Each other and one another is, as far as I know, a problem we all wondered about at least once. Here is what the smart people say:
Each other: 1. each of two or more in reciprocal action or relation
One another: 1. EACH OTHER
So, apparently, the two phrases share a meaning. However, the formally correct usage is using each other when referring to two, while using one another when referring to three or more.
Even so, writers have been using the two phrases interchangeably for a long time. So it’s probably okay to each either one.
Each other – two things (can be used for three or more)
One another – three or more things (can be used for two)