Linear Transformations · Superchallenge problem

Put a real number $f(e)$ on each of the $12$ edges $e$ of a cube. Let $T$ send $f$ to the function on the $8$ vertices whose value at a vertex $v$ is the sum of $f$ over the three edges meeting at $v$. This $T$ is a linear map from $\mathbb R^{12}$ to $\mathbb R^{8}$. What is the dimension of its kernel?