Which day is Miss Zhang's birthday? (Classic reasoning question [reprinted])

Mr. Zhang and Mr. Xiaoqiang are both students of Mr. Zhang. Their birthdate is m month N. Both of them only know the day of Miss Zhang's birthday in the following 10 groups, but do not know the day.
March 4 March 5 March 8 June 4 June 7 September 1 September 5 December 1 December 2
Teacher Zhang told James about M and N to Xiaoqiang. Then he asked them, "Do you know my birthday is that day ?"
James said: "If I don't know, I certainly don't know ".
Xiaoqiang said: "I didn't know either, but now I know ".
James said, "Oh, I know that ".
Determine the day of Miss Zhang's birthday Based on the conversation above.
It is said that the monthly salary of the person is 30 thousand (it is circulated on the Internet that the interview questions for a position with a monthly salary of 30 thousand)

Solution:
James said: "If I don't know, I certainly don't know." In fact, James doesn't know at all. Therefore, this sentence is ambiguous.

If the last half of James's first sentence is true:
James knows that the N value cannot appear only once (2 and 7). It must have appeared twice ). That is to say, the m value cannot be 6 or 12. If it is 6 and 12, then James cannot be sure that Xiaoqiang does not know.
Xiao Qiang also inferred from James's words that since you are sure that I don't know, the N value cannot only appear once, but it must have appeared twice. The m value cannot be 6 or 12.
Then the m value is 9 if it is not 3. If the m value is 3, you may or may not know the small strong. If it is 3-5, then Xiaoqiang does not know; if it is 3-4, 3-8, then Xiaoqiang knows. That is to say, if M is 3, Xiaoqiang cannot say that he knows it. So M is 9(Because Xiaoqiang knows that N is 1). If the N value is 5, John doesn't know it, but John knows it. N is definitely not 5. So N is 1(Because james knows m is 9). This proves that both of them know that Mr. Zhang's birthday is July 22, September 1.

If the last half of James's first sentence is false:
Xiaoming wants to know the teacher's birthday, but does not know whether Xiaoqiang knows it. If Xiaoqiang does not know it, he does not want to know it, I spoke of a tentative lie.
to achieve his goal, James pretended to tell Xiaoqiang that he knew n could not appear only once. No matter whether Xiaoqiang says, he knows it or not. As long as Xiaoqiang tells the truth, he can know the teacher's birthday. It can be inferred that only M = 6 can satisfy James's goal.
Xiaoqiang believes that James is telling the truth, so the previous situation can tell that Xiaoqiang mistakenly thinks it is 9-1. So he said, he didn't know either, but now he knows. James inferred from the ignorance of Xiaoqiang that the N value must have appeared twice. Therefore, the N value is 4. Therefore, the correct birthday is 6-4, not 9-1.