Problem: Coloring Circles (click here to see the problem statement)

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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:

I then fix both the upper-left and lower-right circle and allow three colors:

In the final coloring, I fix only the upper-left circle and allow three colors in the graph:

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.

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.