Clarke courier. [volume] (Berryville, Va.) 1869-1997, 1906-12-26
PUZZLES IN FIGURES
MANY UNsoLvEo MYSTERIES IN TIE
WORLD or NUMBERS

which Child Can Understand.
Though the Greatest Minds Cannot
Master Magic Squares.
Probing into the secrets o. nature Is
passion with all men. only we select
different lines OF research. Men have
spent long lives I'm such attempts "a
to turn the baser metals into gold. TO
discover perpetual motion. to find a
cure for certain malignant diseases. to
navigate the air. Some great nys-
teries have after centuries OF patient
labor been completely solved. others
are at present under investigation.
while many have been demonstrated 70
be quite impossible OF solution.
Let - examine n few cases OF I'M
solved mysteries in the world OF num-
bers-little things the conditions OF
which child can understand. though
the greatest minds cannot master. Ev-
erybody has heard the remark. 1T Is
as hard as squaring the circle" though
many people have very hazy notion
or what IL means. IL Is this: IL Won
nave circular piece OF paper. how are
you TO cut out another piece In the
form OF d square that shall contain
exactly the same area? Well II can
not be done with exactitude. Though We
can get an answer near enough for nil
practical purposes because Il Is not
possible TO say I'm exact numbers what
Is the proportion OF the diameter to
the circumference. But IL Is only I'm
recent times that Il has been Droved
to be impossible. Only cranks now
waste their time In trying 70 solve this
venerable puzzle
Again. we can never measure exact
~ in numbers the diagonal OF n SQUARE
III you have a window pane exactly a
loot on each side. there Is tie distance
from corner TO corner staring Won I'm
the face. yet you can never saw I'm OK.
act numbers what is the length of that
diagonal The novice will at once sug-
gest that we might take our diagonal
hrst. say au exact foot. and then con-
struct our square. Yes. you can do
this. but then you can never say exact
" what is The length OF the side. You
can have IT which way you like. but
you cannot have il both ways.
But let us take few puzzles that
nave not been proved to Ice impossible
but which nevertheless have not been
solved. They will give the reader some
fascinating employment during spare
hours. II be happens TO lie fond OF fig
uring. First. then. take the round table
problem. Nine persons are stoppipg NJ
boarding house. and they all sit down
together TO dinner on twenty sue
cessive nights ut 4 round table. The
rule OF the house is that no person shall
on any two occasions have the two
same neighborS. How is IL TO be done.
II at all?
Here Is another poser. III we write
down the number, composed Of seven
teen ones-11.111.111.111.111.11I-and
ask you TO find some numbcr-other
than l or the number itsclf-that will
divide IL without remainder. the an
swer will give you considerable labor
to discover. We will. however. say at
once that the only numbers that will
divide 1t are 2,O71,723 and 5.363. 222-
30I. Now add two more ones 10 the
number. and we cannot tell you wheth-
er IL can be exactly divided by anv
number or lot. for nobody knows. III
you can find such divisor YOU will
have done something that nobody else
In the world has yet succeeded in do
Ing. And we cannot say that il is i'm
possible
Every one knows what a magic
square Is. Divide square into nine
divisions, or cells, and then place the
numbers 10 0. one number I'm each
cell. so that they shall add up fifteen
In every column, every row and In each
OF the two diagonals. 1t Is quite easy.
ana there is only one way OF doing II.
because we 10 not count as different the
arrangements obtained by merely turn
Ing around the square and rejecting IL
In a mirror
Now. IL we wish TO make a magic
square OF the sixteen numbers. / TO l6.
there are Just $50 different ways OF do
Ing IL, again not counting reVersals and
reflections. This has been finally PROV
ed OF recent years. But how many
magic squares can be formed with the
twenty-five numbers. 1 TO %, nobody
knows. and we shall have to extend
our knowledge In certain directions he
fore we can hope 10 solve this puzzle.
But Ii is startling to find that exactly
1920 such squares may be formed OF
one particular kind only-the bordered
square. in which the inner square OF
nine cells IS In Itself. magic. And the
present writer has shown how this
number may at once he doubled by
merely converting every bordered
square by simple rule into a non
bordered one.
Vain attempts have been made to
construct a magic square by what is
called knights tour" over the chess
board. numbering each square that the
knight visits In succession. 1, .. 3. /,
etc, and II has been done with the ex-
ceptiOn OF the two diagOnalS. which so
far have bafflel all efforts. But II Is
not certain that IL cannot he done.
Here Is one more unSolved problem
in numbers: We all know that a square
number is a number multiplied by It
self. but cube number Is one multi
piled twice by itself-thus. is the
cube of 2. % the cube OF 3. and so on.
Now, some whole numbers are the sum
OF two whole cubeS-as 80 Is the sum
OF the cubes OF and 9-others are the
sum of two fractional cubes-as 43 is
the sum of the cubes OF and 70
while the whole numbers cannot be ex-
pressed as the sum OF two cubes In any
way whatever. 1t Is possible to say OF
any number from to 100 whether IL IS !
or is not the sum OF two cubes. except
&C. Nobody I'm the world can answer
for this number. Can you solve these