| 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.

In solving this puzzle there were two things to be achieved:  first, so to manipulate the counters that the word VICTORIA should read round the cross in the same direction, only with the V on one of the dark arms; and secondly, to perform the feat in the fewest possible moves.  Now, as a matter of fact, it would be impossible to perform the first part in any way whatever if all the letters of the word were different; but as there are two I’s, it can be done by making these letters change places—­that is, the first I changes from the 2nd place to the 7th, and the second I from the 7th place to the 2nd.  But the point I referred to, when introducing the puzzle, as a little remarkable is this:  that a solution in twenty-two moves is obtainable by moving the letters in the order of the following words:  “A VICTOR!  A VICTOR!  A VICTOR I!”