The Bossier banner. [volume] (Bellevue, Bossier Parish, La.) 1859-1952, 1907-01-17
PUZZLES IN FIGURES
MANY UNsOLvED MYSTERIES in THE
WORLD OF NUMBERS.


WORLD OF NUMBERS.
Little Things the Conditions Of
which Child Can Understand,
Though the Greatest Minds Cannot
Master Magic Squares.
Probing into the secrets Of nature is
q passion with all men, only we select
different lines OF research. Men have
spent long lives In such attempts as
to turn the baser metals into gold, to
discover perpetual motion, to find
cure for certain malignant diseases, to
navigate the air. Some great inys-
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 few cases of un-
solved mysteries in the world of num-
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
Of what it means. 1t Is this: II You
have circular piece of paper, how are
you to cut out another piece in the
form Of 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 it 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 square.
II you have window pane exactly a
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 yon like, but
you cannot have 1t both ways.
Bnt let us take few puzzles that
have not been proved to be impossible,
but which nevertheless have not been
solved. They will give the reader some
fascinating employment during spare
hours, If he happens to be fond of fig
uring. First, then, take the round table
problem. Nine persons are stopping at
boarding house, and they all sit down
together to dinner on twenty eight sue
cessive nights at round table The
rule OF the house is that no person shall
on any two occasions have the two
same neighbors. How is it to be done,
| If at all?
Here is another poser. II we write
down the number, composed of seven
teen ones-11 III III. III, III, I'LL an,l
ask you to find some number-oth
than / or the number itself-that will
divide it without. remainder, the an
swer will give you considerable labor
to discover. We will, however, say al
once that the only numbers that will
divide it are 2,O71 ,723 and 5,3G33, ,222.-
357. Now add two more ones to the
number, and we cannot tell you whetll-
er it can be exactly divided by any
number or not, for nobody knows. II
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 it is i'm
possible.
Every one knows what Il magic
square Is. Divide square into nine
divisions, or cells, and then place the
numbers / to 8, one number III each
cell. 50 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 1t
because we do not count as different the
arrangements obtained by merely turn
ing around the square and rejecting II
| In q mirror.
1 Now, il we wish to make magic
square of the sixteen numbers, to 16
there are just $50 different ways Of do
ing 1t, 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, to %, 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
1920 such squares may be formed o1
one particular kind only-the borderet
square, in which the inner square d
nine cells is in itself magic. And th.
present writer has shown how thi!
number may at once be doubled by
merely converting every borderet
square by a simple rule into non
bordered one.
Vain attempts have been made t.
construct magic square by what Il
called ssknight's tour over the ches
board. numbering each square that th'
knight visits IN succession, I. 2, 3,
etc, and Il has been done with the ex
ception Of the two diagonals, which q
far have bafIled all efforts But 1t |
not certain that It cannot be done.
] Here Is one more unsolved problem
In numbers. We all know that a squar
number is q number multiplied by II
!. self, but ube number IS one multi
piled twice By itself thus, 5 is th
! cube Of 2. ~ the cube Of 8, and 50 or
3 Now, some whole numbers are the SUI
: Of two whole cubes as 3J is the SUI
]. | Of the cubes of 2 and 3ofhprs are tn