The phrase "beat one's head against a stone wall" means to struggle with a difficult problem or situation that seems impenetrable. There are several synonyms for this expression, including "banging one's head against a brick wall," "running into a brick wall," or "hitting a dead end." All of these phrases convey the frustration that comes with trying to solve an insurmountable challenge. Other similar expressions include "spinning one's wheels," "going around in circles," and "hitting a roadblock." These idioms all highlight the struggle of trying to overcome a seemingly impossible obstacle and the need to find a new approach or solution.