu st equal at l east g
B ecause the lower central cell is filled with the symbols x 27)
+ 2y and as in this case thi s comb ination m us t equal 9 ; therefore
i f v : 1 then y z 3 or v i c e v ersa
1
U sing these values viz x
z
1
i
and
m
agic
square
Shown
in
F
v
the
familiar
X
y
3
3
3
g
2 26 i s produced
It is i m p o rta nt to r eco gni z e the fact tha tAl tho ugh in F ig 226
the series of numbers u sed has an initial number of 1 and al s o a
ly
con s tant increment o f 1 yet t his may be c o ns idered as o n
a ccidental feature pertaining to this particular s quare the real fact
being that a magic s quare o f 3 X 3 is a lw ays c o mp osed of three sets
o f three nu m b ers t each
The di ff erence between the number s of
each trio is uni form but the di ff erence between the la s t term of one
as
,
_
_
,
,
”
,
.
,
,
,
.
,
.
,
,
,
.
,
.
r
.
,
é
s
,
g
,
.
.
,
1 32
A M A T H E MA T I CA L S T UDY
F ig
F ig
.
.
OF
228
229
.
.
MAGI C SQUAR E S
.
1 34
A M A T H E M A T I CAL S T U DY
OF
M AGI C SQUARES
.
A
MA T H E MA T I C A L S T UDY
any v a l u es to x ,
v,
y
,
OF
MA G I C SQ U AR E S
p erfect
t and s, w i l l yi eld a
co
,
1 35
.
m p o u nd 9 X 9
sq u are
.
V alues
m ay
assigned to x y v t and s which will pro
duce the series 1 to 8 1 inclusive A s stated b efore in connection
with the 3 X 3 square x must naturally equal 1 and in order to
produce 2 one of the remaining symbols must e q ual I In order
to avoid duplicates the next larger number m ust at least equal 3
and by the same proces s the next mu s t not be less than 9 and the
z
1
2
remaining one not less than 2 7 B ecau s e I
I
4
7
3
9
which i s the middle nu m ber o f the series 1
8 1 therefore j u s t
the s e values must be assigned to the five symbols u sed in t he co n
str uctio n o f the square The only symb ol whose value is fi x ed
however is x the other four symbol s may have the values I
3
2
or
assig
ed
to
them
indiscriminately
thus
producing
all
the
n
9
7
possible variations o f a 9 X 9 compound magic s q uare
I f v is first made 1 and y z 2 and a fterwards y i s made I and
v z 2 the resulting squares will be Simply reflections of each other
etc
S ix fundamental forms of 9 X 9 comp ound magic s quares
m ay be con st r ucted as Shown in F igs 2 30 2 3 1 and 2 3 2
O nly S ix for m s may b e made because excluding or whose value
i s fix edf sii:di ff erent couples (o r trios i f x is included ) may be
made from the fou ; sy m b ols T hese si x c el l s being determined the
r
r est o f the square b e c omes fixed
I t wil l be noted that these are arranged in three groups o f tw o
s q uare s each, o n account of the curious fact that the squares in each
pair are mutually convertible into each other by the following
process
I f the homologou s cells of each 3 X 3 sub sq u are b e tak en in t he
order as they occur in the 9 X 9 square a nd a 3 X 3 S q uare m a de
th erefrom , a new magic 3 X 3 square will result A nd i f this process
i s followed with all the cells and the resulting nine 3 X 3 squares are
arranged i n magic square order a new 9 X 9 compound square will
no w
be
,
,
,
.
,
,
.
,
,
,
.
,
—
,
.
,
,
,
,
.
,
,
,
.
.
,
,
,
fi
.
,
‘
,
,
'
g
.
,
.
—
,
-
.
_
.
re s ult
.
example re fe r ring to the upper squa r e i n
if the
numbers in the central cells o f th e nine 3 X 3 sub sq u ares are ar
ranged in magic s q uare order the resulting square will be the
Fo r
,
,
F ig
.
2 30 ,
1 36
A
M A T H E M A T I CAL S T UDY
MAGI C SQUARES
OF
.
cent r al 3 X 3 squa r e in the lower 9 X 9 square in F ig 230 This
l a w holds good in each o f the three groups of two squares (F igs
2 3 0 2 3 1 and 2 3 2 ) and no fundamental fo r ms other than these can
.
.
.
,
be constructed
The question m ay be asked :H ow many variations of 9 X 9
?
compound magic squares can be made
S i nce each sub square may
a ss ume any o f eight aspects without disturbing the general order of
the complete s q uare and Since there are six radically di ff erent or
fundamental form s obtainable the number of possible variation s is
.
,
,
,
6 X 8 !
9
/
e
u
f
t
q
tt
l
’
l
.
magic square as represented in F ig
F ig
.
23 3
.
Fig
.
2 33
F ig
2 34
.
F rom
.
.
.
23 5
our knowledge o f f his
f
F ig
.
.
236
.
qu are and its qualifications we are enabled to write four e q uations
a s follow s :
p y z 5 (S ummation )
a
h
s
k+
K
/
ins
By
terms o f
of t h eir fi nal term
t
F ig
p
s
S
0
n
dI
5
2 33
i
s
a
t
o
n
u
q
it is seen that the sum o f the initial
equal s
so equal s 5
therefore follow s :
I st That the s u m
.
.
o
f
o
,
and likewise that the s um
H ence
.
the terms
s q u are is eq u al
S
5
h
n
c o n tai ned
p
o
i n the i nsi de
5
2
It
.
X
2
f 4X 4
2 d B ecause the middle t erms of the two diagonal column s com
po s e thi s in s ide 2 X 2 square their end term s or the terms in the
fo ur c o rner cells of the 4 X 4 s quare must a lso equal 5 o r :
square o
a
to
.
.
,
,
,
I 38
E
MA TI CA L S T U D Y
or
MAGI C SQUARES
.
in the upper left hand
hich may be represented by
orner a number repre s ented
the lower row we will write
a
and
h
en
in
t
h
e
y
umber repr
x
by g
2 3 4,
-
,
umber s
square
r a num
ay
v,
now be
row s hould be
by law 4 but if
,
r been
,
and therefore
g
may
and
y
div ers e
ec t
in
t,
du cing
thus
s umma ti
l
s s ym
i s to
be per
the cell w h
occupied by a
mu
s
t
be
y
produce with a
y a n
t becau s e
will be g
,
,
,
.
I
x
+
v z
y
+
t
t h e next cell to the
y
a
the two in
'
le
led by a
(
a
x
)
H E MA T I C
S T UDY
M A G I C SQUAR E S
OF
.
I 39
proceed to show what num b ers may be a ss igned
0 15 used in F i g 2 3 5 to p r oduce a pe r fect 4 X 4
1 to 1 6
ai ning the
symbols mu s t e qual I and
must equal 1 and the other
antities being e x cluded
.
.
t be the
larges t,
an
b c g must form
In like manner
,
,
o ther,
at
that
b
i
i
n
e
u
a
t
t
q
we find
z
g
3
,
equal I
b c g ma
,
,
As g v
B y in s pection it is
thi s number to y
,
_
With these Ya
I 3 and
must e q ual
er y or t must eq u
or vice ver s a
to the symbols F ig
n F ig 2 3 6
al
.
,
.
.
.
4
.
A MA T H E M A T I CAL S T U D Y
ST
UDY
OF T
OF
M AGI C SQUARES
[ W
H E P O S S I B L E N UM BE R
M A G I C S Q UA R E S
.
O F V A RI ATI
N S IN
Q
.
K
ha s
been
18
on ]
‘
connect i on l th the 3 X
possible arrangement of n
itute a magic square
all larger squares may
ru cted
eir number o f diver s e
increa s ing in an
every increa s e in the s
the 4 X 4
s olve the problem
o f var
con s tructed with
S ho w hA n
.
c
omp
A s pr evi
fill e
which is equal to a
be comp leted by the
“
fore be termed a ba
n s by which a ba s ic
f four
1 42
A
M AT H
OF
out er cell s contain a
tain b and c and t
and
Cla s se s
eq u ent
II, I I
in
w n
,
v,
MAGI C SQ UAR E S
.
and the two inner cell s con
Fig
.
2 39
.
but one genus ea
ix diff erent types To
h genu s will yield we
s quare
a 4 X 4 square
row
.
,
.
which the
of
Class I G enu s A
see that the s e cell s
cho i c e of writing
g the right hand cell
io u sly given and but
content s of eac h cell
as p rev
.
,
-
,
values assigned to the el e
I , 2,
S
3, 4
e cla ss and
16
Fo r
.
example
from u s ing
central cell s
ing the square we
r ro w
M ultiplying
,
,
.
mX
4 X 4X
fore cl ea r that Class
2
I,
squares
hows a square in which the
A will yield
5 12
po s
.
pr
oduce Class I G enus
lower ro w it is found that the
mu s t be u s ed and there are three
ic
(I )
2
( )
(3 )
o f elements are
n filling the ce ntral
y)
(
a
,
,
ro w
su
fits viz
,
.
,
MA T H E MA T I CAL S T UDY
OF
MA G I C SQUARES
"
.
ever will be found impossible leaving only ( 1
Choosi ng ( 1 ) it will b e seen that there ar
a
ay be located in either the right or 1
;
"
S imilarly i f (2 ) is chosen b
H ence in say the right h
,
m
,
-
,
,
as Shown i
F ig
.
24 1 ,
pairs of elements
fix ed It therefore
ere is a choice of
w
i S a choice of 4
Fo r
o ic e of 4 and for the
lying these choices together
a
ur
is u s ed t
follows
’
re m ainder 0
for the first
16
F0
cell
the central cel
lower ro w the r e is
we have
.
.
.
1 0 24
.
which is the possi
Writing a b
we find that the
to fill the central
S of Class I G enus B
C as given in F ig 2 39
v ) must be ii sed
B
e
.
,
.
,
5 of (a
x
lower row
—
.
and
a
x
+ g
z
b+
+
:
t
‘
v
c
+y
which
p
hese p a i rs however w
and since
two cells
S ix
teen
'
a irs
be made all equ
und unav ai
,
the basic row there is
of the same row there is no
the complements of the first
as i c row there is a choice of 4
,
,
A M A T H E MA T I CA L S T UDY
1 44
which is the po ss ible
P roceeding no w
in F ig 2 42 It
.
row
OF
M A G I C SQUAR E S
.
o f C la ss I G enu s C
ba s ic row may be formed a s
neither a nor g can
but as equivalents of
ati o n s
.
of
.
,
(g + y )
we may
W
of the two couples
(
b
( +<
6
y)
b
(
71
)
may be
F ig
Fo r
roup
Class
.
X
6
X
basic
2
II
X
.
243
.
e two inner cells there i s a
we
4
co ns tituted
ro w
ible to construct a ma
hitherto follo
constructed 0
m
bt actually so on ac
n
lia r relationship between the numbers I 2 3 4 and 0
Clas s I II has for the fi r s t cell of the basic ro w a choi
i
see n i n gl y
i perf ec t
“
,
a re
,
,
,
,
1 46
A
M A T HE M A T I C AL S T UDY
M AGI C SQUAR E S
OF
S ummarizing the preceding results
-
it
-
.
will be seen that there
re in
I G
Class
,
enus A
,
B
,
C
,
I
I
,
u s at lea st
5
12
va r ieties
1 0 24
68
7
2
43 5
s q uare s
the
X
5
5
a s the princi
sq uare s also enter
sq uare s
1
.
may th ere for
be formed by the
the fiv e qualitie
In order that
1 to 2 5 inclu s ive ;
re s pective s ymbols
It
nted
5
component s may
ies in one group
that
of the numbers
ed to the
may
2
C
d:
£
O th er
2
4
might be u s ed
eneral
p
urposes
g
o f thi s sq uare will indi
arge number of
choice
5
o rego in
e probabl y
.
r th e second cel
for the fi ft h there i s
N
ble comb i nat i on Tef t
.
ic
o ic e,
there
have at t
square
.
row may
A MA T H E MA T I C AL S T U DY
OF
MAGI C SQUARES
.
‘
complete the sq uare there are at least four available
resulting squa r es may be designated as Classes I I I I
,
and
,
,
“
,
”
by writing the symbol s of the a
columns acro ss th e s q uare in one
“ ”
and the symbol s of the x
.
,
O PPO S IIC
'
“
dl refiti fi fi és
'
by mak ing
”
move s
Fo r
cted
S clas
h
t
g
’
s
.
F is ;
basic
Fig
’
s
.
25 3
.
ro w
O ne element
p ) run s diagonally while t
m o ves
element
.
cluded in Class IV a r e irregular
of 5 X 5 squares prob ab ly amount
ng to
the figures herein given the number of v
tion s
OF
MAGI C
increa s es
conside
N OT
ES
U MBE R S E RI E S U S E D IN T H E
M A G I C S Q UA R E S
ON N
CO N S T R
U CT ION
OF
.
has long been k nown that ma gic s q uare s may be con s tructed
tical
from a serie s of number s which do not progre ss in arithm e
‘
order E x periment will Show however that any h ap h az ard s erie s
It
'
,
.
,
cannot be u sed for thi s purpo s e but that a de fi nite order o f s e q uence
i s nece ss ary which will entail certain relation s hip s betwee n di ff erent
member s of the s erie s It will there fore be our endeavor i n the
,
*
.
F ig
.
Fig
25 4
.
.
25 5
F ig
.
.
25 6
F ig
.
.
25 7
.
to determine t h e s e relation s hip s and e x pre ss t h e s ame
in de fi nite term s
L et F ig 2 5 4 repre s ent a magi c s q uare o f 4 X 4 By rul e N o 4
“
in the New A naly sis found o n page 1 3 7 it i s seen that the s um o f
p resent art ic le
.
.
.
.
.
”
the terms i n
a ny
tw
o c o n tigu o us corner c el ls
the terms i n the tw o
mi ddle
c ells
i n the
o
equ al
is
pp osi te
to the s um o f
o u tsi de co l u
mn
”
.
and it there fore follow s that
a
d In other word s the s e fou r q uantitie s form a group
v z s
wit h t h e interrelation s hip as Sh own B y the s ame rule (N o 4) it
i s al s o s een that a
p and h ence al s o a l p t giv
t
l
There fore in
,
F ig
.
2 5 4, a
d
s,
v
/
,
.
.
.
—
,
,
,
ing anoth er group o f four number s having the s ame form of inter
“ ”
relation s hip and s ince bot h group s have a a s an initial number
it i s evident that the increment u s ed in one o f t h e s e group s mu s t
be di ff erent from that u sed in the other or duplicate numbers would
,
,
,
4
A M A T HE MA T I CAL S T UDY
90
OF
MAGI C SQUARES
.
numbers I to 1 6 from which the diverse squares F igs 2 62 and 26 3
are formed by the u sual method of con s truction
F ig 2 6 1 shows the arrangement o f an irregular s eries o f sixteen
number s which when placed in the order of magnitude run as
follow s
.
.
-
.
,
,
2
-
79
-
-
1 0 - 1 1 - 1 2 - 1 4- 1 5 - 1 7- 1 8 - 1 9- 2 0 — 2 1 -26 - 3 0 -
33
The magic square formed from this se r ies i s given in F ig 264
In the s tudy of the s e number s erie s the natural que s tion pre s ents
it s el f :Can as many div erse sq uares b e f o rmed fro m o ne s eri es as
fro m ano ther ? Thi s q ue s tion Opens up a w ide and but little ex
l
o
r
ed region a s to the diver s e con s titution o f m agic sq u are s
This
p
idea can there fore be merely touched upon in the p resent article
.
,
.
.
,
e x ample s o f s everal di ff erent plans of construction being given in
illu s tration and t h e fi eld left at pre s ent to other e x p lorers
.
W
F ig
.
262
.
F ig
263
.
F ig
.
.
264
F ig
.
.
265
.
Three example s will be given th fi r
st being what i s s ometime s
”
sq uare or one in which any two number s that
are geometr i cally oppo s ite and equidi s tant from the center o f the
sq uare will be e q ual in summation to any other pair o f number s so
s ituated The s econd ga blew ill be a s quare in which the s um
of every
o f the four sub s quare s of 2 X 2 i s e q ual and
th e t h ird
will b e a sq uare in w h ich the pair s o f number s
h aving s imilar s ummation s are arranged s ymmetricall y in relation
to a perpendicular line through the center o f the square F igs 26 2
26 3 an d 26 5 illust r at e these three exa m ples of squares:
e
,
m
.
“
-
,
.
-3
.
v
,
a
R eturning no w to the question previou sly giv en but little re
flectio n i s re q uired to sh ow t h at it mu s t be an s wered in the negative
for the following reasons Fig 2 64 repre s ents a magic square having
no s pecial qualitie s excepting that the column s horizontal p erp en
,
.
.
.
,
,
A
NI A T H E
MA TI CA L S T UDY
OF
W
MA G I C SQUAR E S
.
and diagonal all have the s ame s ummation viz 66 H ence
a ny se r ies of nu m b ers that can be arranged as shown in F i g 2 5 8 will
yield magic s quares as outlined B ut that it s hall also produce
square s having the qualification s t h at are termed
or may not be the case accordingly a s the s erie s may or may not be
capable o f s till fu rther a r rangement
R eferring to F ig 2 5 4 i f we a m end our definition by no w call
dicular
,
.
,
.
.
.
.
.
F ig
.
ing it a
266
,
F ig
.
.
26 7
.
square we /s hall at once introduce the following
,
and i f we mak e our diagram of magic s quare producing numbers
con form to these new requirements the number o f group s will at
once be greatly curtailed
,
.
Fig
.
268
F ig
.
.
2 69
.
The multiplicity o f algebraical sign s necessary in our amended
diagram is so great that it can only be studied in detail the complete
diagram being a network of minu s and equality sign s
The result will therefore only be given here formulated in the
“
”
following laws which apply in large measure to all perfect s q uare s
I P erfect magic squares are made of as many s erie s or group s
of numbers as there are cell s in a column
,
.
,
.
.
.
'
M
"
?
l
A M A T H EM A T I CAL S T U DY
OF
MAGI C SQUARES
.
series or group is composed of a s many number s a s
there are groups
I I I The differences between any two adj oining number s of a
s eries must obtain between the corresponding numbers of all the
s erie s
IV The initial terms o f the s eries compose another s erie s a s
do the second third fourth terms and s o on
V The di ff erences between any adj oining number s o f the s e
s econdary s eries must also obtain between the corresponding term s
of all the seco ndary seri es
The foregoing rules may be illu strated by the s eries and perfect
square shown in F ig s 2 5 9 and 26 2
F ollowing and con s equent upon the foregoing interrelation s of
”
“
the s e numbers i s the remarkable q uality p o ss e ss ed by the perfect
magic sq uare producing serie s a s follow s
I f the entire s eries is written out in the order o f magnitude and
the di ff erences between the adj acent numbers are written below
the row o f di ff erence s will be found to be geometrically arranged
on each s ide o f the center a s will be s een in the following s erie s
taken from F ig 266
II
E ach
.
.
.
.
.
,
,
,
.
.
'
.
.
.
,
.
.
3
4
-
1 3 1 4 1 8 1 9 2 1 22 2 8 29
-
-
—
-
-
-
-
-
1
3
-
2
3
-
6
6
4
47
3 37
-
-
-
the above example the number 6 occupie s the center and t h e
other numbers are arranged in geometrical order on each side of it
“
”
It i s the belief o f the writer that thi s rule applie s to all perfect
s q uare s whether odd or even
The following example will su f fice to illu s trate the rule a s
applied to a 5 X 5 magic sq uare F ig 268 showing the s erie s and
Fig 26 9 the square
In
.
.
.
,
.
.
1
7 8
4
3
3
1
2
1
2
1
1
2
1
2
1
11
2
1
2
1
1
2
1
2
1
3
3
The diagram Shown in F ig 2 70 i s given to impress upon the
student the idea that a natural series of continuous num b ers may
.
be arranged in a great variety of di f erent magic square p roducing
f
s eries A perfect 9 X 9 sq uarer ma y be produced with any con
.
A MA T HEM A T I CAL S T U D Y
OF
MAGI C SQUARES
.
exactly the same plan and u sing the same b reak m o v es the varia
tions radical as they may appear to b e are only so b ecau se di ff erent
of which series it has
s eri es o f the same nu m bers are empl oyed
been shown there are at least twenty four
I f the student will tak e Fig 2 70 and fill in num b er values
“
making b (successively ) 3 9 and 2 7 he will ac q uire a clear
idea o f the part taken in magic squares by the series conception
The work o f determining the possi b le number o f perfect 9 X 9
,
,
,
,
,
-
.
,
.
,
”
,
,
,
.
magic squares m ay no w b e greatly si m plified for all elements are
thu s determined sav ing o ne i e the nu m b er of p o ss ible m o des o f
:
,
,
p ro gressi o n
.
.
,
.
F ig
2 71
.
.
I may be located in any of 80 cells and progress mav be made
in x ways and 2 4 variants may b e constructed in each case There
fore the possible nu m ber o f di ff erent 9 X 9 squares will be at least
.
,
,
80 X
24
X
.
r
1 920 x
.
A single example will serve to illus rate the possibilities Op en
t
to x the numerical value o f which will be left for the present
others to determine A s previousl y given let
,
.
,
for
L
A P
A MA T HE MA T I CAL S T U DY
OF
Then Fig 2 71 will represent a
arrange m ent o f sy m b ols given in F ig
MAGI C SQUARES
9 X 9
.
.
2 70
.
square based on the
.
right hand
-
and b etw ee
to the right
In
fa
h owever
.
is built up by t
,
e su
moves
ve
,
in the
f ol
co
th e number s 1
uch magic square
iff erent squares b y the
same b reak mo v es
,
rul e,
cells in 11
s to the ri
s eries :
etc
et
v e,
mm o n
may be
.
(1
in at
u c ing series thus givin g
method
,
.
3
L 8 F
0
0
.
"
C HA P T E R
MA GI C
SQ U A RE S
Vl
.
A N D P YT H A G O RE A N N U
.
M
B ER S
h av e c o mp il ed thi s di sco u rse whi c h asks
f o r y o u r co n id er ti o n and p ard o n n o t o n ly b e
t he m att r it s lf i s by n
ca u
m eans
sy to
b e h an dl ed b ut a l so b eca u se the d o c tri nes h erei n
c o nta i n ed a re so m wh at c o ntr a ry t o th o e h eld
by m o st o f the P l at o n i c p hil o sop h er s
P lutarch
I
,
a
s
'
se
e
e
o
ea
,
e
s
”
.
.
H E mysteriou s relat ionships o f numbers have attracted the
minds o f men in
al l
ages The many sided F ranklin whose 200 th
-
.
,
anniversary the philo s ophical s cienti fi c and literary world s have
recently celebrated u s ed to amu s e him s el f with the con s truction
of magic sq uare s and in hi s memoi rs has given an ex ample o f h i s
Sk ill in this direction by showing a very complicated square with
the comment that he believes the sa m e to b e the most magical magic
s q u a re yet const r ucted by any magician
That ma gic sq uares have had in centuries pa s t a deeper mean
ing for t h e mind s o f men than t h at of s imple math ematical curio s
we may in fer from the celebrated picture by A lbe rt D ii rer entitled
”
“
elancolia engraved in 1 5 1 4 T he s ymboli s m o f thi s engraving
ha s intere s ted to a mark ed degree almo s t every ob s erver T h e fi gure
niu s sitting li s tle ss and dej ected amid her un
of the brooding g e
c ompleted labor s the s cattered tool s the swaying balance the flow
ing s and s o f the gla s s and the magic Sq uare of 1 6 beneath the bell
the s e and other detail s reveal an attitude o f mind and a connection
of thought w h ich the great artist never expre s sed in word s but
le ft for eve ry beholder to interp ret for him s el f
Th e di s covery o f the arran gement o f number s in the form o f
magic di agram s wa s undoubtedly k nown to the ancient E gyptian s
,
,
,
,
,
.
M
.
,
.
,
,
,
,
,
—
,
,
.
MAGI C SQ UAR E S A ND PY T H AGOR E AN N U M BERS
.
of the order and the story is told o f an unworthy disciple who re
vealed the secret o f the construction of the dodecahedron inscribed
within a s phere thi s being a symbol of the universe
A mong the best e x po s itions o f the P ythagorean philosophy are
“
”
section s o f the Tim aeu s and R epublic of P lato The s e dia
l o gu es were written after P lato s return from M agna G raec ia where
from contact with A rchyta s of Tarentum and other philo s opher s
he imbibed so much of the Italian s chool that hi s whole s ystem of
philosophy became permeated w ith P ythagorean idea s It i s even
suggested that he incorporated into these dialogues part s of the
lo s t writing s o f P hilolau s who s e works he i s k nown to have pur
cha s ed
N o portion s o f the dialogue s named have been more
puzzling to commentator s than the vagu e re ferences to different
number s s uch a s the number 729 which i s cho s en to e x pre ss the
di ff erence between the k ingly man and the tyrant or t h e so called
“
number of the S tate in th e R epublic or the harmonic number of
“
the s oul in the Tim aeu s of which P lutarch s aid that it would be
an endle ss toil to recite the contention s and di s putes that h ave from
hence arisen among hi s interpreter s
E ither our text of these pa s
s age s i s corrupt o r P lato i s very ob s cure th rowing out indirect hint s
which would be intelligible only to those previou s ly informed P lato
“
“
s tates him s el f in the P h aedru s that all writing s are to be regarded
”
purely a s a means of recollection for him who already k nows and
he therefore probably wrote more for the benefit o f hi s hearer s
,
.
”
.
’
,
,
.
,
.
,
,
-
,
,
”
’
,
.
”
,
,
,
than for distant posterity
It i s upon the principle of a magic square that I wish to inter
“
”
pret the celebrated p assage in the R epublic referring to the number
proceeding
from
this
to
a
discu
s
sion
of
certain
other
numbers
2
79
o f peculiar s igni fi cance in the P ythagorean s ystem
e
f
f
orts
in
y
thi s directio n are to be regarded as purely fanci ful ; the s ame may be
s aid however of the maj ority o f other methods o f interpretation
“
The passage from the R epublic referred to (B ook IX 5 87 8
.
,
.
M
.
,
,
”
-
,
,
tran
s
lation
read
s
a
s
follow
s
)
J
“
A nd i f a person tells the measure o f the interval
S o cra tes
whic h separate s the king from the tyrant in truth o f plea s ure he
’
o w ett s
.
,
MAGI C SQUAR E S AND PY T H A G OREAN N U M BERS
.
will find him when the multiplication is completed living 729 times
more plea s antly and the tyrant more painfully by thi s s ame interval
“
What a wonderful calculation
G la u c o n
S o crates
Yet a true calculation and a number which closely
concern s human li fe if human li fe is concerned with days and night s
and month s and years
The number 729 i s found to be of great importance all through
P lutarch state s that thi s was the number
the P ythagorean s y s tem
belonging to the s un j ust as 243 wa s a s cribed to V enu s 8 1 to M er
cu ry 2 7 to the moon 9 to the earth and 3 to A ntichthon (the earth
Oppo s ite to ours ) The s e and many similar number s were derived
from one of the progres s ions of the Tetracty s
and
The fi gures of the above p roportions were combined by
P lato into one series I 2 3 4 9 8 2 7 (Ti m aeus
P lutarch
“
in his P rocreation of the S oul which is simply a commentary
“
has rep
upon P lato s Tim aeus
resented the numbers in the form
of a tr iangle ; the interior num
ber s 5 1 3 and 3 5 representing
,
”
,
.
”
.
.
.
,
.
.
,
,
,
,
,
.
,
,
,
,
,
,
,
,
.
,
,
’
,
,
,
,
,
the s ums of the Opposite pairs
were also of great importance
T h e deep s ignificance of the
Tetractys in the system O f Py
thago ra s may be inferred from
a fragment O f an oath contained
”
“
i n the G olden V erses
,
.
Fig
.
272
.
.
Na i p a T OV dp ér ep o v tpv xqi w a pa w a r er pa x rbv
IZcbua r 3xo v a a v
nia ew s fi
Ha yav, devafo v d
'
'
.
.
’
'
Yea
,
.
by our Tetractys which giveth the soul the fount
and
”
s ource of ever flowing nature !
O dd numbers were e s pecially favored by the P ythagorean s
and of the s e certain ones such as 3 and its higher powers were
considered to have a higher significance than others and in this way
perh ap s arose the di s tinction between e x pres s ible and ine x pre s sible
or ineffable numbers (dpt ept oi 5117 0 2 m 2 dfifimoi ) N umbers whi ch
e x pressed some astronomical fact also held high places of honor
,
,
.
.
,
I SO
$ 1
“
60
M AGI C SQUARES
AND
PY T H AGOREA N N U M BERS
.
as may be seen from a statement by P lutarch (l o c ci t ) in reference
“
to the Tetracty s
N ow the final member O f the s erie s which is
2 7 has thi s peculiarity that it is equal to th
e sum of the preceding
.
.
,
.
,
,
number s
it also represents the periodical num
ber o f day s in whic h the moon complete s her monthly cour s e ; t h e
”
P ythagorean s have made it the tone O f all their harmonic interval s
.
Fig
.
273
.
Thi s p a ss age indicate s su ffi ciently the s upreme importanc e of
t h e number 2 7
.
If
we construct a magic sq uare
upon the plan of a
to 729 fir s t in numerical
2 7X 2 7
che k er board arranging the numbers 1
order then s hi fting the 9 large s t sq uare s (9 X 9 ) into the po s ition s
indicated in the familiar 3 X 3 square repeating the proce ss with
c
-
—
,
,
I FV
”
’
“
16 2
MAGI C SQUAR E S
PY T H A G OR E A N N U M BERS
AND
.
number 2 7 I believe we may arrive at a s good a solution of the
problem as any that I have seen suggested The following inter
ff
r
i
o
n
o
f
the
ree
k
term
s
i
s
O
ered
e
t
a
t
G
p
,
.
.
i f iz
a u c (f w dt w a t 1 s
/
dvva o r cvéueva i
'
a r
Ka i
t he
s
qu
f th n u m
ti m i t
t
d b y t h i th
t t rm ( f t h
t c ty )
b er
in
p si ; arro ar daet g
flb
o
27
o
oco vv r w v
avo
re
Ka i
Ka i
avf évrw v
o
(
1
z
+ i
the
es
e s
216
¢6w or vw v
ti o
e sa
‘
i
(
e
.
d
or
i g
e c reas n
:
2:
1
.
or
ig
i gl
t o t h e a s c en d n
of
’
i r vra rrp o ar
yyopa na i
a
fl
bnr a m ak es
rr
p
r
i
(
.
t
s
a
ib l
an
su
ur b l
e ss
p ar
t he t r
the
m en s
va v
cihhr la arré n
b
g
p
p
'
i ),
E érrtrp i r o g m i d/un
( v
thi
e
.
8 , 4,
2, 1
g
3 , g.
,
)
27
co m
al l
i ts
su
.
is
e . 2460
s su
i
a nd ex
e
in
e
f
I t m a y a l s o re er
a n d des c en d n g fi u res
m
ea s
i
i i ib l
ly d
6,
'
e ta
hether i nc rea s
w
i ng
ye t
e
ea r
1 0 , 12
in
m
v s
c r ea se
d
e tc
e
.
by
1
,
2,
d
c
d
/
t p ig a
m=
)
by
d di ng 5
m u lti p li d b y 3
X
1
%
=
a nd a
o v vyei g
i
is
bf ndeig
2 46 0
3 , 4. 5 ,
2 46 0
i re i rr
2 1 87
e
f u ti m
wh l
i
f n u m b r u n li k
i ng t h m
b
r
2 7=
e
s
o
V
X
o e s er es
ductofi v rw i }
ica i
s
ra
a nd
G al
p
e
2
,
r ce
c rea se
Te
ér ra pa g dé bpong ha
s ro o
es
firs
r
e
a re o
3 2 80
3 28 5
3 28 5 x 3 =
e
9 85 5
This s olution o f the problem 98 5 5 it will be noted bring s u s
again but by a di ff erent route to the magic number of ou r large
square The s econd part of the passage contains a description of
the number by which the above calculation may be veri fi ed
,
,
,
,
.
.
di mdpla ov ia g
rra p
éx er a c
t
( he
tw
7
75
1 7 1) 1
1)
20 77» i o émc,
’
o
o ne
u mb
er
ar
o
yi ld
h m ni p t
f wh i h i
n
)
c
o
i
p p
z
r fccc
dé,
a
1 00
e
e o
r r rr o
s
a re
e
e
a
s
ar s ,
c
qu
m u lti p li d b y
th
t h r h n id
qu l t th q u
n d th
t h r Ob l ng
s
’
e
a
as
e s
o
e o
o
e
e s
are
o
9X
e
1 0 0=
90 0
MA G I C SQUAR E S AND PY T HA G OR E A N N U M B E RS
.
The remainder of the pa s sage describes the len gth of the
long which we have shown above to be 298 5
a im érr b
o rd
é/ca r c w p év ch
/
di a uérp w v «curl 660 9,
‘
(the
‘
l
) is
o b o ng
ti m
t
i
a
e
r ow
l
deo/u évw v évbg éicéa
11 3 1)
0 17
xp
4
cha
in
»
.
dé dvei v,
.
5
ress
a nd
a nd 2 o f
1 00
.
i g id
h av n
ess o f o n e ea c
e
,
e
.
a
o
e
s
a go n al s o f
(
1 00
id f
h vi ng di
ngl
the
es
rec a
Oh
ib l
e
h
of
t
p ar
f 3
es o
s
s,
X 3=
300
and
the
i
.
e
.
5
the i n ex p res s i
2 85
300
b le
fl
éx a r bv di:xb
wv r
l
u
p
p cédo g
s 1 00
of
ti m
es
the c
ub
e
3
P lato states that the number o f the S tate
represent s a geo
m etrical figure which has control Over the good and evil O f births
F o r when you r guardians are ignorant o f the right sea s on s and unite
b ride and bridegroom out O f due time the children will not be
goodly and happy
The number 98 5 5 expressing a period of
27 years might thus repre s ent the dividin g line between the age s
when men and women s hould begin to bear children to the S tate
“
20 2 7 year s for women 2 7 34 years for men
S
ee
also
R
epublic
(
“
B ook V
A ristotle in hi s P olitic s (V 1 2 8 ) s ay s in
reference to the number of the S tate that when the progre s sion of
number i s increased by
and 5 is added 2 harmonies are produced
giving a s olid diagram Thi s a s may be seen from our analysi s of
the fi r s t part of the pa s sage may have reference to the number
2
2
which
being
repre
s
ented
by
6
8
X
3
3 5 may be s aid to have the
3 5
dimension s of a solid
“
In his R efl ections on M agic S quares D r Carus gives some
very strik ing e x amples Of the relationship between magic squares
and the mu s ical fi gure s of C hladni . I would li k e to touch before
concluding upon a clo s ely related s ubj ect and Show ce rtain connec
tions which exi s t b etween the magic s quare which we have con
structed and the numbers of the Pythagorean harmonic s cale T his
scale had however more than a mu s ical Signi fi cance among the
.
,
’
.
,
,
,
-
-
,
.
,
,
,
.
,
.
,
,
,
,
,
.
.
,
.
,
,
,
n
i
MAGI C SQUARES A N D PY T H AGOREA N N U M BERS
$6 4.
.
G reek philosophers ; it was e x tended to comprehend the harmony
o f planetary movements and above all else to represent the manner
“
”
in which the s oul of the universe wa s composed It is especially
”
in the latter s ense that P lato employs the s cale in hi s Tim aeu s
“
In a treatise by T imaeu s the L ocrian upon the
S oul of the
“
World and N ature we find t h e following pa ss age : No w all th e s e
proportion s are combined harmonically according to number s which
proportion s the demiurge ha s divided according to a scale scien
tific al ly so that a perso n is not ignorant o f what t h ing s and by what
means the soul is combined ; w h ich the deity h a s no t ran k ed after
the s ubstance o f the body
but he made it Older by ta k ing the
first of u ni ti es which i s 3 84 N ow o f these the fi r s t being a ss umed
it i s ea s y to reckon the double and triple ; and all the term s with
their complements and eight s mu s t amount to
T
ran
s
(
lation by B u rge )
P lato s account of t h e combination o f th e s oul i s very s i milar
to the above though h e s eem s to have s elected 1 92
for t h e
fi r s t number P lutarch in hi s commentary make s no mention o f
4
Tim aeu s but s tates that Cranto r wa s the fir s t to s elect 3 84 for t h e
2
reason that it repre s ented the p roduct o f 8 X 6 and i s the lowe s t
number which can be tak en for the increa s e by eighth s without
leaving fraction s A nother very pos s ible rea s on which I have not
s een mentioned i s that 3 84 i s the harmonic ratio of
or
a number which expre ss e s very closely the day s o f the year
.
.
.
,
,
,
.
.
,
.
’
,
,
.
,
,
,
,
.
,
.
:
:
8
3 4
2 43 2 5 6
.
The proportion
was employed by the Pyth
5
which two une q ual s emitone s o f the
ago rean s to mark the ratio
harmonic scale bear to one another
B atteux has calculated the 3 6 term s o f the P ythagorean s cale
starting with 3 84 and his series mu st be Considered correct for it
ful fi l s the conditions speci fi ed by Tim aeu s the number s all footing
'
.
,
W
—
,
u
Cranto r liv ed n ea rly 1 00 y ears a ft er T i m ae s the Lo c ri an
”
“
whi ch b ears the
S o l o f the
o rld an d N at r e,
p o n the
p ro b ab ly b el o ng s to a m c h l at er p er i o d
u
u
u
f“u rth r r f r
P l u t a rc h s P ro c re ti o n
5
Fo r
u
’
a
Of
The
treati se
l att er s
’
name
.
e e en c es
e
.
thi
So ul
to
t he
,
s
”
ra
ti
18
o
.
see
P
l at o s
’
T i m aeu s
,
§
3 6,
and
MAGI C SQUAR E S A N D PY T HA G OR E AN N U M B E RS
s ome re spect s p urely accidental being due to the
,
.
i ntrinsi c harmony
and
therefore
not
implying
a
knowledge
by
the
ancient
s
f
o f magic s quares a s we no w k now them The harmonic arrangement
o
n u mb ers
.
by the G reek s O f number s in geometrical form s both plane and
s olid may however be accepted and P lato s de s cription s o f variou s
’
,
,
,
'
number s ob s cu re and meaningle ss a s t h ey were to s ucceeding gen
eratio ns m ay have been ea s ily compre h ended by hi s hearer s w h en
6
illu s trated by a math ematical diagram or model
D iff erence s between the methods O f notation in ancient and
modern times have nece ss arily p roduced di f ference s i n the c o ncep
tion o f numerical relation s T h e expre ss ion of numbers among t h e
G ree ks by letter s o f the alphabet wa s what led to the idea that every
name must have a numerical attribute but the connection of the
letter s of t h e nam e w a s in many ca s e s lo s t the number being re
garded a s a pure attribute of the Obj ect it s el f A s imilar confu s ion
of symbol s aro s e in the repre s entation o f variou s concept s by geo
metrical f o rm s such as the fi ve letters o f YI EIA and th e symb oliza
tion of health by the P ythagorean s under the form O f the pentalpha
o r fiv e pointed star
,
.
.
,
i
,
.
'
‘
,
-
.
was the great de fect o f the G ree k s chool s t h at in their searc h
for truth methods o f experimental research were not cultivated
”
“
P lato in hi s R ep u
blic (B ook V I I § S3 O S3 I ) ridicule s th e em
It
.
,
~
,
who sought k nowledge by s tudying the s tar s or by com
paring the s ounds o f mu s ical strings and insi s t s that no value i s
“
L et the h eaven s
to be placed upon the testimony o f the s en s es
”
i s h i s con s tant advice
alone and train the intellect
I f the example s set by P ythagora s in acou s tic s and by A rchi
mede s in statics had been generally followed by the G reek philo s
o p hers ou r knowledge O f natural phenomena might have been a d
B ut a s it happened there came to prevail
v anced a thousand year s
but one idea intensi fi ed by both P lato and A ri s totle and handed
down through the schola s tics even to the present time that k nowl
i
i
i
r
c
s
t
s
,
p
,
.
.
,
.
,
,
d escri p ti o n“ o f the n u mb er o f the S tate in the R p ub li c and th at
u l in the T“i maeu s r n d r su ch a mo d e o f rep r nt ti n al mo st
n ec essa ry
P l u t a rc h ( P ro c reati o n o f S o u l
§ 1 2 ) g iv“es an ill u strati o n o f an
h arm ni c di agram 5 X 7 co nt i ni ng 3 5 sma ll squ ar s whi ch co mp reh en d s in
nco rd
f m u si c
its s u b divi si o n s al l the p ro p o rti o ns o f the fi r st
°
The
o f the S o
e
”
e
e
ese
a
”
,
.
o
e
a
co
s o
”
.
o
MAGI C SQUAR E S A N D PY T HA G OR E A N N U M BERS
.
edge was to be sought for only from within H ence came the flood
of idle Speculations which characterized the later Pythagorean and
P latonic schools and which eventually undermined the s tructure of
ancient philosophy B ut beneath the abstraction s of these s chools
.
.
one can di s cover a s trong undercurrent of truth M any Pythago
reans understood by number that which i s now termed natural law
S uch undoubtedly was the meaning O f P hilolau s when he wrote
“
N umber i s the bond of the eternal continuance Of things
a senti
ment which the modern physicist could not expre s s more fittingly
“
A s the first study of importance for the youth of his R epublic
P lato selected the science of numbers ; he cho s e as the s econd ge
o m et ry and as the third a s tronomy but the point which he emp ha
sized above all was that these and all other science s s hould be
“
studied in their mutual relation s hips that we m ay learn the nature
“
”
”
“
of the bond which unite s them
he s tate s will
F o r only then
a pursuit of them have a value for ou r obj ect and the labor w h ich
might otherwi s e prove fruitles s be well be s towed
N oble utter
ance ! and how much greater need O f this at the pre s ent day with
our comple x ity of science s and tendency toward s narrow speciali
.
.
,
.
,
'
.
,
,
,
,
.
,
zatio n
.
the spirit o f the great ma s ter whom we have j ust quoted
we may compare the physical univer s e to an i mmense magic s q uare
I solated investigator s in di ff erent areas have di s covered here and
there a f ew s eemingly re s tricted laws and paying no regard to the
territo ry beyond their confine s are as yet obliviou s of the great
pervading and unifying B ond which connects the s cattered parts
and binds them into one harmoniou s syste m O mar the astron
omer poet may have had such a thought in mind when he wrote
In
.
,
,
.
,
-
,
,
“
lue
Co u ld yo u b ut fin d it to th tr su r h o u
A n d p era dv nt u r to th M st er to o ;
Yes ;
i gl A lif w r
an d a s n
W
e
e
—
e
e
h o s ecret p r
R u nn i ng q u i k sil
esen c e,
e s
c
t he
e e
v erl ik e
e
c
e-
ea
se
a
thro u gh r ti o n v i n s
y o u r p ai n s ; tc
el u d
c
ea
’
s
e
”
es
“
e
.
When P lato s advice is followed and the mutual relationships
between our sciences are understood we may perchance find this
clue and having found it be surpri s ed to discover as great a S im
’
”
,
“
m?
1
68
MAGI C SQUARES
”
A ND
PY T H AGOREAN N U M BERS
.
underly i ng the whole fabric O f natural phenomena a s e x i s ts
in the construction of a magic square
i
i
l
c
t
y
p
.
MR
.
B RO
W
N
ES
’
S
C
Q UA R E A N D
L
USUS
NU
.
A B
.
.
MER ORU M
.
The 27 X 2 7 square Of M r C A B rowne Jr is interesting
because in a ddito n to its arith m etical qualities commonly possessed
by m agic squares it represents some ulterior significance of our
.
.
.
.
,
,
,
calendar system referring to the days o f the month a s well as the
days O f the year and cycle s o f years It is wonderful and at fi rst
s ight my s ti fying to observe how the course o f nature reflects even
to intricate details the intrinsic harmony of mathematical relation s ;
and yet when we consider that natu re and pure thought are simply
the result o f conditions first laid down and then consistently carried
out in definite functions O f a distinct and stable character w e will
no longer be puzzled but understand why science is pos s ible why
man s rea s on contain s the clue to many problems O f nature and
generally speaking why reason with all its wealth o f a p ri o ri
thought s can develop at all in a world that at first s ight seems to be
a mere chaos of particular facts The purely formal relation s O f
mathematics materially con s idered mere nonentities consti tute the
bond o f union which encompasses the universe stars as well a s
mote s the motions o f the M ilky Way not less than the minute com
b inatio ns o f chemical atoms and also the construction o f pure
thought in man s mind
M r B rowne s square is o f great in tere s t to G reek scho l ar s b e
cause it throws light o n an obscu r e passage in P lato s R epublic t e
ferring to a magic square the center of which is 36 5 the number of
.
,
,
,
,
’
,
‘
.
,
,
,
,
,
’
.
’
.
’
,
,
day s in a year
The con s truction o f M r B rowne s square i s based upon the
s imple s t s quare Of O dd numbers which is 3 X 3 B ut it becomes
somew h at complicated by being extended to three in the third power
which is 2 7 O dd magic squares as we have seen are built up
.
’
.
.
.
,
,
by a p rogression in staircase fashion but since those numbers
that fall outside the square have to be transferred to their cor
,
f
Lo
M A G I C SQUAR E S A N D PY T HAGOREA N N U M BERS
w
.
immanent harmony of numbers is natu rally impre ss ed by it s ap
arent
occult
power
and
so
it
happens
that
they
were
deemed
super
p
natural and have been called magic
They s eem to be the product
o f some secret intelligence and to contain a message o f ulterior
meaning B ut i f we have the key to their regularity we know that
the harmony that pervades them is neces s ary and intrinsic
N or is the regularity limited to magic s quares There are
other number combinations which exhibit surprising qualitie s and
I will here select a few striking cases
,
,
’
.
.
.
.
,
.
we write down all the nine figure s in a s cending and descend
ing order we have a number which i s equal to the square o f a num
ber consisting o f the figu re 9 repeated 9 times divided by the s um
O f an ascending and descending series of all the fi gu res thu s :
If
,
, 23 45 6 78 9 876 5 43 2 ,
The secret of this mysterious coincidence is that 1 1 X 1 1 : 1 2 1 ;
1 1 1 X 1 1 1 : 1 2 32 1 ;
1 11 1 X 111 1
1 2 3 43 2 1
etc and a sum o f an
=
.
,
,
ascending and descending series which starts with 1 i s always
I +2+
equal to the square o f its highest number
etc which we will illu strate by o ne more
.
.
,
in s tance o f the sam e kind as follow s
,
There are more instance s of numerical regularities
A ll number s con s isting of s ix e q ual figures are divi s ible by 7
and also a s a matter o f course by 3 and I I as indicated in the
following list
.
,
,
,
,
1 746
22222 2
I
6
47 9
3 333 3 3
::o
2
;
444444 7
4
3 9
::
6
:
5 5 5 5 5 5 7 793 s
=7=
=
66666 6 :
7
2
zz
z
777777 7
888888 :
2
8
95 3
1 1 1 1 1 1
MA G I C SQUAR E S AND PY T HA G OREA N N U M B E RS
F inally
nu m ero ru m
we will O ff er two more strange
.
c o i néi denc es
of a
l us u s
.
9+
1=
1
X 9+
2=
11
OX
1
+3
1 23 X 9+ 4
1 2 3 4X 9 + 5
1 2 3 45 X 9 + 6
1 2 3 45 6 X +
9 7
1 2 3 45 6 7X 9 + 8:
1 2 3 45 6 78 X 9 +
9
1 23 45 6789 X 9 +
12X 9
=
11
=
1
=
=
=
=
1 2 34X 8
1 2 3 45 6 789 X
I
I11
1 I I 11
1 I I 1 1 1
1 1 1 1 I 1 1
1 I I 1 1 I I I
1 1 1 1 1 1 I 11
1 I 1 1 I I111
.
+ 4 98 76
=
8 + 9= 9 876 5 43 2 1
wonder that s uch strange regularities impress the human
mind A man who k nows only the e x ternality o f these result s will
naturally be inclined toward occultism The world o f numbers as
much as the actual univer s e is full o f regularities which can be
reduced to definite rule s and law s giving u s a key that will unlock
their mysteries and enable u s to predict certain re s ults under defi
nite conditions H ere is the k ey to the significance o f the a p ri ori
athematics i s a pu rely mental con s truction but its compo
O n the contr ary it i s tracing the results of
sitio n i s not arbitrary
our own doings and tak ing the con s e q uence s of the condition s we
have created Though the scope o f our imagination with all its
po ss ibilitie s be in fi nite the re s ult s O f our con s truction are defi nitely
determined as soon a s we have laid their foundation and the actual
world i s simply one realization of the infinite potentialitie s of being
Its regularities can be unraveled as s urely a s the harmonic relation s
of a magic square
No
.
.
,
M
.
.
,
.
.
,
,
.
.
MAGI C SQUARES A N D PY T H AGOREAN N U M BERS
.
,
are j ust a s much determined a s our th ough t s
can but gain a clue to their formation we can s olve the p
their nature and are enabled to predict their occurrence 2
F acts
,
,
times even to adapt them to our own need s and purpo s es
.
A s tudy o f magi c sq uare s may have no practical a]
but an acquaintance with them will certainly prove
were merely to gain an in s ight into the fabric o f regul arit
k ind
.
SO ME
1 74
intere s ting
A n ci ent
C
URIOUS MAGI C SQU AR E S
6 X 6
a nd
NAT
A
eo
.
work entitled G am es
is here rep r oduced in
t w o co r ner diagonals
n sp o siti o n of some of
square i s illustrate
Ori enta l b y G
ION S
.
It
will be seen
o f thi s square do n o t su m
c o rrec t ed a s shthis i m perfection
o w n in F i gt 2 79:
ther transpositions are also possible which will e ff ect the same
F ig
.
2 78
.
—
—
=
—
—
,
'
F ig
.
278
.
result The p eculi arity O f this square consis ts in its b eing divided
into nine 2 X 2 squares in each o f the fou r subdiv i s i ons of which
the nu m bers follow in arithmetical sequence and the 2 X 2 squares
a r e arranged in the orde r o f a 3 X 3 magi c square accordin g to
.
,
,
the p r ogressive value Of the nu m bers 1 to 3 6 The construction
o f this 6 X 6 square i s regular only in relation to the total s of the
2 X 2 squares as shown in F i g 2 80
.
.
,
F ig
F ig
.
.
280
.
is a remarkable 8 X 8 square which is given on page
0
0
O
f
the
above
m
entioned
book
and
which
is
presented
by
M
r
3
“
F a lk en er as the most perfect m agic squa r e O f 8 X 8 that can be
constructed
S o m e o f its p r operties a r e as follows ;
5
1
The whole is a magic square o f 8 X 8
2
E ach qua r te r is a
.
28 1
W
.
,
.
.
.
Pub li h d by L
s
e
o n g m an s
Gr
een
C0
.
,
Lo n
d
.
o n an d
é
N ew
Y rk
o
,
1 892
.
SOM E
C
URIOUS MA G I C SQUAR E S AND
C
OM B I NA T ION S
I
.
7S
2
2
The
sixteen
sub
squa
r
es
have
a
constant
su
m
mation
X
3
of 1 30
4 E ach quarte r contains fou r 3 X 3 squa r es the corne r nu m b ers
of which su m 1 3 0
A
ny
squa
r
e
which
is
contained
within
the
8
8
square
X
X
5
5
5
has its co r ne r nu m b ers in a r ith m etical sequence
A very interesting class o f squares is r e fe rred to in the sam e
as
follows
w o r k o n pp
8
and
3 3 7 33
33 9
The R ev A H F r ost while a m issionary fo r many years in
India o f the Chu rch M issionary S ociety interested himsel f in his
-
.
.
.
.
.
.
-
.
.
.
.
,
.
,
,
Fig
.
28 1
Fig
.
leisure hours in the stud y of these squares and
a rticles which he pu b lished on the subj ect gave
N asik fro m the town in which he resided H e
N asik cub es in the S outh Kensin gton M useu m
has a vast mass of unpublished materials of an
m o st ca r efully wo rked out
’
‘
.
‘
’
.
282
.
cube s and in the
them the name of
has also deposited
L
ondon
and
he
)
(
exhaustive nature
,
”
.
M r Kesson has also treated the sa m e su b j ect in a di ff erent way
and m ore popular form in the
H e gives the m the very
appropriate name o f Cai ssan S quares a na m e given to these squares
.
‘
'
,
,
he says by S ir William Jones
“
The p r ope r name however for such squa r es should rathe r b e
Indian
ah m ins b een known to b e g r eat
f o r n o f only have the B r
.
,
,
‘
,
’
,
Pu b li h d
s
e
in L o n
d
o n,
E ngl an d
.
SOM E
1 76
C
URIOU S M A G I C SQUAR E S
AND
W
C
OM BI N A T ION S
.
adepts in the formation of such squares from time immemorial not
only does M r F rost give his an I ndian na m e ané - lso n la -Kee
i
o
u
g
B
n o t only is o ne of these
Bi m
squares represented over the gate o f G walior while the natives of
I ndia wear them as a m ulets but L a Lo ub ere who wrote in 1 69 3
expressly calls the m f Indian S quares
M
.
M
,
,
,
,
,
,
,
’
.
“
In
these Indian squares it i s necessary not merely that the
su m mation of the rows colum ns and dia gon als should be alike but
that the nu m bers o f s u ch sq u ares s ho uld b e s o harm o ni o us ly b a l
,
a n c ed
as
,
tha t the s u mm a ti o n o f
i n the m o v es
An
o
b is ho p
fi
a
f
ex a m p l e o f
a ny
ei g ht n u m b ers
kni g ht
or a
in
s ho u l d a ls o ,
di r ecti o n
o ne
be
a li ke
”
.
Of theSe
squares is given in F ig 282 and
slight exa m ination will show it to be o f the same o rder as the
Fig
Jaina
1 25
.
2 83
o ne
.
F ig
.
.
284
.
square described by D r Caru s in a previou s chapter (p
but having enlarged characteristics consequent o n its in
.
.
crease ln size It will be seen t hat the extraordinary p rOp erti eS as
q uoted above in italics exist in this square so that starting fro m any
cell in the square a ny eight numbers that are covered by eight c o n
In addition to this the num
s ec u tiv e similar moves will sum 2 60
bers in every 2 X 2 square whether taken actually or constructively
z
I
1
and
I
2
1
1
6
6
6
0
8
sum 1 30 ; thu s 1
+ 55
3
+
5
5 +
'
.
,
f
a
e
ed
t
,
.
,
,
,
: 1 0 etc
urthermore
as
in
the
1
F
0
al so I
8
4
(
3
3
5
Jaina square ) the properties o f this s q uare will necessarily remain
unchanged i f columns are taken from o n e side and put on the other
1 30 ,
r
.
,
,
/
1 78
SOME
C
URIO U S M AGI C SQUAR E S
AND
C
OM B I NA T ION S
.
any of the nine sub centers contains numbers in its corner
cells that will sum 1 3 0 excepting when the diagonal s of
any of the four sub squares o f 4 X 4 form o ne side of the
parallelogram
-
,
-
.
A ny octagon Of two cells
any
o
f
of
2 60
NO
a side that is concentric with
the nine sub centers will have a constant su m mation
on
,
-
.
less than
1 92
columns
of
eight consecutive numbers
may be found having the constant summation
follows ( S ee F ig 2 86 )
of
2 60
.
Fig
.
285
Fig
.
.
286
.
H orizontal columns
P e rpendicular colu m ns
P erpendicular zig zags (A to A I )
H ori zontal zig zags (A to A 2 )
Corne r diagonal s
-
-
Constructive diagonals (D to D I )
B ent diagonals (as in F r anklin squa r es ) (T to T and
16
T to T )
Columns partly st r aight and pa r tly zig zag (as V tOV ) 88
Columns partly diagonal and p artly zig zag (as P to D ) 3 2
16
D ouble bent diagonal columns (as M to N )
I
2
—
l
-
1
Total
as
SOM E
Mr
C
URIOUS MAGI C SQUAR E S AND
C
OM B I NA T ION S
1 79
.
has also constructed an 8 X 8 square shown in
F ig 2 87 which i s still more curiou s than the last one in that it
pe r fectly comb ines the salient features of the F ranklin and the In
F rierson
,
,
.
dian squares viz
.
,
Fig
,
the
b ent
287
.
diago na ls
c o nti nu o u s
and the
besides
,
F ig
.
.
e x hibiting many othe r interesting properties some of which may
b e mentioned as follows
,
M
d
s
q
uare
has
a
constant
summation
of
X
The corner cells of any 3 X 3 square which lies wholly to
the right o r le ft of the ax i s AB sum 1 30
The corner cell s of any 2 X 4 2 X 6 or 2 X 8 rectangle
perpendicular to AB and symmetrical the r ewith sum 1 3 0
The corne r cells of any 2 X 7 o r 3 X 6 rectangle diagonal
1
6
1
to AB sum 1 30 as 1 2
2
0
o
49 +
5 + 45 + 3
3
1 9 + 46
I 30 etc etc
squa
r
e
contain
numbers
in
The
corne
r
cells
o
f
any
5
5 X 5
arithmetical p r ogression
A ny constructive diagonal column sums 2 60
A ny b ent diagonal sums 260
A ny reflected diagonal sums 260
A ny
2
1 3 0, w
2
e
/
.
,
.
,
,
.
,
.
.
.
.
.
.
ec t ed
( N OTE :R efl
di go nal s a r sh o w n
a
e
in
d o tt d li n
e
es o n
F ig
.
B y dividing this s q uare into quarters and sub dividing each
,
quarter into
X
:
f o Ll l
2
2
squares the nu m bers will be found sym
,
W
1 80
SOM E
C
URIOUS M AGI C SQUARES
A ND
C
OM B I NA T ION S
.
metrically arranged in relation to cells that are Similarly located in
diagonally Opposite 2 X 2 s q uares in each q uarter thu s :64
1
6 5 57
8 z 6 5 etc
,
.
,
Fi g
A nother
.
289
8 X 8
Fig
.
.
sq uare by
M F rier s on
290
.
i s g i ven i n
r
.
F ig
.
2 88
which i s alike remarkable for its constructive simplicity and for
its curiou s properties
L ik e F ig 2 87 thi s square combines the
principal features o f the I ndian and the F ranklin square s in its
.
.
b ent
and
c o nti nu o us
F ig
.
diagonal column s
29 1
.
F ig
.
.
292
.
T o render it s structure grap h ically plain the numbers
to 3 2
The number s in the complete square are
,
are written within circles
.
1
arranged symmetrically in relation to the two heavy horizo n
tal lines
so that when the numbers in the
firs t half
o
f
the seri es are entered,
SOM E
C
URIOUS M A G I C SQUARES A N D
C
OM B I NA T ION S
.
1
2
4
and 294 are ingen iou s combinations O f 4 X 4 squares
also d evised by M r F rierson F ig 293 is a magic cross which
F igs
2 93
.
.
.
.
possesses many unique features It contains the almost incredi b le
number O f
di ff erent columns O f twenty one numbers which
sum 1 471
.
-
.
F ig
.
294
.
S ome O f the propertie s found in the m agic penta gr am
mav be stated as follow s
E ach 4 X 4
F ig
.
2 94
rhom b us is perfectly magic with summations of
1 62
It therefore follows that from any point to the next the num
b ers sum 3 24 and also that every bent row o f eight numbers which
i s parallel with the row s from point to point sum s 3 24
In each 4 X 4 rhombu s there are five other s o f 2 X 2 whose
numbers sum 1 62 also four oth ers o f 3 X 3 the corner numbers o f
which sum 1 6 2
In each 4 X 4 rhombus every number ends with o ne o f tw o
numbers viz O and 1 2 and 9 3 and 8 4 and 7 5 and 6
,
.
,
.
,
,
.
,
,
.
,
,
,
,
,
.
SOM E
C
URIOUS MA G I C SQUAR E S
A ND
C
OM BI NA T ION S
.
18
3
M odifications of the concent r ic magic squares (de s cribed in the
fi r st chapter ) have b een devised by Mr F r ierson two examples o f
.
which are Shown in
Fig
.
295
F igs
.
2 95
and 2 96
,
3
.
F ig
.
.
296
.
The constructive plan u s ed in building the 6 X 6 magic square
illu strated in F ig 2 79 has been applied to the 1 0 X 1 0 square in
Fig 2 97
The 2 X 2 squares (each containing four consecutive
.
.
.
Fig
.
297
.
F ig
.
298
numbers ) are arranged in the order of the 5 X 5 s q uare shown in
F ig 2 98 which is constructed with the total s of the 2 X 2 sub
s q uares F ig 297 is interesting in the fact that ten out O f its
.
,
.
.
l7
1 84
SOM E
C
URIOUS M AGI C SQUARES
A ND
eighteen constructive diagonal s possess the
the Indian squares as shown by dotted lines
,
C
OM B I NA T ION S
co nti nu o us
,
.
f ez
an d-it -isp o ss
N O TE S O N VARIOUS C ON S T RU CT IVE PLA N S
W
.
another plan for the F ranklin squares another f o r the
o r c o nti nu o u s squ ares and so forth
so that a knowledge o f these
plans mak es it easy to classi fy all 4 X 4 s q uares S ix of the eleven
,
,
.
plans given by M r
the remaining
.
F rierson
cover distinct methods
plans being made up
PLA N NO
.
of
of
arrangement
various combinations
,
.
I
this plan which i s probably the Simpl e
st o f all the pairs o f
numbers that sum n + 1 are arranged symmetrically in adj acent
In
,
.
2
F ig
.
299
F ig 3 00
.
.
cells for m ing two ve r tical columns as shown in
,
,
grammatically
in F i g 3 0 0
.
This plan di ff ers from
.
30 1
.
2
F igs
.
.
.
and dia
.
Fig 3 02
.
2
0
3
of
.
in adj acent columns as
,
.
PLA N
F ig 3 03
2 99 ,
N o I only in the fact that the pairs of
.
and
1
0
3
.
.
numbers are placed in alternate instead
seen in
F ig
.
PLA N N O
F ig
.
NO
.
3
.
F ig 3 0 4
.
N O TE S O N VARIOUS C O N S T R U C T I VE P LA NS
A ccording to this plan the pairs
.
of
numbers are arranged sym
m etrically O u each side of the central axis one hal f of the ele m ents
being adj acent to each other and the other half constructively a d
j acent as shown in F igs 30 3 and 30 4 This arrangement furnishes
the F ranklin squares when expanded to 8 X 8 providing that the
nu mbers in a ll 2 X 2 sub squa r es a r e arranged to sum 1 30 ( S ee
I f this condition is not fulfilled only hal f of
F igs 30 5 and
-
,
,
.
.
,
-
.
,
F i g 3 05
.
F ig 306
.
.
.
the b ent dia gonals will h ave proper summations A n imperfect
F ranklin square of this type may b e seen i n F ig 2 88
.
.
PLA N
NO
.
.
4
.
this plan the pairs of nu m bers are a r ranged adj acent to each
other diagonally producing four centers of equili b riu m ( S ee F igs
30 7 and
In
.
F ig 3 08
Fig 3 0 7
.
.
.
.
M agic squares constructed on this plan exhibit in pa rt the fea
tures of the
F ranklin
and the I ndian squares
.
N O T ES ON VARIOUS C ON S T RU C T I V E PLA NS
PLA N
NO
.
5
.
.
The pair s o f numbers in this plan are ar
cells in the diagonal columns and it produces
alternate
s s q uares
,
which have been termed Jaina N asik and
squares F ig 3 0 9
i s the Jaina square as modified by D r Caru s (F ig 2 22 p 1 2 7)
and F ig 3 1 0 s h ow s the arrangement o f the pairs o f number s The
.
.
,
.
,
.
.
.
.
F ig 3 1 0
.
.
square F ig 282 is a Simple expansion o f
F ig 3 1 0 and the diagram o f the F rierson square F ig 287 shows
a design like F ig 3 1 0 repeated in each o f its four quarters
diagram
.
.
.
,
.
.
PLA N
NO
.
6
WW
.
U nde r thi s plan the p airs of numbers are balanced
around the center
common to all
ri c ally
F ig 3 1 1
.
m
m
of
4 X 4
of
the square and
s q uares tho —
,
,
Fig 3 1 2
.
F ig 3 1 1
.
square the diagrammatic plan be
,
met
“ this arrangement i s
.
.
sym
show s
.
tho na tal
g
.
1
2
3
form
.
d
“
”
N O TE S O N VARIOUS CO N S T RU CT IV E PLA N S
PLAN N O
This is also a com b ination
F igs 3 1 9 and 3 2 0
of
.
IO
plans
.
.
2
and
and is illu strated in
3
.
.
F ig
.
3 19
Fig 32 0
.
.
.
PLA N
W
ance with plan N O
a r e located apart by knight s moves which
a lk
hitherto considered
.
’
.
F ig 3 2 1
.
,
F ig 3 22
.
W
W
t
57
f
t
f
m
e
j
f
/
2
.
0
4
m
.
and
2
1
3
the
Show
22
3
exhibit
squares with
J3 5
a
'
°
¢a
-
“
.
to any plan
u ra lly
r 24
r
a
erent
anta
.
2
3 960 ,
i s di ff
m
F igs
v
'
2?
¢4 w
7
y
7
,
3 36
m
fi t
£ 7J g g
7
zzz
m za
3 44i ; 3 4g ;
ca
/ a
r
.
afl
m
fi
“
W7
zy
s sv
75 ?
NO TE S ON VARIO U S CO N S T RU CT IV E P LA N S
.
:3 /
F igs
pages
19
and
and
24
.
F ig
.
are identical with 6 X 6 squares shown on
A ll squa r es of this class have the same charac
m
2
3 9
3LT
WM
‘
“
F ig
¢
¢
.
m
L ‘ U
L L
‘
Q
u
x v
v n
1 \
1 u
u
o
I
PLAN NO
This is also a co m b ination
F igs 3 1 9 and 3 2 0
.
of
.
I
O
plans
L
A
LV O
.
.
2
and
3
and i s illust
.
F ig
.
3 19
.
PLA N
ance with plan N O
.
2,
but in
the pairs
o
f
i
N O T ES
1 92
“
ON
VARI OUS CO N S T RU CT IVE PLAN S
.
num
ers
located
b
f
i
r
worthy of notice in havi
a
s
p
BQ
apart b y knight s moves F igs
and g illu strate another
It will be seen
6 X 6 square with its plan and numerical diagram
1
’
.
.
.
295 3 3 !
Fi g s -
F ig saw
P i g gy: 3 3 3
F ig gear
F is
m
.
3 3 2
.
_
3 3
.
y
F ig 5
3
.
that the latter i s symmetrically balanced on each side di f fering in
t his r espect from the nume r ical diagrams o f the 05 4413 9 1 9; 6 X 6
squares as described in Chapter I
,
1
.
C HA
T E R
P
MA TH E MA TI CA L
TH E
lx
f o l lo w m g
q uotations bearing
M
O il
from a paper entitled
on a Chessboard by M aj or P A
ac
pu b lished in P r o ceedi ngs o f the R o yal
V o l X V II N o 96 pp 5 0 6 1 F eb 4
.
.
M Mh
-
.
.
,
.
,
,
.
the ab ove subj ect are Co p
agic S quares and O ther P rob lems
“
i ed
MA GI C
V A LU E O F
SQ U A RE S
HE
.
.
,
a
o n,
R
.
A
I nsti tu ti o n
18
9
2
.
o
,
D
f
G
.
Se
.
,
rea t
B ritai n
,
.
“
The construction o f m agic squa r es is an amuseme nt o f great
antiquity ; we hear o f thei r b eing con st r ucted in India an d China
b e fo r e the Christian e r a while they appea r to have b een introduced
into E u rope b y o sc o p u l u s who flourished at Constantinople early
in the fifteenth centu ry
M
,
”
.
“
H oweve r what was at fi r st m erely a p r actice o f magicians
and talisman mak ers has n o w f o r a long time b eco m e a serious
,
stu dv
,
f o r m athe m aticians
N ot that they have i m agined that it
.
would lead the m to anything Of solid advantage b ut b ecau se the
theo r y was seen to be fraught with di f ficulty a nd it was considered
possible that some new p r operties Of n umbers might be discovered
which mathematicians could turn to account
This has in fact
o
r
oved
t
b e the case for fro m a ce r tain point o f view the su b j ect
p
has b e en found to b e algebraical r ather than arith m etical and to be
inti m ately connected with great departments of science such as the
i nfin iteSi m al calculus the c a l c u l u s b f Operations and the theory
o f groups
“
N o pe r son living knows in ho w many ways it is p ossi b le to
fo r m a m agic square o f any orde r exceeding 4 X 4 The fact is
,
,
.
,
”t
‘
’
,
‘
‘
’
’
,
’
.
.
that before we can atte m pt to enumerate magic squares we must
see o u r way to solve problem s o f a far more si m ple character
.
T H E MA T H E MA T I CAL V A L U E
OF
MA G I C SQ U AR E S
.
To say and to establish that p r ob le m s o f the genera l nature
i e m agic s q uare a r e intimately connected with the i nfinitesi
calculus and the calculus O f finite di ff erences i s to sum the
up
3,
.
is therefore evident that this field of study is b y no means
ed and i f this m a y b e said in connection with magic squa r e s
statement will naturally apply with a larger meaning to the
deration of m agic cubes
It
,
.
I N D EX
1 98
G a m es A nci ent
F a lk ener ),
en
1 59
Pr
a n,
y l
Gw li o r
Ro
a
a,
,
c
o
’
ssb o a
u mb r
,
v
e
gi c qu ar
s
of,
a
sp
m agi c
en
Mp
47ff
,
h res vi
th o d o f c nstr u t
qu re 34ff ; o dd
e
.
,
,
13
1 22
s,
a
s
es,
c
o
.
c
di
In di
an
qu r
1 79 ,
1 77,
.
,
W
,
1 2 7, 1 77, 1 88
K n ingt n M
K n M 76
K h j u r h (In di
esso
r
,
.
a
a
in
1 25
1
,
,
a
1 83
,
;
a
O
a
,
a,
a
,
.
o s,
s
o
es
,
,
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ca
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e
a
.
s
se
.
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,
.
,
,
.
.
a
,
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s,
a
o
12
e s,
o
M
M
S o uth,
1 75
e of
P
,
1 5 6,
a i na
) J
,
i nscri p ti
on
a es,
s
s
a e
,
i ct u re
of,
A l b rt
by
e
.
1 75
of
M gi qu r
o dd
1 93
So
,
.
157
tr u cti ng
do n ),
f ma
( B ac h et de ) m eth o d
’
Mu u m
o
.
o s c o p u l u s,
.
.
gi c qu r
m agi c qu r s 1 93
s
va l u
;
1 22
t u dy
m ati c a l
M yr 3
M l h ly
a
.
,
e
D ii rer,
.
s,
,
.
a
e
H 0 The
of
1 29 ff
,
,
se
a
a es, I 7
c s
.
.
u th K
i gt o n
en s n
(L o n
,
.
.
K i lh rn P r f
K night m v in
o
e
,
ec o
a
c o ns
a
es
,
a
a
.
o
.
s s
ezeri a c s
u seum ,
o
s
e
.
’
e an c
‘
,
,
s
M th
.
,
,
.
,
.
1 75
,
a
Co nc em
,
,
a
.
illi am
o n es S ir
J
1 58 161
o w ett
J
,
,
e
a e
s
e
di fi d
mo
c
,
,
e s
Mp
,
a e,
a
.
tri
ea
a
44ff ;
,
a
.
,
s
a na ,
s
n
,
.
o
i nsc ri p ti o n i n K h aj u rah o In
1 76
87 94 1 2 5 ff
dia 1 24 ; qu r
qu r m o di fi d by D r C r u
1 77 ;
i
J
u d
yth go r n
lt rn ti o n
-
1 76,
a es,
P
the
on
’
.
.
,
,
1 93
em s
; by
1 75
o
o
e
1 76,
1 25 ,
1 23 ,
a,
m agi c s
1 87 1 88
1 85
e
o
o
.
.
o
s, 1 73
tru ti n f 1 4 5 4ff 1 85 ff ;
D fi niti n f 1 ; E rli t r
rd f
1 ; E v en
1 8ff
3 4ff ; F rank li n 88
8gif 94 95 1 1 1 ; F r nk li n n ly z d
by D r Car u s g6 ti ; F ri er so n 1 77 ;
F ri r o n s a naly i of 1 29 ; In di n
(La Lo ub é ) 1 76 ; Ji n 87 94
1 2 5 ff
1 76
1 77 ;
in ym b l
1 20 ;
M th em ti l t u dy f 1 29 ff ;
Nu m b r ri in 1 48 ff ; O dd 1 ff ;
K night s m o v e in 4 5 7; V ri
ti n in 1 40 ; with p r d et erm i ned
u m m ti n 5 4
re
17
1 4,
-
.
i ti o n
and
;
,
.
r bl
p
Co nc en
.
a
In
1 93
,
Co mp o
;
.
e
1 63 ,
rd
th er
e s, 1 5 6 ff
Co ns
la ) m e
e
e s
in g
the
of
on
a
,
he
o
.
an d c o m b na
a es,
an d
181
es,
,
H rm y
H ir (D
HO
c s
tri c
.
ma
a
1 0 2 ff
yth ago rean
1 64
c
n
.
.
P
ec a
a
c
1 57
es,
sca e,
c
o
a
1 93
,
the
Oi
,
.
H rm ni fi g u r
H rm ni
l
a
di ng s
o c ee
o
M gi r t ngl
M gi qu r
if ;
.
tit u t e f
1 76
I n di
Ins
a
.
.
e s es,
ea
(G eo
ern
o
1 74
G ld V r
G r t B rit i
o
Md
and
.
o
,
’
o
s
.
1 25
1 24,
,
ma
e
.
1 75
,
gi c qu r
s
N a si k sq u ares and c u b es
N u mb er seri es 1 48ff
a es,
The
,
.
.
4)
O dd m a
La Lo ub ere,
Letters
.
.
a nd
i
en a m n
j
F
r
ank
li
n
)
,
.
Li f e
li n
Lo
B
P hi l o s op hi ca l
on
G enera l
64ff
t
n o es
.
gi c squa res
iac s m eth d
B ac h et de
Mez
o f c o n tr u c ti n g
o
I 7 ; B reak mo v es in
7 ; E x amp l es
of b
ak m o es in 8 ; G en er a l p ri n
O dd m a
er
,
1 ff
.
;
’
s
,
,
a nd
Ti m es
am es
(B y J
g an
M
f B enj a mi n F rank
P a rt o n ) 96
o
,
89 9 1
Lo h The S c ro ll o f ,
L s s N um ero rum
,
r
.
,
,
uu
.
,
M Mh
1 22
v
,
1 68 ,
1 71
1 1 3 ff
.
o n,
.
.
,
.
,
,
,
O dd, 64ff
.
.
,
.
,
7
.
tro no m er
fi g u res (0
t
the a s
,
e s
.
v
c ip l es o f ,
r
O rd r
M aj o r P A 1 93
M agi c c u b es Ch aract eri sti c s o f 64 ;
G eneral no tes o n 84ff ; Ev en 76ff ;
a
re
Om a
.
,
ac
,
84ff
o n,
.
P apers
(B y
j
S u b ec ts
89
1 76
gi c c u bes
of
,
,
po e
,
1 67
ro ,
i,
ri,
.
P rt Jm 96
P nt gr m M gi
“
P h dru f P l t
P hil l u 5 8 67
a
o n,
e
a
ae
o a
a
a
es,
a
,
”
s
s,
1
,
c,
1 00
1 82
a o,
o
,
1
.
.
.
1 58
.
.
) 79
,
I N DE X
P hi l o s o phi ca l S u bj ects by B enj ami n
and
F ra nkli n, L etters
In
P yt h g r
a
;
.
(D r
Chines e
P hi l o s o phy,
o
P apers
.
1 58
ea n,
o n,
89
a
s
P u l C ru
a
.
)
1
a on c
sc
a c
cs
1
,
ne
s
s
o
a n,
on o
o c ea
1
,
10
a
o n s,
1
.
c
a
ns
e
o
a c
,
1
,
ea
o
1
,
a e
a e
1 09
an sp o s
iti
1
,
1
c
1
,
1
o n,
oo
.
So
A lt rn ti
e
Re
es,
e
1 1 3 ff
a
on
So
u li
1 6 7,
Ro y a
Pr
l
1 68
c
s
es,
79, 87,
l er
S chefl
P
l t
1 5 8,
a o,
1 63 ,
1 66 ,
tit u t
di ng
s
’
s
st ruc t ing
e
Gr t
of
of,
(Pr
W
the
o
,
i gt
do n )
1 75
of
m agi c
B
ea
1 93
rit i n
a
,
qu r
a es,
e,
.
e,
rld
,
,
N a t u re
an d
Mu u m
on
(L o n
se
.
h r H arm ny o f th vi
S tif li u 92
S u mm ati o n
M gi squ r s with
p r d t rm i n d 5 4
S ymb l
M agi c qu r in 1 20
Sp
e es,
o
s,
s,
e
e,
.
.
a
e e
e
c
,
o s,
a e
.
s
T etracty s
u
i
T m ae
66n
1 5 9,
,
”
a es
1 61
of
s
,
.
P
.
l at
o,
r
of
14
.
co n
1 5 8,
1 5 9,
1 64,
b y,
1 06
.
a n sp o s
,
1 64
.
iti o n A lt rn ti
e
,
a
on
.
Va ri o u s c o nstru ctiv e p l ans N o t es
1 84ff
.
Ventr D B 86
V ers G ld en 1 5 9
V irgil 1 24
.
es,
,
.
.
.
Y g d yi
Y ih K i g
an
.
o
,
.
) m eth o d
s
.
.
en s n
es ,
.
In s
o c ee
‘
,
a
of
c
of
u th K
1 12
.
.
Rep b
ul
e
o
.
,
.
a
.
T i m aeu s the L o c ri an
a
s
o
o
,
12
e,
(T i m aeu s ) 1 64
“
.
t ngl M gic 1 8 1
flc ti n o n m gi qu ar
a
a
.
o c ea
1 66
.
.
,
,
“
1
,
,
o
l
1 75
e
e
,
o
T
R ec
e
ac
.
.
.
The,
u een,
o
1
,
-
e s,
Q u t rn t tr
b y,
12
,
o so
a
Q
12
,
c
e
.
o n c sc a e o
a
o
a s,
o
e e
,
.
,
.
Pyth g r v 3 4 5 7 66 ;
H rm i
l f 63 64 ; S h l
f
f
5 7 ; P hil
p hy
58
Pyth g r n n u mb r 5 6ff
a
c
.
a
o
e
1 66 n
1 64,
1 5 9,
1
e,
e
,
c
.
ea
o
0,
1
,
,
n
o c ee
.
s o
a es
s
1
,
o
e e
e
.
oo
1
,
”
o
1
.
hilli ng P ro f 1 24
S hl i rm h r 6 1 n
S hn i d r 1 6m
S h u b ert P ro f H erm nn 44 I 6 1 n
S ro ll f Lo h Th
2
S m ith P r f D vid E u g n
1 24
S u l P r r ti o n
f th
1 59
1 64
Sc
c
.
P l t 5 8ff ; 69
P l t i h l 67
P l u t r h 5 6 5 9 6 64 66n
“
f A ri t tl
P liti
3
P r d t rm i d u mm ti
M gi
qu r with 5 4
P r di g f th R y l I tit u t
f G r t B rit i
93
“
P r r ti f th S u l ( P l u t r h )
a o,
.
an
n
,
n,
1 22
1 22
.
.
.
P O R T R A IT S O F
MI N E N T MA T H E MA T IC IA N S
E
hree p o rtf li s edit ed b y D AV ID E U GENE S MI T H Ph D P o f sso r 0
M ath em ti c s in T c h e s C ll ege C l u m b i U n iv rsity N e Y k C ity
In r esp o nse to
wid esp re d d em n d fr m th o s i nt erest ed in m t h em ti c s
n d the hi t ry o f e d u c a ti n P r f esso r S m ith h s
dit ed three p o rtf l i s o f the
p tr it s o f s m e o f the m o st em i n en t o f the w rl d s c ntri b u t rs to the m t h
m ti l sc i en es
A c c o m p nyi ng e c h p rtr it is a b ri ef b i o gr p h i c l sk et c h
with c c si o n l n t es o f i nt e st c nc ern i ng th rti st s rep r s nt d T he
of
S i ze t h t
ll o w s f o fr m i ng (1 1 1 4) it b ei ng t h e h p e th t
p i ct u r s
ne
i nt ere t in m th em ti cs m y b e ro u sed t h r u gh th d ec o r ti n f c l
r o m s b y the p o rtr it s f th se ho h el p d to c re t e th sc i en c e
T
o
o
a
,
ea
’
r
o
a
a
a
s o
o
or
a
o
a
ca
c
o
a
o
re
a
a
o
a
a
h a l es P yth go r
N a p i er D esc a rt es
o
,
,
1
.
a
’
o
o
o
a
a
a
a
a
x
a
,
e
.
o
,
e
a
a
e
a
o
o
0
T
0
c
Fe
e
ea
e
,
a
,
’
o
a
ca
a
a
a
.
a
eo
,
,
a
a
a ro
,
a
a ss
a
o
o
o
sa ,
a
a
,
a
,
a
a
o
,
a
,
a,
.
i n fin it i m l
l c u l u s : C v lli eri Jh nn n d Jk b B rno u lli P
L H p it l
B r w L p l c e L gr ng E u l r G u s M o ng e n d N i
T rt gli a
es
e
a
e e
o
e
a
o
e
w
or
w lv e gr t m th em ati c i n s d w n t 1 700 A D
E u lid A rc hi m ed s L n rd o f P i
C rd n V i et
rm t N ewto n Leib n i z
as,
a
a
O
P OR TFO Ll O NO
o
r
w
,
e
r
,
e
o
a
.
e
a
a
o
a
,
e
o
a
s
T
a
,
.
a
w
a
o
a re
e
o
,
.
a
a
a
e,
o
a
e
s
a
e
a sc a l ,
,
a
,
l
cco o
.
rtf li s esp e i lly a d p t ed f o h igh s h l s n d a c d em i es i nc l u di ng
p rt r it s o f
T H A L E S with wh o m b g n th
t u dy f sc i enti fi c g eo m try ;
PY T HA G OR A S ho p r v d the p r p siti n f the s q u
n th h y o t n u s e
E U C LI D w h e E l m nt f G o m try f rm the b i o f l l m d rn t x t b k s
A R C H IM E D E S w h e tre t m ent o f the c irc l e c o ne c yli n d er nd S p h ere
i n flu n o u w rk t d y ;
D E S C A R T E S t w h m e a e i n d eb t d f o the gr p hi c lg eb r in o u high
s ho l s ;
NE T O N ho g en r l i zed the b i n m i l th o rem n d i n v nt ed th e c l u l u s
N A P IE R ho i n v en t ed l g rit h m s n d ntri b u t ed to trig n m etry ;
ho di sc o v ed the
PA S CA L
M ysti c H ex gr m t th e g e o f s i x t een
po
o
o
a
o
c a
e
—
os
—
e
—
W
c
e
oo
a
o
e
o
,
e
o
o
e
o
a re o
o
as s
a
o-
a
o
e
s O
o
o
—
c
e s
os
r
c es
a
o
w
—
e
r
a
a
e
o
e
e
e
oo
a
,
,
p
a
r
w
e
r
a
a
r
a
o
—
—
e a
w
—
o
a
o
w
a
e
e
co
a
o
er
w
a
a
a
a
:
a c
o
a
.
P R ICE S
f
tf
tf
tf
i
li
li
li
P o rt
ol o 1 or 2
Po r
o
Po r
o
Po r
o
e
dg
o 1 or 2
o
3
o
3
(1 2
(1 2
i
ll
iz
i
it
(8 p o rt ra
(8 p o rt ra
i
v
Ja p a n es e
s )o n A m e r c a n
t s )o n
i
i
el l u
m
a nd
to
i
i
p l a t e p a p e r , s ze
b ot h f o r
b oth f or
i
s ng l e
e 1 1 11 1 4 ,
s
,
1 1x 1 4,
s
i
p o rt ra t s , 5 0 cen t s
ng e p o rt ra t s , 3 5 c
i l
i
.
.
i ss u e o f thi s fin e ll ec ti o n is q u lly c r dit b l t th x p ert k n w
f th e e dit r P r f es r D vid E u g n S m ith
n d di c ri m i n ti n g t st e
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