The Topeka state journal. [volume] (Topeka, Kansas) 1892-1980, 1907-09-06
GREATEST PUZZLES ON EARTH
The World of Numbers Presents Many
Mysteries.

Probing into the secrets of nature
is a passion with all men; only we
select differen- 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
a cure for certain malignant diseases,
to navigate the air. Some great mys-
terieS have. after centuries or 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-little things, the conditions OF
which a child can understand. though
the greatest minds cannot master.
Everybody has heard the remark: 1T
is as hard squaringv the circle, =
though many people have very hazy
notion of what it means. 1t is. this:
If you have a 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 z Well, it can
not be done with exactitude, though
we can get an answer near enough for
all practicable purposes, because it is
not possible to say in exact numbers
what is the proportion of the diameter
to the circumference But 1T 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 ex-
actIy in numbers the diagonal of a
square. If you have 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 exact numbers what is
The length of that diagonal. The novice
will at once suggest that We might
take our diagonal first, say an exact
foot and then construct our square.
Yes, you can do this. but then you can
never say exactly what is the length
of the side You can nave it which
way you like, but you cannot have 1t
both ways.
But let us take a few puzzles that
have not been proved to be impossible.
but which. nevertheless. have not been
solved. First is the Round Table prob
lem. Nine persons are stopping at a
boarding house. and they all sit down
together to dinner on 2S successive
nights at a round table. The rule of
the house is that no person shall on
anv two occasions have the two same
neighbors. How is it to be done, If
at all?
Try to write out the arrangements
for the nine persons. The present
writer first propOunded the question,
and showed how six persons may be
seated in ten ways and has recently
solved the puzzle in the difficult case
of seven persons to be seated in hf-
teen days The solution for eight
persons on twenty-one days has also
O--
show how we are to seat those nine
persons on twenty-eight days without
anybody having the same pair of
neighbors twice?
Here is another poser. If we write
down the number composed of II ones
Hill, III, III, Hill 1,-and ask you
to find some mum ber-other than
or the nunbe, itself-that will divide
it without remainder. the answer Will
give you considerable labor to discover.
We will however. say at once that the
only numbers that will divide it are
20T1,723 and 5.363,222,351. Now add
two more ones to the number and we
cannot tell whether it can be divided
nv any number or not. for nobody
knows If you can find such divisor
you will have done something that
nobody else in the world has yet sue
ceeded in doing AND We cannot say
that 1T is impossible.
Every one knows what a magic
square is. Divide a square into nine
divisions or cells. and then place the
numbers 1 to 9, one number in each
cell so that they shall add UP IS 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 ARRANGEMENTS obtained by merely
turning round the square and reflect
ing 1t in q mirror
Now. if we wish to make a magic
square of the 6 numbers 1 to L. there
are lust $50 different ways of doing it
again not counting reversals and re-
Aecfions. This has been finally proved
of recent years. But how many magic
squares can be formed with the 25
numbers l to 25 nobody knows. and
we shall have to extend our know
ledge in certain directions before we
can hops. to solve this puzzle
Vain attempts have been made to
construct q magic square by. what is
called knights tour" over the chess
board, numbering each square that the
knight visits in succession 1. 2, 8. 4,
etc, and it has been done with the
excention of the two diagonals which
50 far have baff7ed all efforts But il
is not certain that 1t cannot be done.
Here is one more unsolved problem
in numbers We al know that a.