Question 167
The sides of an equilateral triangle are divided into $n$ equal parts by $n-1$ points on each side. Through these points one draws parallel lines to the sides of the triangle. Thus, the initial triangle is divided into $n^2$ equal equilateral triangles. In every vertex of such a triangle there is a beetle. The beetles start crawling simultaneously, with equal speed, along the sides of the small triangles. When they reach a vertex, the beetles change the direction of their movement by $60^\circ$ or by $120^\circ$.
a) Prove that, if $n \ge 7$, the beetles can move indefinitely on the sides of the small triangles without two beetles ever meeting in a vertex of a small triangle.
b) Determine all the values of $n \ge 1$ for which the beetles can move along the sides of the small triangles without meeting in their vertices.
a) Prove that, if $n \ge 7$, the beetles can move indefinitely on the sides of the small triangles without two beetles ever meeting in a vertex of a small triangle.
b) Determine all the values of $n \ge 1$ for which the beetles can move along the sides of the small triangles without meeting in their vertices.