
Answer :
The number of ways
in which nine things may be
arranged in a row without any restrictions is
1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9
= 362,880.
But we are told that the two circular rings must never be
together; therefore we must deduct the number of times that this would
occur.
The number is
1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 = 40,320 × 2
= 80,640, because if we consider the two circular links to be
inseparably joined together they become as one link, and eight links
are capable of 40,320 arrangements; but as these two links may always
be put on in the orders AB or BA, we have to double this number, it
being a question of arrangement and not of design. The deduction
required reduces our total to 282,240.
Then one of our links is of a
peculiar form, like an 8.
We have therefore the option of joining on
either one end or the other on every occasion, so we must double the
last result.
This brings up our total to 564,480.
Every link may be
put on in one of
two ways.
If we join the first finger and thumb of our left hand
horizontally, and then link the first finger and thumb of the right
hand, we see that the right thumb may be either above or
below.
But in
the case of our chain we must remember that although that 8shaped link
has two independent ends
it is like every other
link in having only two sides—that
is,
you cannot turn over one end without turning the other at the same time.
Assume that each
link
has a black side and a side painted white.
Now, if it were stipulated
that (with the chain lying on the table, and every successive link
falling over its predecessor in the same way, as in the diagram) only
the white sides should be uppermost as in A, then the answer would be
564,480, as above—ignoring for the present all reversals of
the completed chain.
If, however, the first link were allowed to be
placed either side up, then we could have either A or B, and the answer
would be 2 × 564,480 = 1,128,960; if two
links might be placed either way up, the answer would be
4 × 564,480; if three links, then
8 × 564,480, and so on.
Since, therefore,
every link may be placed either side up, the number will be 564,480
multiplied by 2^{9},
or by 512.
This raises our
total to 289,013,760.
Any given
arrangement three of the other arrangements may be obtained by simply
turning the chain over through its entire length and by reversing the
ends.
Thus C is really the same as A, and if we turn this page upside
down, then A and C give two other arrangements that are still really
identical.
Thus to get the correct answer to the puzzle we must divide
our last total by 4, when we find that there are just 72,253,440
different ways in which the smith might have put those links
together.
In other words, if the nine links had originally formed a piece of
chain, and it was known that the two circular links were separated,
then it would be 72,253,439 chances to 1 that the smith would not have
put the links together again precisely as they were arranged before!
