Each of the nine dots in this figure is to be colored red, white or blue. No two dots connected by a segment (with no other dots between) may be the same color. How many ways are there to color the dots of this figure?
Ok, so for the left thre dots, there are $3\cdot2\cdot1$ ways to connect them. Thus, our solution is $3\cdot2\cdot1\cdot3=18.$ I think I am missing something here, thank you in advance!
Best Answer
I've made a labeled diagram.
To make thing show up better I'm using red, green and blue as the colors. Black vertices haven't been colored yet. If we color $\triangle ABC$ as shown, then E must be colored green or blue and $D$ must me colored blue or red. Because of this, the color assigned to $F$ uniquely determines the coloring of $\triangle DEF$. Now the same argument shows that however we color $\triangle DEF$, we have three choices for $\triangle GHI$. Of course, there are six ways to color $\triangle ABC$, giving $6\cdot3\cdot3=54$ ways in all.