Directions in computer algebra
The largest problems are now tractable on common computers:

The idea is to calculate the number of terms (not the number of components!) in the covariant Ricci tensor (Ra b) in n dimensions. The most general form of the metric tensor is used, that is n(n+1)/2 functions of n variables. The number of terms grows rapidly with n.
Dimension (n)2345
Diagonal components174169,990298,134
Off-diagonal components1751913,280410,973
CPU (s) on notebook01141720


The following example (n=5) was run on my notebook.

It is not difficult to invent intractable problems: e.g. case n=6 (>23 CPU days (at 3 GHz), >3 GB RAM).