. . . . +---+---+---+---+---+---+---+ . . . .

| | | | | | | |

. . . . +---+---+---+---+---+---+---+ . . . .

| | | | | | | |

. . . . +---+---+---+---+---+---+---+ . . . .

| | | | | | | |

. . . . +---+---+---+---+---+---+---+ . . . .

| | | | | | | |

. . . . +---+---+---+---+---+---+---+ . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

Each square can have at most one piece on it, represented by a letter - I'm using different letters so that I can say which piece I'm talking about, there is otherwise no difference between pieces with different letters. A piece may move by jumping over an horizontally or vertically (but not diagonally) adjacent piece, removing it from the board.

A very trivial question is how many pieces I need on the board to be able to get one on the top row - the answer is obviously just one piece starting on the top row. But suppose I am restricted to not having any pieces on the top row to start with - how many pieces do I then need to be able to get one on the top row? That's almost as trivial - two pieces will do, one on the second row and one immediately below it on the third row:

. . . . +---+---+---+---+---+---+---+ . . . .

| | | | | | | |

. . . . +---+---+---+---+---+---+---+ . . . .

| | | | A | | | |

. . . . +---+---+---+---+---+---+---+ . . . .

| | | | B | | | |

. . . . +---+---+---+---+---+---+---+ . . . .

| | | | | | | |

. . . . +---+---+---+---+---+---+---+ . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

Piece B jumps over piece A to land on the top row, removing piece A in the process.

OK, make it harder by not allowing any pieces on the top two rows to start with - how many pieces do I then need on the board to start with? It's fairly easy to eliminate the possibilities of doing it with one, two or three pieces, but four is doable:

. . . . +---+---+---+---+---+---+---+ . . . .

| | | | | | | |

. . . . +---+---+---+---+---+---+---+ . . . .

| | | | | | | |

. . . . +---+---+---+---+---+---+---+ . . . .

| | | A | B | C | | |

. . . . +---+---+---+---+---+---+---+ . . . .

| | | D | | | | |

. . . . +---+---+---+---+---+---+---+ . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

Piece D jumps over piece A and removes it from the board, then piece C jumps over pieces B and D to land on the top row.

In increasing order of difficulty:

* What's the smallest number of starting pieces I need on the board to get one to the top row if they're not allowed to start on the top, second or third rows?

* What's the smallest number of starting pieces I need on the board to get one to the top row if they're not allowed to start on the top, second, third or fourth rows?

* What's the smallest number of starting pieces I need on the board to get one to the top row if they're not allowed to start on the top, second, third, fourth or fifth rows?

Not my original puzzle, by the way - but I'll acknowledge my source after people have had a go at solving it.

Gengulphus