chessboard problems

Question

I recently looked into some of these problems and found out they are so amazing.

If a chessboard were to have wheat placed upon each square, such that one grain were placed on the first square, two on the second, four on the third, and so on (doubling the number of grains on each subsequent square), how many grains of wheat would be on the chessboard at the finish?

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A and B alternately put white and black knights on the squares of a chessboard, which are unoccupied. In addition, a knight may not be placed on a square threatened by an enemy knight (of the other color). The loser, is the one who cannot move any more. Who wins?

Can anyone suggest to me more such chessboard problems?

Answer 1

A classic.

Remove two diagonally opposite corners of the board. Can you cover it with dominoes each of which covers two adjacent squares?

Answer 2

Starting and ending on diagonally opposite corners of the board, can a knight visit each square exactly once?

Answer:

No. It must make exactly $63$ moves to do the above, but must start and end on a square of the same color. However, the color of the square the knight is on changes every time it makes a move.

Answer 3

How many ways are there of placing 8 queens on a chessboard so that no two threaten each other?

And I can't resist: in ordinary chess, who has a winning strategy or who can force a draw?