9.--_The Two Errand Boys._

All that is necessary is to add the two distances at which they meet to
twice their difference. Thus 720 + 400 + 640 = 1760 yards, or one mile,
which is the distance required. Or, put another way, three times the
first distance less the second distance will always give the answer, only
the first distance should be more than two-thirds of the second.

100.--_On the Ramsgate Sands._

Just six different rings may be formed without breaking the conditions.
Here is one way of effecting the arrangements.

A B C D E F G H I J K L M
A C E G I K M B D F H J L
A D G J M C F I L B E H K
A E I M D H L C G K B F J
A F K C H M E J B G L D I
A G M F L E K D J C I B H

Join the ends and you have the six rings.

Lucas devised a simple mechanical method for obtaining the _n_ rings that
may be formed under the conditions by 2_n_+1 children.

101.--_The Three Motor-Cars._

The only set of three numbers, of two, three, and five figures
respectively, that will fulfil the required conditions is 27 x 594 =
16,038. These three numbers contain all the nine digits and 0, without
repetition; the first two numbers multiplied together make the third, and
the second is exactly twenty-two times the first. If the numbers might
contain one, four, and five figures respectively, there would be many
correct answers, such as 3 x 5,694 = 17,082; but it is a curious fact
that there is only one answer to the problem as propounded, though it is
no easy matter to prove that this is the case.

102.--_A Reversible Magic Square._

[Illustration:

11 77 62 29
69 22 17 71
27 61 79 12
72 19 21 67 ]

It will be seen that in the arrangement given every number is different,
and all the columns, all the rows, and each of the two diagonals, add up
179, whether you turn the page upside down or not. The reader will notice
that I have not used the figures 3, 4, 5, 8, or 0.

103.--_The Tube Railway._

There are 640 different routes. A general formula for puzzles of this
kind is not practicable. We have obviously only to consider the
variations of route between B and E. Here there are nine sections or
"lines," but it is impossible for a train, under the conditions, to
traverse more than seven of these lines in any route. In the following
table by "directions" is meant the order of stations irrespective of
"routes." Thus, the "direction" BCDE gives nine "routes," because t