Is there a difference in using で and に in these two example sentences?

Question

逃げる以外に方法がない。

逃げる以外で方法がない。

I know the sentence means roughly 'There is no other way except to run away' (per this site; https://www.japaneseforbeginners.info/2016/07/igai.html), but I don't understand what would be the difference in alternating between these two particles. Can't they both 'adverbialize' a noun? I just want to better understand the nuance between で and に when they are modifying a verb.

Answer 1

When a word takes で even without 以外, it still takes で with 以外:

• 現金以外で支払えますか？
• 東京以外で働きたい。
• 炊飯器以外でお米を炊く方法

Notice that we can say 現金で支払う, 東京で働く, 炊飯器でお米を炊く. This explains 病気などの理由以外で, too (理由 safely takes で).

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Next, what about 逃げる以外で? 逃げるで of course makes no sense because 逃げる is not even a noun. This type of で is a kind of condition marker, and 逃げる以外で sounds like you are imposing some constraint on your request/question. Something like this is perfectly fine:

• 逃げる以外で何か方法はありますか？ _{(逃げる以外に is also fine)}
• 逃げる以外で対策を考えよう。 _{(逃げる以外に is acceptable but less natural)}
• お勧めの観光地を教えてください。京都以外で。 _{(京都以外に sounds weird)}

However, in your case, the sentence is not asking for something in the first place, so 逃げる以外に方法がない sounds more natural to me. Other examples where に is more appropriate:

• 大学に入ったら運転免許以外にノートPCも欲しい。_{(運転免許以外で is acceptable but less natural)}
• 日本語を学ぶなら漢字を覚える以外にない。_{(漢字を覚える以外でない is wrong)}