# More Chessboard

You are free to comment your solutions, questions and remarks..

You have two kinds of allowed moves. One move is to jump with a piece over another piece in a horizontal or vertical direction like this:

and to jump with a piece over another piece in (any) diagonal direction like this:

Challenge 1:

Is it possible to move four pieces from the lower right corner to the upper right corner like this:

Challenge 2:

Is it possible to move four pieces from the lower right corner to the upper left corner like this:

Challenge 3:

Is it possible to move nine pieces from the lower right corner to the upper right corner like this:

Challenge 4:

Is it possible to move nine pieces from the lower right corner to the upper left corner like this:

????

For details see this
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### 3 Responses to More Chessboard

1. Annika says:

4 colors for number 4. Solved.

2. Tjl says:

1 is trivial (just separate the columns and keep hopping). 2 reduces to two instances of 1. 3 is not possible because number of pieces on black/white squares changes and neither of the moves allows them to change.

4 seems less trivial… at least I don’t see any easy way to prove it impossible

3. Annika says:

4 is impossible. It can be proved by coloring the board with four colors (say, red, blue, yellow, green), so that even if the piece is moved, the color underneath it stays the same. Let’s assume the board is colored this way now. Now, we can see, that the upper left 3×3 corner has different number of either red, blue, yellow or green squares than the lower right 3×3 corner. The amount of x-colored squares covered with the playing pieces should stay the same even if we move the pieces, so it’s impossible to move the 9 pieces from the lower right corner to the upper left corner.