The Richmond palladium. (Richmond, Ind.) 1906-1907, 1906-12-27
PUZZLES IN FIGURES
MANY UNSOLvED MYSTERIES IN THE
WORLD or NUMBERS.
Little Things the Conditions OF
which . Child Can Understand,
Though the Greatest Minds Cannot
Master Magic Squares


Master Magic Squares.
| Probing into the secrets of nature is
a passion with all nleu. 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 perpa-lal motion. 10 find
cure for certain malignant diseases, to
navigate the air. Some great mys-
| teries have after centuries of patient
labor been completely solved. others
ara at present under Investigation,
while many have been demonstrated TO
be quite impossible of solution.
Let us examine 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 squarin. the circle" though
many people have a very hazy notion
of what ii means. 1t Is this: II you
have a circular piece of paper, how are
you to cut out another piece in the
form OF q 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 all
practical purposes. because IL 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
10 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.
If 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
ly 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
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
a boarding house, and they all sit down
together to dinner on twenty-eig 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 it to be done,
IL at all?
Here is another poser. If we write
down the number, composed OF seven
teen ones-11 ! II, Ill Ill III, III - and
ask you to find some number-other
than I or the number itself-that will
divide it 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 5,363, 222.-
357. Now add two more ones to the
er it can be exactly divided by any
number or not, for nobody knows. IL
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 1t Is i'm
possible.
Every one knows what a magic
square is. Divide a square into nine
divisions. or cells, and then place the
numbers 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 IL |
In mirror.
Now, if we wish to make a magic |
square Of the sixteen numbers, to 16, |
there are just $50 different ways Of do |
ing it, again not counting reversals and |
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
174.24 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. Al the
present writer has shown how this
merely converting every bordered
square by a simple rule into a non
bordered one.
Vain attempts have been made to
construct a magic square by what is
called a 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-
reption of the two diagonals, which so
Car have baffled all efforts. But it is
not certain that it cannot be done
Here is one more unsolved problem
N numbers: We all know that square
number is number multiplied by it
self. but a cube number is one multi
piled twice by itself-thus, s is the
cube of 2. % the cube of 8, and so on.
Now, some whole numbers are the sum
IF two whole cubes-as 80 is the sum
IF the cubes of and 3-others are the
sum of two fractional cubes-as 43 is
he sum of the cubes of u and T-2-
while the whole numbers cannot be ex-
pressed as the sum OF two cubes In any