Question

A teacher thinks of two consecutive numbers between 1 and 10. The first student knows one number and the second student knows the second number. The following exchange takes place:
First: I do not know your number.
Second: Neither do I know your number.
First: Now I know.
What are the 4 solutions of this easy number puzzle?

Answer 1

2-3
3-4
7-8
9-10

Answer 2

ANSWER:
Consider the numbers between 1-10: 2,3,4,5,6,7,8,9

The fact that neither of them knows the other person's number so they cannot have the extreme values, that are, 2 & 9.

So we are left with 3-8.

Case 1:

If A can immediately guess the number when B doesn't have the solution that means A has a 3 or a 8 and B a 4 or a 7 respectively.
Answer: A=3, B=4
OR
A=8, B=7

Case 2: Similarly A and B can interchange positions
Therefore,
A=4, B=3
OR
A=7, B=8

So these are the four cases.
Hope it answers your question.