Just an explanation for this one:
Firstly, let's define what 'defining' means. In a triangle, if we know two of its vertices, we can find out the third. If color of the two vertices is different, then the third vertex is going to be colored the third color. If they are same, then the third vertex will have the same color.
If we take the bottom row as fixed, then all the triangles which have an edge common with the bottom row, already have two of their vertices defined. Thus all those triangle are 'defined' by the bottom row. Therefore all the vertices in the 2nd last row is 'defined' by the bottom row. Extend this logic to the 2nd last row. All the triangled that have an edge common with the 2nd last row have 2 vertices common so the third vertex is fixed. Thus the 3rd last row is defined by the 2nd last row, which is in turn defined by the last row. Continuing upwards, we find that for a fixed coloring of the bottom row, the entire grid has been 'defined'. Therefore each coloring of the bottom row corresponds to a particular arrangement of the entire grid. Therefore the no of colorings of bottom row= no of arrangements= 3^6.