+--------+------------+-----------+-----------+
| 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.

418.—­SUCH A GETTING UPSTAIRS.

Number the treads in regular order upwards, 1 to 8.  Then proceed as follows:  1 (step back to floor), 1, 2, 3 (2), 3, 4, 5 (4), 5, 6, 7 (6), 7, 8, landing (8), landing.  The steps in brackets are taken in a backward direction.  It will thus be seen that by returning to the floor after the first step, and then always going three steps forward for one step backward, we perform the required feat in nineteen steps.

419.—­THE FIVE PENNIES.

[Illustration]

First lay three of the pennies in the way shown in Fig. 1.  Now hold the remaining two pennies in the position shown in Fig. 2, so that they touch one another at the top, and at the base are in contact with the three horizontally placed coins.  Then the five pennies will be equidistant, for every penny will touch every other penny.

420.—­THE INDUSTRIOUS BOOKWORM.

The hasty reader will assume that the bookworm, in boring from the first to the last page of a book in three volumes, standing in their proper order on the shelves, has to go through all three volumes and four covers.  This, in our case, would mean a distance of 91/2 in., which is a long way from the correct answer.  You will find, on examining any three consecutive volumes on your shelves, that the first page of Vol.  I. and the last page of Vol.  III. are actually the pages that are nearest to Vol.  II., so that the worm would only have to penetrate four covers (together, 1/2 in.) and the leaves in the second volume (3 in.), or a distance of 31/2 inches, in order to tunnel from the first page to the last.