Chess problem 3

Chess Month continues with another very chaotic chess challenge.

1. What should white do?
   
2. Bengt has come up with another peculiar game. It's rather zen-like, or so Bengt thinks, anyway. Others might say "boring" is a more fitting description. Anyway. There are three pieces, all identical. The board is a chessboard, and it is empty when the game starts. In each turn, the player can add a piece to the board, or remove one, or move one from any square to any other empty square. You are not allowed to make a move so that you get a situation on the board which you have had before. (A different piece in the same square counts as the same situation. You can't add a piece after it has been removed.)
   How many turns can the game last, at the most?
   
3. This game could also be played with two players. If the winner is the first person who is not able to make a move, who wins - the player who starts, or the other one? (Assume that they are using optimal strategies, that they want to win, and so on.)
   
4. How many turns will it last?
   
5. Do questions b, c and d again, but with n pieces instead.
   

Chess problem 2

Bengt doesn't often play chess. It's not that he doesn't like it, but he gets upset when he loses, and people don't like Bengt when he's upset. So most of his friends don't play with him anymore. But he still likes to make up weird chess problems, usually involving completely unrealistic situations, that would never appear in an actual game. Here's another one:

1. What should white do? Bengt can come up with a rather simple solution that leads to mate in four moves, and a more unusual one that leads to mate in only two moves.
   
2. Since Bengt doesn't have anyone to play with, he makes up games and challenges for himself. His latest idea is this: You place a number of white pawns in any of the 16 squares forming one quadrant of the board, let's say the upper left. Then you place a number of black pawns in the quadrant diagonally opposite, in this case the lower right. When you've placed all the pawns, you suspend the board, balancing it on the diagonal between the quadrants, so in this case you lift the lower left and the upper right corner, so the board can pivot around that diagonal. The aim of the game is to get as many more black pawns than white pawns as possible (black pawns minus white pawns should be as big as possible), but so that the board does not tip over to the black side. How many is it possible to get?
   Assume that the pawns are pointlike and stand in the middle of the square. You are allowed to use more than one chess set, that it, more than eight of each type of pawn.
   

Chess problem 1

There are many kinds of good problems, not just ones involving complicated maths and physics calculations. So, this month, it's Chess Month! Here's the first one:

1. What should white do?
   
2. We can't avoid adding a little bit of maths after all. Think of the chessboard as a mathematical pattern. Each point on the board is either black or white. The colours are equivalent; in other words, if you invert the colours and rotate the board, you get the same thing. The length of the side of a square is 1, and there are no points outside the squares. Now, imagine drawing a line so that every point on the line is white. What is the upper limit for the length of such a line?
   
3. Is there any maximum length at all?