Answer :
Place this rather lengthy number
on the
ribbon, 0212765957446808510638297872340425531914393617.
It may be
multiplied by any number up to 46 inclusive to give the same order of
figures in the ring.
The number previously given can be multiplied by
any
number up to 16.
I made the limit 9 in order to put readers off the
scent.
The fact is these two numbers are simply the recurring decimals
that equal 1/17 and 1/47 respectively.
Multiply the one by seventeen
and
the other by fortyseven, and you will get all nines in each case.
In transforming a vulgar fraction, say 1/17, to a
decimal fraction, we
proceed as below, adding as many noughts to the dividend as we like
until
there is no remainder, or until we get a recurring series of figures,
or
until we have carried it as far as we require, since every additional
figure in a neverending decimal carries us nearer and nearer to
exactitude.
17) 
100 
(.058823 

85 

—— 

150 

136 

—— 

140 

136 

—— 

40 

34 

—— 

60 

51 

—— 

9 
Now, since all powers of 10 can only contain
factors of the
powers of 2
and 5, it clearly follows that your decimal never will come to an end
if
any other factor than these occurs in the denominator of your vulgar
fraction.
Thus, ½, 1/4, and 1/8 give us the exact decimals,
.5, .25,
and .125; 1/5 and 1/25 give us .2 and .04; 1/10 and 1/20 give us .1 and
.05: because the denominators are all composed of 2 and 5
factors.
But
if
you wish to convert 1/3, 1/6, or 1/7, your division sum will never end,
but you will get these decimals, .33333, etc., .166666, etc., and
.142857142857142857, etc., where, in the first case, the 3 keeps on
repeating for ever and ever; in the second case the 6 is the repeater,
and in the last case we get the recurring period of 142857.
In the case
of 1/17 (in "The Ribbon Problem") we find the circulating period to be
.0588235294117647.
Now, in the division sum above, the successive
remainders are 1, 10,
15,
14, 4, 6, 9, etc., and these numbers I have inserted around the inner
ring of the diagram.
It will be seen that every number from 1 to 16
occurs once, and that if we multiply our ribbon number by any one of
the
numbers in the inner ring its position indicates exactly the point at
which the product will begin.
Thus, if we multiply by 4, the product
will
be 235, etc.; if we multiply by 6, 352, etc.
We can therefore multiply
by
any number from 1 to 16 and get the desired result.
The kernel of the puzzle is this: Any prime
number, with the
exception of
2 and 5, which are the factors of 10, will exactly divide without
remainder a number consisting of as many nines as the number itself,
less
one. Thus 999999 (six 9's) is divisible by 7, sixteen 9's are divisible
by 17, eighteen 9's by 19, and so on.
This is always the case, though
frequently fewer 9's will suffice; for one 9 is divisible by 3, two by
11, six by 13, when our ribbon rule for consecutive multipliers breaks
down and another law comes in.
Therefore, since the 0 and 7 at the ends
of the ribbon may not
be removed, we must seek a fraction with a prime
denominator ending in 7 that gives a full period circulator.
We try 37,
and find that it gives a short period decimal, .027, because 37 exactly
divides 999; it, therefore, will not do.
We next examine 47, and find
that it gives us the full period circulator, in 46 figures, at the
beginning of this article.
If you cut any of these full period circulators in
half and
place one
half under the other, you will find that they will add up all 9's; so
you
need only work out one half and then write down the complements.
Thus,
in
the ribbon above, if you add 05882352 to 94117647 the result is
99999999,
and so with our long solution number.
Note also in the diagram above
that
not only are the opposite numbers on the outer ring complementary,
always
making 9 when added, but that opposite numbers in the inner ring, our
remainders, are also complementary, adding to 17 in every case.
I ought
perhaps to point out that in limiting our multipliers to the first nine
numbers it seems just possible that a short period circulator might
give
a solution in fewer figures, but there are reasons for thinking it
improbable.
