Add basic problems from algebraic engine testing

Test-problems.md

@@ -2,6 +2,41 @@ Here are some objects one might explore and problems one might solve with dyna3.
 
 ## Basic elements and constraints
 
+### Points on a sphere
+
+#### Statement
+
+Choose a whole number $n \ge 1$. Make a sphere and constrain $n$ points to lie on it.
+
+#### Notes
+
+In a generic solution, the points are all different, but there are degenerate solutions where some or all of the points coincide. These are valid solutions to the problem as stated here, but it's also interesting to test whether one can gravitate toward generic solutions or deform a degenerate solution into a generic one.
+
+### Mutually tangent spheres
+
+#### Statement
+
+Choose a graph $G$. Make a sphere for each vertex of $G$, and constrain the spheres to be tangent whenever the corresponding vertices are connected by an edge.
+
+#### Notes
+
+Here are some interesting tangency graphs:
+
+- Triangle
+- Tetrahedron
+- 5-cell
+- Triangular bipyramid
+
+### Points at the intersection of two spheres
+
+#### Statement
+
+Choose a whole number $n \ge 1$. Make two spheres and constrain $n$ points to each lie on both spheres.
+
+#### Notes
+
+The spheres are allowed to coincide, and the points are also allowed to coincide,
+
 ### Rigid hexagon
 
 #### Source