We had solutions to this problem from pupils at The Bishops' School and from Ruth who goes to Swanborne House School. Ruth sent us a very detailed solution:

When you overlap two circles which are the same size, they make a shape like a pointed oval. This is made up of two arcs which are the same length and the same width. They look like a reflection of each other along a line in the centre.
The bits left over are crescents.

When you overlap more than one circle of the same size, you can make a pattern that looks like a daisy. This is made up of lots of pairs of arcs. If you put in the centre lines of the pairs of arcs and also draw lines between the points of the arcs, you get a number of triangles. We made equilateral triangles, which combined to make a hexagon.

If you overlap two circles which are not the same size, you get a lopsided pair of arcs which are the same length, but they are not the same width. The smaller circle makes the deeper arc, because the curve of the arc is tighter. The larger circle has a more gently curving arc and this is therefore shallower.

Well done Ruth - you have investigated this very thoroughly.