I am reading about a problem where we have a rectangular board, and we have to show that it is impossible to complete a circuit of the board if both sides have odd length.
Circuit is a sequence if moves of $1$ square vertically or horizontally, which visits all the squares exactly once and ends to the starting square.
My question is about the solution provided which goes as follows:
The number of moves made must equal the number of squares. After an odd number of moves, the color of the current square is different from the color of the starting square. So for odd number of moves we can't form a circuit as we can't return to the starting square.
So the solution uses the approach of coloring the chessboard, and although I understand it, it seems to me something is missing. I.e.
How does the explanation cover the case that after an odd number of moves we could end up adjacent to the starting square and hence form a circuit? In theory it seems to me that it could happen i.e. hasn't been dis proven. Or is that already covered in a subtle way I am not understanding?