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.
no contest. n. in criminal law, a defendant's plea in court that he/she will not contest the charge of a particular crime, also called nolo contendere. While technically not an admission of guilt for commission of the crime, the judge will treat a plea of no contest as such an admission and proceed to find the defendant guilty as charged.