**THE GRASSHOPPER PUZZLE.— solution**

Move the counters in the following order. The moves in brackets are to be made four times in succession. 12, 1, 3, 2, 12, 11, 1, 3, 2 (5, 7, 9, 10, 8, 6, 4), 3, 2, 12, 11, 2, 1, 2. The grasshoppers will then be reversed in forty-four moves.

The general solution of this problem is very difficult. Of course it can
always be solved by the method given in the solution of the last puzzle, if we
have no desire to use the fewest possible moves. But to employ a full economy of
moves we have two main points to consider. There are always what I call a lower
movement (L) and an upper movement (U). L consists in exchanging certain of the
highest numbers, such as 12, 11, 10 in our "Grasshopper Puzzle," with certain of
the lower numbers, 1, 2, 3; the former moving in a clockwise direction, the
latter in a non-clockwise direction. U consists in reversing the intermediate
counters. In the above solution for 12, it will be seen that 12, 11, and 1, 2, 3
are engaged in the L movement, and 4, 5, 6, 7, 8, 9, 10 in the U movement. The L
movement needs 16 moves and U 28, making together 44. We might also involve 10
in the L movement, which would result in L 23, U 21, making also together 44
moves. These I call the first and second methods. But any other scheme will
entail an increase of moves. You always get these two methods (of equal economy)
for odd or even counters, but the point is to determine just how many to involve
in L and how many in U. Here is the solution in table form. But first note, in
giving values to *n*, that 2, 3, and 4 counters are special cases,
requiring respectively 3, 3, and 6 moves, and that 5 and 6 counters do not give
a minimum solution by the second method—only by the first.

Total No. of Counters. |
L MOVEMENT. | U MOVEMENT. | Total No. of Moves. | ||

No. of Counters. |
No. of Moves. |
No. of Counters. |
No. of Moves. | ||

4n | n - 1 and n | 2(n - 1)² + 5n - 7 | 2n + 1 | 2n² + 3n + 1 | 4(n² + n - 1) |

4n - 2 | n - 1 " n | 2(n - 1)² + 5n - 7 | 2n - 1 | 2(n - 1)² + 3n - 2 | 4n² - 5 |

4n + 1 | n " n + 1 | 2n² + 5n - 2 | 2n | 2n² + 3n - 4 | 2(2n² + 4n - 3) |

4n - 1 | n - 1 " n | 2(n - 1)² + 5n - 7 | 2n | 2n² + 3n - 4 | 4n² + 4n - 9 |

Total No. of Counters. |
L MOVEMENT. | U MOVEMENT. | Total No. of Moves. | ||

No. of Counters. |
No. of Moves. |
No. of Counters. |
No. of Moves. | ||

4n | n and n | 2n² + 3n - 4 | 2n | 2(n - 1)² + 5n - 2 | 4(n² + n - 1) |

4n - 2 | n - 1 " n - 1 | 2(n - 1)² + 3n - 7 | 2n | 2(n - 1)² + 5n - 2 | 4n² - 5 |

4n + 1 | n " n | 2n² + 3n - 4 | 2n + 1 | 2n² + 5n - 2 | 2(2n² + 4n - 3) |

4n - 1 | n " n | 2n² + 3n - 4 | 2n - 1 | 2(n - 1)² + 5n - 7 | 4n² + 4n-9 |

More
generally we may say that with *m* counters, where *m* is even and
greater than 4, we require
^{(m² + 4m - 16)}/_{4} moves; and where
*m* is odd and greater than 3,
^{(m² + 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.

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