Two logicians who each pick a number secretly between 1 and 30

Question

Recently a logic question has become quite popular. Here it is:

Two logicians (A and B) each secretly pick a whole number from 1 to 30 inclusive. Each knows the other has done this.

A: "Is your number double mine?"

B: "I don't know. Is your number double mine?"

A: "I don't know. Is your number half mine?"

B: "I don't know. Is your number half mine?"

A: "I don't know."

B: "I know your number."

All the statements are true. What is A's number and how did B know?

My though process is outlined below.

1. Assume for the sake of contradiction that A is a number between 16 and 30. Then, A knows that double their number cannot be B, as B is between 1 and 30. However, assuming A is a logical logician, why would they ask such a question they already know the answer to? This implies A's number is between 1 and 15.

2. B first replies with "I don't know". Assume for the sake of contradiction that B has an odd number. Then, they would know that their number cannot be double any value of A, and would thus reply with "No". Thus, a contradiction, and B must be an even number. B then asks if A's number is double B. By the previous section, this implies that B's number is between 1 and 15. So, the possible values of B are 2, 4, 6, ..., 14. However, if B's number is 8, 10, ..., 14, then B already know that A cannot be twice B's number, since A is between 1 and 15. Thus, B's number must be 2, 4, or 6.

3. Here's my first problem with this logic puzzle. By saying "I don't know", A is implying that their number is 4, 8, or 12; otherwise, A would reply with a definitive "No". Then, A asks if B's number is half of A's. Why would A ask this? The only answer to A's question from B's side is "I don't know"; if B answered "Yes" or "No", that indicate that B knows A's number, which I don't think is possible with just the first 3 statements.

4. And here's where I'm just confused. B says "I don't know", as expected. Then B asks if A's number is half of B's number. Wouldn't this be asking if A's number is 1, 2, or 3? Isn't this a contradiction with the previous statement where A implies their number is 4, 8, or 12? So shouldn't A reply with "No"? Instead, A replies once again with "I don't know", which makes no sense to me.

I likely made a silly error somewhere, please correct me if so! Thanks in advance.

Answer 1

We will call A's number $a$ and B's number $b$. From the first three statements we know $a$ is a multiple of $4$ and $b$ is even. From B's second we know $b \le 14$ then from A's third we know $a \le 7$, so $a=4$

Answer 2

This puzzle is malformed at the very beginning. it can only be solved by assuming A can ask a question he doesn't have, because why not.

When A>15, then asking "is B=2A" means that 'A' is asking a question but he is not having that question (since he already knows that 2A > 30) So, the common arguments made are to decouple the questions "he has" from the questions "he asks" in order for the puzzle to have a solution, by mimicking the same erroneous assumptions as the puzzle author. It's simpler he asks questions whose literal answer, even he, doesn't know. (which can only happen when A =< 15)

A question existing is one thing, while a question being useful is another thing. Even if the question does legitimately exist in his mind when A =< 15, it's still a useless question. It can exist, since the answer to it's literal wording is unknown even by A, but it's useless since B can't answer it with precision. At this point, and only this point, can we apply the reasoning of "Let's ask it anyway even though B can't possibly know the answer, at least he can give us a clue if he has an odd or even"