What would be the difference between 書こう and 書きよう when used in this sentence?

Question

EDIT: Apparently I read the answers wrong, and the correct answer is actually the one I chose, i.e. 書こうにも.

Then my question would be focused on the difference between these two forms. I guess in this fixed usage 書こう would be the grammatical one to use, while 書きよう would be used when the speaker wants to make a proposal in general, even though both are "volitional forms"? Or is my understanding incorrect.

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Question from Shin Kanzen Master N1 Grammar:

手を怪我してしまった。これでは急ぎの書類を＿書けない。

I chose 書こうにも but the answer was 書きようにも

However, there was a real question from the 1995 N1 exam that was:

こう毎日レポートや試験に追われていては、国の両親に手紙を書こうにも書けない。

which used 書こうにも

Therefore this question was quite confusing to me.

I guess a more general question would also be, is there any difference between these two forms for any word, or are they both volitional forms which are interchangeable in most situations?

Answer 1

I believe the correct answer should be 「書こうにも」.

Grammatical sentences are:

「手を怪我してしまった。これでは急ぎの書類を書こうにも書けない。」
「こう毎日レポートや試験に追われていては、国の両親に手紙を書こうにも書けない。」

「書きようにもかけない」 is not correct.

The formula is: 「volitional form 書こう + にも + potential negative 書けない」.

書きよう is not the volitional form. It's 書き+[様]{よう}, "way of writing".

To use 「書きよう」 you could say:

「～の書きようがない。」 "There's no way to write..."