Inspired by the rather lengthy discussion on [the low-dimensional topology blog][1], there's a rather basic question for 3-manifold algorithmics that is still unsolved, as far as I know. 

* If a 3-manifold $M$ admits a complete hyperbolic structure of finite volume, does it admit an ideal triangulation?  i.e. the kind of triangulation where the software [SnapPy][2] could find its hyperbolic structure. 

  [1]: https://ldtopology.wordpress.com/2013/04/23/when-are-two-hyperbolic-3-manifolds-homeomorphic/
  [2]: http://www.math.uic.edu/t3m/SnapPy/

It would be nice to either understand the hyperbolic manifolds that SnapPy (in its current state) can not deal with.  Or if such manifolds do not exist, have a sense for how complicated the triangulation needs to be in order to find the hyperbolic structure.