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 × 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 × 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 there are three ways of getting from B to C, and three ways of getting from D to E. But the "direction" BDCE admits of no variation; therefore yields only one route. 2 two-line directions of 3 routes -- 6 1 three-line " " 1 " -- 1 1 " " " 9 " -- 9 2 four-line " " 6 " -- 12 2 " " " 18 " -- 36 6 five-line " " 6 " -- 36 2 " " " 18 " -- 36 2 six-line " " 36 " -- 72 12 seven-line " " 36 " -- 432 ---- Total 640 We thus see that there are just 640 different routes in all, which is the correct answer to the puzzle. 104.--_The Skipper and the Sea-Serpent._