# Tiling by trapezoids

An equilateral triangle with sidelength $L$ can be tiled by trapezoids with sidelengths $2,1,1,1$.
What are the possible values for $L$?

An equilateral triangle with sidelength $L$ can be divided into $L^2$ equilateral triangles with sidelength $1$. A trapezoid with sidelengths $2,1,1,1$ covers three of these triangles, so $L^2$ must be divisible by $3$. $L$ has to be an integer (because if it is tiled by the trapezoids, its sidelength must be the sum of sidelengths of the trapezoids), so $L$ is divisible by $3$.
Note that three of these trapezoids can tile an equilateral triangle with sidelength $3$. Then, for any equilateral triangle with a sidelength that is a multiple of $3$, we can divide it into triangles with sidelength $3$ and tile each of those with the trapezoids.
which means L is