Recreational chess questions based on the knights

Question

I basically know whether the following statements are true, but I would like to know how they are proved.

1. A knight kept anywhere on an empty chess board can not reach its adjacent square in exactly 8 moves.
2. It is possible to fill the entire board with black and white knights such that no knight can kill another knight of the other colour.
3. A knight that reaches a square in $n$ moves can not reach it in $n+1$ moves but can reach it in $n+2$ moves in atleast $n$ ways.

P.S. I'm not sure if my tags are ok.

Answer 1

Think about the coloring of a chess board. If a knight starts on a white square, what color of square is it on in one move? In two moves? In $2n$ moves? In $2n+1$ moves?

If you go through this carefully, this should answer all of the questions.

Answer 2

For part three you need to think about the $n$ steps you take to get to the final square. How can you use each of these steps to make a new path containing two additional steps (that will give you $n$ ways). [sorry that doesn't quite do it, I'll think further, the paths I was thinking of need not be distinct, but you can use the fact that every square is adjacent by knight's move to at least two others to fix my idea]

Part $2$ should probably refer to attacking knights of the same colour. Some simple observations about a knights tour of the whole chessboard show that if there are knights of both colours, there must be a change of colour along the tour.