What are the means of permissible entertainment available to a Muslim who is committed to his religion, memorizes the Quran, and maintains prayer, steering clear of forbidden things whose prohibition he knows, such as movies, songs, dominoes, and backgammon, and those about which there is disagreement, such as chess?

There is no harm in entertaining oneself with permissible recreational activities to restore a Muslim's energy and vitality. One may even be rewarded for it if done with proper intention (ihtisab). Ibn Taymiyyah mentioned that an intelligent person has four hours: one for supplicating to his Lord, one for self-reckoning, one for keeping company with friends who inform him of his shortcomings, and one for permissible, beautiful pleasures that aid him in the other hours. Allah created pleasures and desires for the complete benefit of creation. Permissible means of entertainment are many, including: hunting, swimming, parks, trips, various types of sports, computer-based mental entertainment programs, and others. As for cartoon films, it is permissible to watch them if they are free from شرعية prohibitions. Songs that include music are prohibited.