I will give my complete working of the solution, so that readers may see
how easy it is when you know how to proceed. And first of all, as there
is an even number of rings, I will say that they may all be taken off in
one-third of (2^(n + 1) - 2) moves; and since n in our case is 14, all
the rings may be taken off in 10,922 moves. Then I say 10,922 - 9,999 =
923, and proceed to find the position when only 923 out of the 10,922
moves remain to be made. Here is the curious method of doing this. It is
based on the binary scale method used by Monsieur L. Gros, for an
account of which see W.W. Rouse Ball's _Mathematical Recreations_.
Divide 923 by 2, and we get 461 and the remainder 1; divide 461 by 2,
and we get 230 and the remainder 1; divide 230 by 2, and we get 115
and the remainder nought. Keep on dividing by 2 in this way as long as
possible, and all the remainders will be found to be 1, 1, 1, 0, 0, 1,
1, 0, 1, 1, the last remainder being to the left and the first
remainder to the right. As there are fourteen rings and only ten
figures, we place the difference, in the form of four noughts, in
brackets to the left, and bracket all those figures that repeat a
figure on their left. Then we get the following arrangement: (0 0 0 0)
1 (1 1) 0 (0) 1 (1) 0 1 (1). This is the correct answer to the puzzle,
for if we now place rings below the line to represent the figures in
brackets and rings on the line for the other figures, we get the
solution in the required form, as below:--
O O O OO
-------------------------
OOOO OO O O O
This is the exact position of the rings after the 9,999th move has been
made, and the reader will find that the method shown will solve any
similar question, no matter how many rings are on the tiring-irons. But
in working the inverse process, where you are required to ascertain the
number of moves necessary in order to reach a given position of the
rings, the rule will require a little modification, because it does not
necessarily follow that the position is one that is actually reached in
course of taking off all the rings on the irons, as the reader will
presently see. I will here state that where the total number of rings is
odd the number of moves required to take them all off is one-third of
(2^(n + 1) - 1).
With n rings (where n is _odd_) there are 2^n positions counting all on
and all off. In (1/3)(2^(n + 1) + 2) positions they are all removed. The
number of positions not used is (1/3)(2^n - 2).
With n rings (where n is _even_) there are 2^n positions counting all on
and all off. In (2^(n + 1) + 1) positions they are all removed. The
number of positions not used is here (1/3)(2^n - 1).
It will be convenient to tabulate a few cases.
+--------+------------+-----------+-----------+
| No. of | Total | Positions | Positions |
| Rings. | Positions. | used. | not used. |
+--------+------------+-----------+-----------+
| 1 | 2 | 2 | 0 |
| 3 | 8 | 6 | 2 |
| 5 | 32 | 22 | 10 |
| 7 | 128 | 86 | 42 |
| 9 | 512 | 342 | 170 |
| | | | |
| 2 | 4 | 3 | 1 |
| 4 | 16 | 11 | 5 |
| 6 | 64 | 43 | 21 |
| 8 | 256 | 171 | 85 |
| 10 | 1024 | 683 | 341 |
+--------+------------+-----------+-----------+
Note first that the number of _positions used_ is one more than the
number of _moves_ required to take all the rings off, because we are
including "all on" which is a position but not a move. Then note that
the number of _positions not used_ is the same as the number of _moves
used_ to take off a set that has one ring fewer. For example, it takes
85 moves to remove 7 rings, and the 42 positions not used are exactly
the number of moves required to take off a set of 6 rings. The fact is
that if there are 7 rings and you take off the first 6, and then wish to
remove the 7th ring, there is no course open to you but to reverse all
those 42 moves that never ought to have been made. In other words, you
must replace all the 7 rings on the loop and start afresh! You ought
first to have taken off 5 rings, to do which you should have taken off 3
rings, and previously to that 1 ring. To take off 6 you first remove 2
and then 4 rings.