Is there a simple mathematical algorithm where you can get a CLOSED knight' tour on an n x m chessboard? I need a way to prove that it is mathematically possible or impossible to have a closed knight's tour on such a board.
I have puzzled over many different websites such as this one, but I can't seem to understand the techniques used and how you can explain those. Maybe it's just me being daft, but I will really appreciate a simple, easy explanation of the techniques used in one of these puzzles!
I've understood that you cannot have a closed tour on a chessboard where n x m is odd because the knight always moves to a different-coloured square every move, and if there are an odd number of turns the knight won't be able to to return to the same coloured square that it started with.
After weeks, I still don't fully understand the problem!
How should I solve and attempt this difficult puzzle?
NOTE: This diagram is the one that confused me -- the dotted lines don't make any sense to me -- how can a knight get back to the dotted lines without getting in the way of the next steps and filling in all the gaps later:
I'll explain the diagram you ask about.
It illustrates how to extend a closed or open tour on an $n\times m$ board to such a tour on an $n\times(m+4)$ board. To do this you need an open tour on an $n\times4$ board that starts and ends in the two adjacent squares just above the bottom left corner.
Here it is illustrated for the specific case of extending a $6\times5$ closed tour to a $6\times 9$ closed tour.
One move out of the corner of the board is removed, and the two squares of that move are connected to the two ends of the open tour on the $n\times 4$ board. This same method works for a tour on any $n\times m$ board with $n\ge 6$ because there are $n\times 4$ open tours with the required end points for all such sizes.