Minimal hypergraph transversals
The problem is NP-hard by a reduction from hitting set. Thus, there is no efficient or "easy" algorithm in this sense. Of course, any algorithm you might execute with pen and paper can also be realized on a computer.

With that being said, just try all possibilities by hand. As you observe, it is easy to see if the solution is of size one or two. It can get much more tedious after that, and the reason is NP-hardness.