The problem in "The Twenty-One Balloons" is that Professor Sherman and the inhabitants of the island, Krakatoa, must find a way to escape the island before a massive eruption destroys the entire island.
The Professor and the families must work together in order to escape the island. The resolution of the problem occurs when the families and Professor Sherman escape on a flying platform held aloft by twenty balloons. The families parachute off the flying platform and start new lives in different parts of the world.
Professor Sherman remains on the platform, dropping into the ocean, where he is rescued.