|Spoiler Inside: Solution to Puzzle||SelectShow>|
It is possible to visit each square exactly once. One possibility is to start at a corner square, walk to the squares on one side, then walk up one row and walk back, repeating this last step until the entire board has been covered. Any other route is longer, because it has duplicate visits.
You start having visited one square. When you have finished walking, you have visited 63 more squares so the shortest route is 63 square-to-square walks long.