"At the beginning of the century" or "in the beginning of the century"?

Solution 1:

In general, "at" marks a spot and "in" marks a space. Obviously, there are idiomatic exceptions, but this really isn't one. The beginning of a period of time is a "spot", the period of time itself is rather "spacious".

Solution 2:

To answer the question: I would never say "in the beginning of the century". I think Peter Eisentraut's argument is essentially correct: "the beginning of the century" is notionally a point, not a period (even though in practice, "at the beginning of the century" may in context cover a period of several years).

In fact the only instance I can think of of "in the beginning" is the opening of St. John's Gospel. That phrase is now archaic, and would not be used except in imitation of that specific use.