Obviously when Riku was running a towel out to the Prince he didn't think he needed to bring one of the wooden swords with him.
Riku accepted the sword from Abe with a nod and began to slowly circle the Prince. "Technically, I should be facing Conrad now, as he was the winner of your match," Riku teased.
He didn't wait for the Prince to make the first move, he attacked first, and then ran away through the water. The Prince was going to have to chase him a little if he wanted to fight him.