You are standing in the very corner of a 10 X 10 grid of dots. How many different ways are there to get to the opposite corner of the grid? You must travel through every node once, and only once. You cannot travel diagonally, and you may not go outside of the overall perimeter.

The question comments the movement among dots and not checkers. For the dots, one could travel horizontally and vertically as one likes. One might travel a longer journey from left lower corner to the right lower corner by means of travelling vertically upward to the top of the dot & turn to the right so as to move horizontally to the left corner top of the top. From then, make a downward turn vertically to reach the right hand corner of the dot. The travelling path can be done in such a way that one could move to one upward node & turn to the right node & move upward again to advance slowly & finally turn back to advance each node at a time to reach the final destination of right hand lower dot. Thus, I could imagine that there are quite a number of ways to travel instead of restricting to 1 or 80 or 102 ways.

Using the chessboard to view the question for solution. it is not practicable in the sense that the checkboard is made up of black & white squares instead of merely black dots. The question does not mention white dots or white box. Thus, the travelling from one dot to another vertically is possible due to there is no line or restriction to limit its path. For a chessboard, one could presume that travelling from black dot or box to white dot & box is not perssible, this could be end up that the trip could be turned up to zero. The reason is simply that on the condition that one could not travel diagonally & one could not move from black dot or box or square to white dot or box or square, it makes it impossible to advance. As the previous people that comment the travelling between dots through the use of chessboard, they use the chessboard to view the movement of the question should be considered correct from their point of view.

However, the question mentions merely black dots only but not white dots or boxes or squares, we should have the same view as the writer of this puzzle that we should not link up the black dots to the chessboard.

Thus, there are numerous ways in travelling from black dots to black dots. Whether the black dots are in even number or odd, we could still draw line from one dot to another to reach its final destination.