Expands on the original eight queens problem in which you try to figure out how to put eight queens on a chess board in such a way that none of them can attack any of the others. NQueens@home
attempts to find solutions if you increased the number of boards and queens to the value N, which most recently is 26. Really, you wouldn't want to try to figure that out without the help of a distributed computing network.
This isn't the only project devoted to figuring out chess problems. Chess960@home focuses on Fischer Random Chess, a twist on classical chess in which pieces start in different positions.