You know I haven’t studied logic and I have no good way to annotate, but I will make a start…

Start Moment:
A knows that B has 12, that A has 12 or 15, that B sees either 12 or 15 and no other number.
A says No.
B knows that A has 12 and that A has seen either 12 or fifteen on B. He knows he must have 12 or fifteen. If A has seen 15, then he is thinking either I have 9 or 12. If A has seen 12, then A is thinking I have either 12 or 15. B knows this is what A is thinking.
B says no.
A knows now that if B has seen 12 he is thinking that he either has 12 or 15. While at the same

I can imagine where one takes into account the limited possibilities and what the other must be thinking that at some point an elimination happens. But I cannot hold it in my head.

Though I suspected such to be the intent of the puzzle, I also suspect the proposed solution to be fallacious. I currently don’t think that it can be solved that way, although misleading into the illusion of a solution. We’ll see.

A and B cannot both be in 1. Either A is in 1, B is in 1, or Neither are in one
Either A or B, or both A and B is in group 2.

F is undefined. E is undefined. C is undefined.

D is undefined. F is undefined. A is undefined.

Either A or E, or both A and E are in group 1.

D is undefined. E is undefined. C is undefined. B is undefined.

Either D or B, or both D and B are in group 2.

No solution.

If a statement is wrong, only one component may be wrong. Therefore, F is undefined.
For instance, in 2. F may be in 2, because only the end of the statment may be wrong. Therefore F is undefined

Solution to (“Perfect Logicians”)
[tab]A: Since he has 12, that means I do not have 9.
Case 1. I have 15. This means he thinks he has 12 or 9.
Case 2. I have 12. This means he thinks he has 12 or 15.
If I say no, that will tell him that he does not have nine, and that I know he does not have nine. Because if he had nine, that would mean I have 15. But I am unsure if I have 15.
“No.”
B: Him saying no means I do not have 9. Because if I had nine, he would know that his is 15. So he knows i do not have nine.
Case 1. I have 15. This means he thinks he has 12 or 9.
Case 2. I have 12. This means he thinks he has 12 or 15.
If I say no, that will tell him that he does not have nine, and that I know he does not have have nine, and that I know he does not have nine. Because if he had nine that would mean I know I have 15. But I am unsure if I have 15.
“No.”
A: Case 1. I have 15. Since I already made him know he doesn’t have 9, this means he would think he has twelve. Since he doesn’t know if he has twelve, this means Case 2 is true, that I have twelve.
“I have twelve.” Three turns[/tab]

[tab]A: HIS IS 12 SO MINE MUST BE EITHER 12 OR 15. IF HE THINKS I SEE A 12 HE WILL THINK HIS IS EITHER 12 OR 15. IF HE THINKS I SEE A 15, HE WILL THINK HIS IS EITHER 9 OR 12.

“NO”

B: HE SAID NO, SO THAT MEANS I DON’T HAVE A 9. IF I HAD A 9 HE WOULD KNOW FOR SURE THAT HIS WOULD BE 15.
HOWEVER I STILL DON’T KNOW IF MY NUMBER IS 12 OR 15.

“NO”

A: WE BOTH KNOW THAT WE’RE NOT 9 AND THEREFORE THE OTHER EITHER 12 OR 15, BUT DON’T KNOW WHICH. WE CAN’T BOTH BE 15, SO IF I HAD 15 HE WOULD KNOW FOR SURE HIS IS 12. SINCE HE ANSWERED NO, I MUST HAVE A 12.

You are on the right track - that means: You can go on, because there is no logical error; only the answer is false, but the logical track is right so far.

[tab]And if so, then you merely have to follow this track for a longer time, with more patience, and especially with more consequence!

Cue: Recursive conclusion.[/tab]
Is it okay for you now, or shall I give you more information?