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(Unclassified Problems.)
The illustration represents one of the most ancient of all mechanical
puzzles. Its origin is unknown. Cardan, the mathematician, wrote about
it in 1550, and Wallis in 1693; while it is said still to be found in
obscure English villages (sometimes deposited in strange places, such as
a church belfry), made of iron, and appropriately called "tiring-irons,"
and to be used by the Norwegians to-day as a lock for boxes and bags. In
the toyshops it is sometimes called the "Chinese rings," though there
seems to be no authority for the description, and it more frequently
goes by the unsatisfactory name of "the puzzling rings." The French call
it "Baguenaudier."
The puzzle will be seen to consist of a simple _loop_ of wire fixed in a
handle to be held in the left hand, and a certain number of _rings_
secured by _wires_ which pass through holes in the _bar_ and are kept
there by their blunted ends. The wires work freely in the bar, but
cannot come apart from it, nor can the wires be removed from the rings.
The general puzzle is to detach the loop completely from all the rings,
and then to put them all on again.
Now, it will be seen at a glance that the first ring (to the right) can
be taken off at any time by sliding it over the end and dropping it
through the loop; or it may be put on by reversing the operation. With
this exception, the only ring that can ever be removed is the one that
happens to be a contiguous second on the loop at the right-hand end.
Thus, with all the rings on, the second can be dropped at once; with the
first ring down, you cannot drop the second, but may remove the third;
with the first three rings down, you cannot drop the fourth, but may
remove the fifth; and so on. It will be found that the first and second
rings can be dropped together or put on together; but to prevent
confusion we will throughout disallow this exceptional double move, and
say that only one ring may be put on or removed at a time.
We can thus take off one ring in 1 move; two rings in 2 moves; three
rings in 5 moves; four rings in 10 moves; five rings in 21 moves; and if
we keep on doubling (and adding one where the number of rings is odd) we
may easily ascertain the number of moves for completely removing any
number of rings. To get off all the seven rings requires 85 moves. Let
us look at the five moves made in removing the first three rings, the
circles above the line standing for rings on the loop and those under
for rings off the loop.
Drop the first ring; drop the third; put up the first; drop the second;
and drop the first--5 moves, as shown clearly in the diagrams. The dark
circles show at each stage, from the starting position to the finish,
which rings it is possible to drop. After move 2 it will be noticed that
no ring can be dropped until one has been put on, because the first and
second rings from the right now on the loop are not together. After the
fifth move, if we wish to remove all seven rings we must now drop the
fifth. But before we can then remove the fourth it is necessary to put
on the first three and remove the first two. We shall then have 7, 6, 4,
3 on the loop, and may therefore drop the fourth. When we have put on 2
and 1 and removed 3, 2, 1, we may drop the seventh ring. The next
operation then will be to get 6, 5, 4, 3, 2, 1 on the loop and remove 4,
3, 2, 1, when 6 will come off; then get 5, 4, 3, 2, 1 on the loop, and
remove 3, 2, 1, when 5 will come off; then get 4, 3, 2, 1 on the loop
and remove 2, 1, when 4 will come off; then get 3, 2, 1 on the loop and
remove 1, when 3 will come off; then get 2, 1 on the loop, when 2 will
come off; and 1 will fall through on the 85th move, leaving the loop
quite free. The reader should now be able to understand the puzzle,
whether or not he has it in his hand in a practical form.
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The particular problem I propose is simply this. Suppose there are
altogether fourteen rings on the tiring-irons, and we proceed to take
them all off in the correct way so as not to waste any moves. What will
be the position of the rings after the 9,999th move has been made?

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