Answer
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.

.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.

##### FIRST METHOD.

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 |

##### SECOND METHOD.

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.