Learning about different problem solving strategies and when to use them will give you a good start. Problem solving is a process. Most strategies provide steps that help you identify the problem and choose the best solution. There are two basic types of strategies: algorithmic and heuristic.
It should be 'No vacancies' (plural). Vacant means empty/free. Vacancy is the state of being empty, therefore 'no vacancies' = no empty rooms.
As used in the phrase null and void, refers to something that binds no one or is incapable of giving rise to any rights or duties under any circumstances.