If I understand it in the right way (and the circles are compound paths with gradient fills), you may:
1) Create a horizontal path through the centres of all circles and extending past the sides;
2) Select all, Pathfinder>Divide, and Object>Ungroup;
This should give you three sets of halve circles with the gradients as in the image;
3) In the Layers palette/panel, change the stacking order of the half circles to get the right ones behind the others.
I'm sorry, I'm feeling more an more like a noob now.
I'm having trouble with step four (move to fit average)
Just saw that you updated... thanks... I'll give it a go... one sec.
You are welcome, Jenna.
Cool technique.. Thank you, Jacob!