Is the ground that was trodden upon by a shoe with which its wearer entered the bathroom, then exited and walked with it throughout the house, considered impure and must it be purified three times?

The basic principle concerning the floor of the bathroom, slippers, and the splashed water in it is that they are pure. Their impurity is not determined except by certainty. Therefore, it is not obligatory to purify the house just because one walks with bathroom slippers inside it. If the floor becomes impure, pouring water on the impure spot once purifies it and is sufficient to remove the impurity.