Funny chess exercise

Can you solve them?

I got this question from my math teacher, and it took me a while to solve the problem:

A knight, as the only piece on the board, stands on a1. Is it possible for the knight to visit every single square on the board once, and end up on h8? If yes, please show the route.

When I solved it I got another exercise:

In a special version of chess, the player makes two moves in a row on each turn. Explain why black, in theory, has no chance of winning.

The answer to the second problem is actually very simple.

If you check someone on your first move, then for your second move can you take their king?

SOLUTION TO TASK 1
A chess board consists of 64 squares. The knight stands on one square, meaning there are 63 squares left. Whenever a knight moves, it moves from  one colour to antoher. a1 is a black square, so after 63 moves (an odd number of moves), the knight lands on a white square. But since h8 is a black square, there exists no route as described in the task.

SOLUTION TO TASK 2
Before the game starts, there are two outcomes:

• It exists a not-losing strategy for white, meaning the game always ends in a draw or win for white. In this case, black cannot win.
   
• Else; black has a winning strategy. But since white moves first, he/she can move his knight to for instance f3 and then back to g1 (which is one move). This will reverse the game, like black moves first. Now white can use blacks' winning strategy. Again, black cannot win.

Fortunately, theory and practice are two different things, and black can win in one move:

1. f3 and g4, e6 and Qh4 mate.

lol white would have to be dumb.

but to skf95, couldn't black also copy whites idea with Nf6-Ng8?

As pointed out, black sure has winning chances (in practice) since we are just talking about perfect theory, which none players play.

rofl never thought of that... lol good point