Subdividing Equilateral Triangles into Smaller Equilateral Triangles

For which values of $n$ can we subdivide an equilateral triangle into $n$ sub-triangles where each sub-triangle is also equilateral?

This problem focuses on dissecting geometric figures into smaller congruent shapes, specifically subdividing an equilateral triangle into smaller equilateral triangles. At first glance, this problem may appear to be purely geometric, but it involves deep connections with concepts in combinatorics and number theory. Understanding how complex geometric shapes can be partitioned into simpler, congruent components requires thinking about symmetry, tiling patterns, and divisibility.

An equilateral triangle, by definition, has equal sides and equal angles. The fundamental challenge here is determining when such a triangle can be split into smaller triangles that maintain these properties. The approach often involves considering the possible side lengths of the smaller triangles relative to the original. Key mathematical ideas relate to finding a strategy that either directly constructs such subdivisions or employs a proof strategy to show the conditions under which such a subdivision is impossible or possible.

The problem also invites the application of mathematical induction or an examination of modular arithmetic to assess the feasibility of subdividing an equilateral triangle into a given number of equivalent sub-triangles. This exploration encourages students to see beyond the initial geometric visualization and appreciate the numeric and algebraic reasoning underpinning geometric dissection.