And Tolkien took that "thirdings" idea a step further by having "Farthings" in the Shire.

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"whereas in the UK the term is not used to define the region."

I don't know where you get that, Hugh - normally when we refer to a county we are meaning the geographical territory. Or do you mean that typically in England the word "county" tends to be left out, except where it coincides with the name of a city (as with Durham)?