As it rolls around the grid, the tetrahedron will always present the same face to particular triangle of the grid. In fact the grid can be subdivided into four classes of triangles, each of which can only be landed on by one particular face of the tetrahedron. That being the case, a selection of painted triangles is solvable if and only if it contains one each of the four classes of triangles. Grid 3 is the only such grid.

Interestingly, so long as the grid is solvable, it doesn’t matter where you initially place the tetrahedron, and you can also decide to end up on any particular triangle of the grid (even if it isn’t initially painted).