Johnstown weekly Democrat. (Johnstown, Cambria County, Pa.) 1889-1916, 1890-01-31
SOME ANCIENT PUZZLES.
THEY SEEM TO BE INVESTED WITH
A VIGOROUS IMMORTALITY.


That Are Played with Numerels-Inter-
estlng Problems In Arithmetic.
There are certain problems, chiefly
arithmetical, endued with a vigorous
Immortality No matter how often the
solution is printed, or how widely an ex-
haustive answer is published, the ques-
tion comes up again, before the ink is
fairly dried, to the lips of hundreds who
have not seen the reply, or who either
cannot understand it or will not accent
it. There are several of these which we
have printed SO often, but which still
kept coming, that to save further time
we struck off hundred proofs of each,
and mailed one to the inquirers in sue
. cession without comment. These proofs
are exhausted, and we have accumulated
from a score or more of correspondents
$the same old questions, with urgent re-
quests for fresh solution. We notice
that The Brooklyn Eagle has been strug-
Cling with one Of these. The editor who
has charge of that department is very
clever, and we think he is playing a little
with his inquisitor.
The original question sent to us forty
years ago and involving the same point
submitted to The Eagle was, how to find
the product of e19 los. I'D. 3f. multi
piled by itself. OF course if the parts of
the pounds were stated as fractions, and
the pounds as whole numbers, then
10 959-960ths could be multiplied by
itself. But money of account has not
two dimensions. If a table is feet wide
and feet long, then 4x4=16 feet
and we have the number of square
feet on the surface Five times five
pounds are e25, but five pounds times
five pounds is unmeaning, as money
does not measure itself in that fashion.
Twice two children are four children,
but twice children two children has no
meaning. So nineteen pounds, nineteen
shillings, eleven pence, three farthings
times nineteen pounds, nineteen shill-
ings, eleven pence, three farthings." is
utter nonsense.
TROUBLESOME MILLS.
The next puzzle on the list, and one
which comes the oftenest to our desk, in
some form of a problem which proposes
to divide whole sum into fractions that
together did not make the dividend.
The original of this in our columns was
an answer to an actual case where a
man in his will had devised one-third,
one-fourth, one-fifth and one-sixth of
his property respectively to his four
children, supposing that he had thus de-
xriGn hn NrrhI Of
tions mentioned only made nineteen
twentieths of a whole number. This is
easily seen if they are reduced to a COM
mon denominator. One-third is twenty
sixtieths, one-fourth is fifteen-sixtieths,
,one-fifth is twelve-sixtieths, and one
sixth is ten-siXtieths, which together
make but fifty-seven siXtieths, leaving
three-sixtieths (or 1-20th) to make up
the whole number.
This puzzle reappears in some form
every few days the year round. 1t is
answered on the same principle involved
in the Interpretation of the Arabs will.
He had fifteen horses and four sons. He
devised his estate, giving one son a half,
another quarter, another an eighth
and the last a sixteenth. They found it
impossible to agree on a division. The
eldest son insisted that as seven horses
would not be half of fifteen he should
have eight; but the other sons objected,
and as neither one-half, one-fourth, one
eighth nor one-sixteenth would give
either son an even lot they had fierce
dispute over the division. A venerable
sheik rode up just as the quarrel was at its
height, and to compose their differences
dismounted and generously offered to
add his mare to the fifteen belonging to
the estate, agreeing that each should
take his allotted share from the whole
sixteen, only stipulating that his should
be the last selected The addition made
an easy solution of the difficulty. The
first then took eight as his half of
the sixteen, the next took four for his
quarter, the third took two for his
eighth, and the fourth took one for his
sixteenth. AS this made but fifteen the
sheik mounted his mare and rode away.
The Arab boys regarded it as miracle,
and exclaimed that Allah had given a
horse to the sheik for his generous inter-
ference. In spite of this oft told tale the
problem still survives and annually puz-
zles hundreds of our countrymen.
JUGGLING WITH FRACTIONS.
A more recent problem which we have
already answered several times, but
which is repeated every week from some
quarter, is the division of one fraction
by another. The original question which
we answered several years ago was:
"What is the quotient of two-thirds di-
vided by one-half?" The unthinking per-
son would say that as the half of two
thirds is one-third, this must be the solu-
tion of the problem, but Daboll will east
ly refute it. The quotient of 2-3 divided
by is 1-3; that is, will go in 2-3 one
and one-third times. The last form of
the problem, received as we write this,
is to find the quotient of 1 divided by !,
two partners in leading banking house,
having disputed, as they say, all one day
over the result, the senior maintaining
that l divided 10y is ], and defying any
one to refute it. We answer that when
1 is divided by ! the quotient is 2; that
is, will be found two times in !. If 6
be divided by J the answer is 12; that is,
there are twelve halves in six. We
should beg pardon of our readers for re-
peating these demonstrations if it were