The N-Queens problem is a classic example for a constraint satisfaction problem
: given a chessboard
of size NxN, place N queens on the board such that none of them can hit any other in one move.
There are 12 distinct solutions for the 8-queens problem - two solutions are not distinct if you can obtain one from another by rotating or flipping the chessboard. Clearly, there are no solutions for the 2-queens and 3-queens problems.
There’s a nice on-line N-Queens demo at http://kaminari.scitec.kobe-u.ac.jp/PrologCafe/Queens.html, where you can set N and let a Java applet find the solutions for you.
At the time of writing, Google returns about 12,000 hits when searching for ‘n-queens’, and there are dozens – if not more - of methods and algorithms for finding solutions.