Solution:

Enter the number to each field how many different way we can get it from the left upper field. We can step to a field from the next above field or from the left next field (if they exist), so we can obtain this number for the actual field to sum the numbers from above and left fields. On this basis, if we fill in the fields as shown in the picture, we can obtain the result: 3432

Solution:

The two opposite field of chessboard have same color. The horse change the color of the field in each step. The horse starts from the upper left field and in the 63rd step arrived to the lower right field. It should be different color then the first field, but it is NOT. So horse can’t carry out the above conditions.