"Nor did I" or "Neither did I" [duplicate]

These two questions are from a mobile app:

  1. I didn't pass my driving test.
    correct answer: "Nor did I."
  2. He didn't pass his driving test.
    correct answer: "Neither did I."

Why do we use nor in the first sentence and neither in the second? Is there any exact rule to follow?

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A: John doesn't smoke
B: Neither do I
C: Nor do I
D: I don't either
E: Me neither

They're all correct, E is "nonstandard English", I wouldn't recommend writing it in a formal paper or in an exam, but it's fine in conversation.

A: John didn't pass the test
B: Neither did I
C: Nor did I
D: I didn't either
E: Me neither

All these answers are perfectly acceptable. We use neither and nor + auxiliary/modal verb + subject to mean ‘also not’:

A: John does nothing all day, he hardly works
B: Neither does Alice
C: Nor does Alice

We can use subject + auxiliary + not + either to mean ‘also not’,

D: Alice doesn't either

But the expressions me too and me neither are only used in very informal speech, and can only be used with first person singular. So ...

E: *Her neither

In this case E is ungrammatical.

To sum up: The app was wrong in marking neither did I as being incorrect. Both Nor did I and Neither did I are acceptable short answers.

Sources: Cambridge Dictionaries Online and BBC learning English
See also this question Is “Neither I you” Correct?

Here's another source: (Collins COBUILD Active English Grammar. HarperCollins, 2003)

"You use 'so,' 'neither,' or 'nor' with an auiliary, modal, or the main verb 'be.' The verb comes before the subject.

You were different then. -- So were you.
I don't normally drink at lunch. -- Neither do I.
I can't do it. -- Nor can I.

You can use 'not either' instead of 'neither,' in which case the verb comes after the subject.

He doesn't understand. -- We don't either


I'm gonna have to disagree with the app.

Neither is the negative of either.

Both the questions can be answered:

I didn't pass it either.

Which can be phrased as Neither did I

Nor is the negative of or.

I don't think Nor did I would not be appropriate in either situation, (even though it sounds acceptable because, like neither, it is widely used to introduce a further negative statement).

For nor to be applicable, there should be a corresponding phrasing using or.

Or I didn't pass it doesn't sound like something anyone would reply

Nor would be appropriate in sentences where a negation of or is needed, like.

I didn't pass my cooking test, nor did I pass my driving test.

Alternatively, I didn't pass my cooking test or my driving test.