The Stark County Democrat. [volume] (Canton, Ohio) 1833-1912, 1907-08-16

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 exact numbers what is
the length of that diagonal. The
novice will Mt 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
have It which way you like, but you
cannot have II both ways.
But let us take few puzzles that
have not been proved to be impossi.
/ble, but which, nevertheless, have not
been solved. First is the Round
Table problem. Nine persons are
stopping at a boarding house, and
they all sit down together to dinner
on 2S successive 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 1t to be done, II at all?
Try to write out the arrangements
for the nine persons. The present
writer, first propounded the question
and showed how six person may be
seated in ten ways, and has recently
solved the puzzle in the difficult case
of seven persons to be seated in fif-
teen days. The solution for eight per.
sons on twenty one days has also been
discovered. But can the reader show
how we are to seat those nine persons
on twenty eight days without any
body having the same pair of neigh
bors twice7
Here is another poser. II we write
down the number composed of II
ones-11,111, i'll III, i'll, ,111-and ask
you to find some number-other than
l or the number itself--that will
divide II 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 II are 2,071,72 and s..
1863, 3,222,357. Now add two more ones
to the number and we cannot tell
whether IL 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 IL is
impossible.
Everyone knows what magic
square is. Divide square into nine
divisions or cells and then place the
numbers / to o, one number In each
cell, so that they shall add up is in
every column every row and In each
OF the two diagonals. II is quite easy,
and there is only one way of doing
Il, because we do not count as differ
ent the arrangements obtained by
merely turning round the square and
rejecting II in a mirror.
Now, II we wish to make n magic
square OF the 16 numbers / TO IS,
there are Just $50 different ways OF
doing it-again not counting reversals
and reflections. This has been finally
proved OF recent years. But how
many magic square can be formed
with the rs numbers l to 20, nobody
knows. and we shall have to extend
our knowledge in certain directions
before we can hope to solve this puz
zle.
Vain attempts have been made 20
construct a magic square by what is
called q knights tour" over the
chessboard. numbering each square
that the knight visits in succession /,
2, 3, 4, etc. and II has been done with
the exception OF the two dlagonals,
which so far have bafied all efforts
But II is not certain that 1t cannot be
done.
Here is one more unsolved prob.
lem in numbers. We all know that
square number is a number multi
piled by itself; but a cube number Is
one multiplied twice by itself. Thus
is the cube OF 2, 27 the cube OF 8
and so on. Now, some whole mum.
bers are the sum OF two whole cubes
AS 80 is the sum of the cubes of z
and 3--others are the sums of two
fractional cubes-as 43 is the sum OF
cubes OF l.2 and 7-2-while other
whole numbers cannot be expressed
as the sum Of two cubes In any way
whatever. 1t is possible to say OF
any number from l to 100 whether
II is or is not the sum OF two cubes
except 66. Nobody in the world can
answer for this number? Can you