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.

[Illustration]

[Illustration: 

o o o o o * *
{-------------

o o o o * o
1{------------- o

o o o o   o
2{-------------
o   o

o o o o   * *
3{-------------
o

o o o o     *
4{-------------
o o

o o * o
5{-------------
o o o

]

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?

418.—­SUCH A GETTING UPSTAIRS.

In a suburban villa there is a small staircase with eight steps, not counting the landing.  The little puzzle with which Tommy Smart perplexed his family is this.  You are required to start from the bottom and land twice on the floor above (stopping there at the finish), having returned once to the ground floor.  But you must be careful to use every tread the same number of times.  In how few steps can you make the ascent?  It seems a very simple matter, but it is more than likely that at your first attempt you will make a great many more steps than are necessary.  Of course you must not go more than one riser at a time.