Coloring number with this polygon

Question

I ended up 3 * 2 * 2 * 2 * 2 * 2 = 96. But it was wrong. How can you find out coloring vertices in this polygon?

Answer 1

(I assume that you have 3 different colors and that connected vertices cannot have the same color.)

Say that we have three different colors: R, G and B.

Step 1) Pick any color for the leftmost vertex, say R (3 different choices).

Step 2) Pick any color for the rightmost vertex, say G (2 different choices).

Step 3) Consider two bottom vertices: there are only 3 acceptable combinations of colors: GB, GR and BR (3 different choices).

Step 4) Consider two top vertices: again, there are only 3 acceptable combinations of colors: GB, GR and BR (3 different choices).

So the total number of colorings is $3\times2\times3\times3=54$.