A tetrahedron is regular if every face is an equilateral triangle, and the dihedral angles are also equal. We will show that the smallest such $k$ number is seven. If $k=6$, it is possible for the tetrahedron to have two faces that are equilateral triangles, and these two faces, for example, enclose a $90^{\circ}$ angle. In this case, the tetrahedron is not regular.

If $k=7$, then by the pigeonhole principle, there are at least two faces with at least two angles of $60^{\circ}$, and in addition, there is at least one face with at least one angle of $60^{\circ}$. This latter face shares two sides with the previously mentioned congruent equilateral triangular faces, so it is isosceles, and since it has one $60^{\circ}$ angle, it will be equilateral. The fourth face is formed from the edges of the three previously described congruent equilateral triangles, so this face and the tetrahedron itself are - regular.

Breuer János (Budapest, Apáczai Csere J. Gymn., 10th grade) Csóka Endre (Debrecen, Fazekas M. Gymn., 8th grade) Reviczky Ádám János (Budapest, Szent István Gymn., 10th grade)
What is the smallest natural number $k$ for which the following statement is true? "If a tetrahedron has $k$ edges with an angle of $60^{\circ}$, then the tetrahedron must be regular."