Mathematics is not only about learning problem-solving methods; it is also about gaining a deeper understanding of their purpose, advantages and disadvantages. Frequently, the same problem can be solved by applying several different methods. Mathematics education should also include clues to those best fitting to the person's aim. To achieve that students are encouraged to find as many correct solutions to problems as possible. Then, properties of individual solutions are discussed. In the paper we exemplify our approach by solving the problem: How many four- digit numbers not containing 3, 6, and 9 are divisible by 3? The following methods are shown: (1) An estimation giving an approximate result, (2) Constructing a finite automaton recognizing the "divisibility by 3" property, (3) Forming a formal grammar capable of generating these numbers, (4) Writing a computer program producing all the numbers, (5) Using dynamic programming to speed up the execution of the program. Comparing different solutions, the students disclose not only what their solutions can do and what they cannot do; they are also encouraged to generalize their observations. One of these discussions is also described. (Author)