Find the fundamental group of the space comprising a collection of six 2-spheres touching pairwise in a cyclic order, and each touching a common plane.
Touching means having one point in common. I am thinking about using Van Kampen's Theorem... But having trouble executing it...
Interesting problem! My approach would be to deform the space through a sequence of homotopy equivalences into a space whose fundamental group is easily computed, instead of using van Kampen's theorem directly.
First, wherever two spheres, or a sphere and the plane, are touching, I elongate the contact point into a line segment. This space deformation retracts onto our original space, and so has the same homotopy type. Then since the plane is contractible, I can contract away the plane to a single point. At this point, I get something like the following:
Each sphere is connected to a line segment at three places. Pick a contractible arc on each sphere passing through the three contact points and contract away each arc. We now get a space consisting of a graph on seven vertices with a sphere attached to six of the vertices.
Now contract a spanning tree of the graph. We end up with a space homotopy equivalent to $\bigvee_{i=1}^6 S^1 \vee \bigvee_{j=1}^6 S^2$.
Hopefully you can continue on from this point and compute the fundamental group of this space. Write back if anything is unclear!