The Richmond palladium. (Richmond, Ind.) 1906-1907, 1906-11-19
PUZZLES IN FIGURES
MANY uNsoLyED MYSTERIES IN THE
= WORLD OF NUMBERS
Little things the Conditions OF
which q Child Can Jnderstand,
Though the Greatest Minds Cannot
Master Magic Squares.

1 ThO0Rh the sxBGAIC0t HAn08 SaaaVs
Master Magic Squares.
| Probing into the secrets of nature is
| a passion with all men, only we select
dffferent lines Of research Men have
spent long lives in such attempts as
| TO turn the baser metals into gold, to
discover perpeSlal motion. to find a
eure for certain malignant diseases, to
| navigate the air. Some great mys-
teries have after centuries of patient
labor been completely solved. others
are at present under Investigation,
| while many have been demonstrated to
be quite Impossible Of solution.
Let us examine a few cases Of I'M
solved mysteries in the world Of num-
bers-litt1e things the conditions Of
which a child can understand, though
the greatest minds cannot master. Ev-
erybody has heard the remark. 1T Is
as hard as squarins the circle" though
many people have very hazy notion
Of what it means. It Is this: If you
have q circular piece OF paper. how are
you to cut out another piece in the
form OF a square that shall contain
exactly the same area? Well, it can
not be done with exactitude, though we
can get an answer near enough for all
practical purposes, because it is not
possible to say In exact numbers what
is the proportion OF the diameter to
the circumference But it is only in
recent times that 1t has been proved
to be impossible. Only cranks now
waste their time IN trying to solve this
venerable puzzle.
Again, we can never measure exact
~ in numbers the diagonal Of a square.
II you have a window pane exactly
foot on each side, there is the distance
from corner to corner staring you in
the face, yet you can never say in ex-
act numbers what is the length Of that
diagonal. The novice will at once sug-
gest that we might take our diagonal
first. say an 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 it both ways.
But let us take a few puzzles that
have not been proved to be impossible,
but which nevertheless have not been
solved. They will give the reader some
CAhpiOy llChL oolihS SPdtC
hours, il he happens to be fond of fig
uring. First, then. take the round table
problem. Nine persons are stopping at
a boarding house, and they all sit down
together to dinner on twenty eight sue
cessive nights at a round table. The
rule OF the house is that no person shall
on any two occasions have the two
same neighbors. How is 1t to be done,
II at all?
Here is another poser. II we write
down the number, composed Of seven
teen ones-11 II ], III. Ill Ill, Ill and
ask you to find some number-other
than I or the number itself-that will
divide 1T 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 it are 2,O71, 723 and $500 .222.-
357. Now add two more ones to the
and we cannot tell you wheth-
er it can be exactly divided by any
number or not. for nobody knows. II
you can find such a divisor you will
have done something that nobody else
in the world has yet succeeded in do
ing. And we cannot say that it Is i'm
possible.
Every one knows what q magic
square il. Divide square into nine
or cells, and then place the
numbers I to 8, one number in 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,
and there is only one way of doing it,
because we do not count as different the
obtained by merely turn
ing around the square and rejecting 1t
In a mirror.
Now, if we wish to make q magic
square Of the sixteen numbers, 1 to 16
there are just $50 different ways Of do
ing X, 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 -ive numbers, 1 to Is, nobody
knows. and we shall have to extend |
our knowledge In certain directions be- |
fore we can hope to solve this puzzle |
But 1t is startling to find that exactly |
174.24O 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 |
number may at once be doubled by |
merely converting every bordered |
square by a simple rule into a nom
bordered one.
Vain attempts have been made to I
construct q magic square by what is
called knights tour" over the chess
board. numbering each square that the
knight visits in succession, i, 2. 3. 4.
etc. and it has been done with the ex-
ception OF the two diagonals, which so /
far have baff1ed all efforts. But it Is /
not certain that it cannot be done.
Here is one more unsolved problem |
IN numbers: We all know that square
number is number multiplied by it
self. but a cube number is one multi
piled twice by itseif-thus. 8 is the
cube of .. 81 the cube Of 5. and so on.
Now. some whole numbers are the sum |
Of two whole cubes-as 80 is the sum
Of the cubes Of ? and 3-others are the
sum Of two fractional cubes-as As is
the sum Of the cubes of II and 1-2- |
while the whole numbers cannot be ex-