Fundamental group a collection of six 2-spheres touching pairwise in a cyclic order, and each touching a common plane. 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...

 A: 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!  
