Problem: Coloring
Circles (click here to see the problem statement)
My solution.
I can first observe that the center circle is initally irrelevant
in that it is constrained only by the color of the circle in the
upper-left to which it is connected and in no way constrains the
coloring of any other circle. Therefore, I can focus on the four
circles that are connected in a cycle.
I begin by fixing the color of the upper-left circle and the
lower-right circle and constrain the choices to two colors.
In this way, I obtain six colorings:
![](../ColoringCircle/2Colors.gif)
I then fix both the upper-left and lower-right circle and
allow three colors:
![](../ColoringCircle/3Colors2.gif)
In the final coloring, I fix only the upper-left circle and
allow three colors in the graph:
![](../ColoringCircle/3Colors1.gif)
In this way, I have obtained 18 colorings. However, at this
point, I need to consider the center circle. The coloring of the
center circle is constrained only by the color of the upper-left
circle and given three colors, any individual color can be matched
with either of the other two colors.
![](../ColoringCircle/CenterCircle.gif)
So each of the 18 the colorings obtained above has two additional
possibilites for center circle coloring, resulting in a total
of 36 distinct colorings.