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The problem is: I've stumbled on some Guardian articles that said "twentieth-first", but I'm not sure if this is the right wording. Is there any possibility that such expression is also acceptable? Thanks.
Comments 
Bruce TorresIs there any possibility that such expression is also acceptable?
Anything is possible. You would have to give us the full quotation of several sentences before we could make a judgment.

CJ
"This title provides an insightful and informative account of this intriguing and backward country trapped between the middle ages and the twentieth-first century."

"Both boil down to an unobjectionable concern for treating people fairly, transposed into the language of twentieth-first century liberal academic orthodoxy: for both authors ‘empire’ is inevitably bad, ‘interdisciplinarity’ just as unquestionably good."

These are some. Are they enough or do you need more quotes? Thanks.
Bruce TorresAre they enough
Yes. That's enough to see that it's an editorial mistake.

I suspect the writer wrote:

between the middle ages and the twentieth century.

The editor realized that we are now in the twenty-first century, so he marked the text in pencil with "-first". Then when the text was revised, the typist saw the correction, but forgot to change "twentieth" to "twenty" when he added "-first".

So as written, it's wrong.

CJ