How many color patterns? 
Pixie has three colors of paint, which she uses to pain the pattern below. Each of the three colors are used at least once. If pixie is willing to paint adjacent region (regions sharing a common edge) in the same color, then how many color patterns can she make?

 
My Attempt
I am novice to this kind of problem and I realize that my approach is very naive.
There are six regions and three colors, so every region can be chosen in six ways and every color in three ways. However, depending on which region is chosen, there are different number of adjacent regions. From this point  on I am lost, and any help would be appreciated. Thanks in advance!
 A: There are six regions and three colors, so the number of ways to color the grid equals $3^6$. However, we must subtract all cases in which either only one or only two colors are used. As such, the number of ways to color the grid in a valid way equals:
$$3^6 - {3 \choose 2} (2^6 - 2) - {3 \choose 1} = 540$$
A: Since there are six regions and three ways to paint each region, we could form $3^6$ color patterns if there were no restrictions. From these, we must exclude those in which fewer than three colors are used.
There are three ways to exclude one of the colors and $2^6$ ways to paint the regions with the remaining two colors.  Hence, there are 
$$\binom{3}{1}2^6$$
color patterns if one color is excluded.
However, if we subtract $\binom{3}{1}2^6$ from the total, we will have subtracted those cases in which two of the colors twice, once for each of the ways we could designate one of the colors as the excluded color.  Therefore, we must add those cases back.
There are three ways to exclude two of the colors and one way to paint all six regions with the remaining color.  Hence, there are 
$$\binom{3}{2}1^6$$
color patterns in which two of the colors have been excluded.
By the Inclusion-Exclusion Principle, the number of color patterns in which all three colors are used at least once is 
$$3^6 - \binom{3}{1}2^6 + \binom{3}{2}1^6$$
A: There is a second way of viewing this problem: We have to count the surjective functions $f:\>[6]\to[3]$, namely: For each of the distinguishable six regions one of the three colors has to be specified, whereby all colors have to be used at least once. Now such a coloring induces a partition of $[6]$ into three nonempty blocks. The number of such partitions is called the Stirling number of the second kind $S(6,3)$ (there are various notations around). There are tables for these numbers, and systems like Mathematica or Maple know them. Given such a partition of $[6]$ we can assign the three colors to the three parts in $3!$ ways. It follows that there are
$$S(6,3)\cdot 3!= 90\cdot 6=540$$
admissible color patterns for this painting.
