In "An Béal Bocht" by Myles na gCopaleen he writes -"Ach tar éis an tsaoil, fuair sé fuascailt na noide sin".
It has been translated in Clubleabhar.com "After all, he found a solution to the problem" Is this correct?
I'm having a problem with the use of ''oide" in this context !

Could it actually be this word:
http://www.teanglann.ie/en/fgb/nod
note that at the end it says there is a feminine variant (of which the genitive would be na noide)

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ЯГОН ТОҶИК НЕСТ ИНҶО???

Oide?
Fuascailt na n-oide sin? (impossible without a hyphen or upper-case O, na nOide)

It is most probably an nod, feminine gen. na noide, fuascailt na noide.
(in Dinneen’s dict. nod was invariably feminine with a genitive noide/nuide)