TITLE: Knight tour problem??
QUESTION [5 upvotes]: Consider an n × n chess board. For what values of n is it possible to find a knight’s tour around the
board which uses every possible move just once (in one direction or the other).
Here on what factors does n depends??
Any Hints.
Is every possible move means, we have to visit every possible path on chess board 
If that's the meaning that the problem can be converted in euler's tour problem for which necessary and sufficient condition is every node must have even number of edges. And this condition is true for chess board.

REPLY [4 votes]: I think the question is, draw a graph where the vertices are the squares of a chessboard, and there's an edge between two  vertices if the corresponding squares are a knight's move from each other; then find a path on this graph that uses each edge exactly once. 
So we are being asked to find an Eulerian path in the graph. This can be done if and only if there are at most two vertices of odd degree. 
The cases  $n\le3$ are trivial. 
If $n\ge4$, then any square next to a corner has degree $3$, and there are $8$ such squares, so no Eulerian path.