You can mathematically prove that the puzzle is impossible: Assign each reachable tile a black or white colour as in a chess grid. You end up with 57 white and 55 black squares (assuming a white upper left corner). Any walk that only touches each grid cell once and only goes up/down/left/right must alternate between stepping on a white or a black cell. You start on a white cell, which means any path must have an equal number of white/black cells or one more white than black. However, in the room there are 2 more white than black cells, therefore a path touching every cell only once is impossible. QED.
