The purpose of this article is to suggest a few problems for exploration which could motivate pupils to think and work independently on their solution. The article consists of three parts, describing experinces with pupils of different age groups: Part I describes a problem, related to solving the equation $ax+by=n$, with $a,b,n\in N$ in non-negative integers $x$ and~$y$. This problem has been studied with pupils aged 14--15. Part II is devoted to sequences and combinatorial investigations of problems designed for 16--18 years old youngsters. In Part III the importance of introducing youngsters at an early age to independent work is underlined. This will be done by providing problems for children aged 10--13.