FIRST METHOD. +----------+---------------------------+-----------------------+-----------+ | Total No.| L MOVEMENT. | U MOVEMENT. | | | of +-------------+-------------+----------+------------+ Total No. | | Counters.| No. of | No. of | No. of | No. of | of Moves. | | | Counters. | Moves. |Counters. | Moves. | | +----------+-------------+-------------+----------+------------+-----------+ | 4n | n-1 and n |2(n-1) squared+5n-7 | 2n+1 |2n squared+3n+1 |4(n squared+n-1) | | 4n-2 | n-1 " n |2(n-1) squared+5n-7 | 2n-1 |2(n-1) squared+3n-2|4n squared-5 | | 4n+1 | n " n+1 |2n squared+5n-2 | 2n |2n squared+3n-4 |2(2n squared+4n-3)| | 4n-1 | n-1 " n |2(n-1) squared+5n-7 | 2n |2n squared+3n-4 |4n squared+4n-9 | +----------+-------------+-------------+----------+------------+-----------+ SECOND METHOD. +---------+--------------------------+-------------------------+-----------+ |Total No.| L MOVEMENT. | U MOVEMENT. | | | of +-------------+------------+----------+--------------+ Total No. | |Counters.| No. of | No. of | No. of | No. of | of Moves. | | | Counters. | Moves. | Counters.| Moves. | | +---------+-------------+------------+----------+--------------+-----------+ | 4n | n and n |2n squared+3n-4 | 2n | 2(n-1) squared+5n-2 |4(n squared+n-1) | | 4n-2 | n-1 " n-1 |2(n-1) squared+3n-7| 2n | 2(n-1) squared+5n-2 |4n squared-5 | | 4n+1 | n " n |2n squared+3n-4 | 2n+1 | 2n squared+5n-2 |2(2n squared+4n-3)| | 4n-1 | n " n |2n squared+3n-4 | 2n-1 | 2(n-1) squared+5n-7 |4n squared+4n-9 | +---------+-------------+------------+----------+--------------+-----------+ More generally we may say that with m counters, where m is even and greater than 4, we require (m squared + 4m - 16)/4 moves; and where m is odd and greater than 3, (m squared + 6m - 31)/4 moves. I have thus shown the reader how to find the minimum number of moves for any case, and the character and direction of the moves. I will leave him to discover for himself how the actual order of moves is to be determined. This is a hard nut, and requires careful adjustment of the L and the U movements, so that they may be mutually accommodating. 216.--THE EDUCATED FROGS. The following leaps solve the puzzle in ten moves: 2 to 1, 5 to 2, 3 to 5, 6 to 3, 7 to 6, 4 to 7, 1 to 4, 3 to 1, 6 to 3, 7 to 6. 217.--THE TWICKENHAM PUZZLE. Play the counters in the following order: K C E K W T C E H M K W T A N C E H M I K C E H M T, and there you are, at Twickenham. The position itself will always determine whether you are to make a leap or a simple move. 218.--THE VICTORIA CROSS PUZZLE.