Abstract 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 $a,b,nın N$ 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.
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