No. The sum you gave as a solution was false. And you would have known this, if you had considered the premise. Therefore I reminded you of the peremise.
[tab]Your solution was the sum 27 (read your posts again), but the sum 27 is not possible as a solution, because the sum has to be 24. Do not think too much about what you would think if you were A and B, although it is not absolutely irrelevant. Remember what I said to you in this post. Or, … wait …, here comes the quote:
You should go on with that. (7), (8), (9), … and so on. Do you understand? If yes: Can you do that?[/tab]
I started to include that, but it got complicated.
As soon as you said “if he saw 15, he would know his own number was 12 because…”, you implied that each person knew that the other had already disqualified “9”.
The entire game is figuring out what the other person must know. That is why it is in the category of “Perfect Logicians”, else one couldn’t be certain of what the other might deduce.
They both know that they don’t have a 9 from the start. But neither knows that the other knows that because one might have a 15 and thus the other would have a 12 or a 9. Each can see a 12, but they can’t know they don’t have a 15 and thus the other person thinking that he might have a 9 or a 12.
If the person saying “no” suspects that he might have a 9, the reasoning that he said “no” because he realizes that he must have either 12 or 15, doesn’t hold. When the 15 is discounted, he would still wonder if he had 12 or 9.
Any of them would immediately say yes if they could see a 9. This is obvious to both.
If both of them say no on the 1st round, then none have 9.
This is evident.
[tab]In the beginning A knows that (1) a = 12 or a = 15, and (2) B knows that b = 12 or b = 15.
Okay. But A does not know that B (2) knows, and B does not know that A (1) knows. So the statement above is not suited for the recursive conclusion.
But both A and B know all of the following statements and that each of them knows that the other one knows them:
(3) a = 24 - b or a = 27 - b and (4) b = 24 - a or b = 27 - a.
Now, from the first “No” of A and from (4) follows (5) b < 24, because if b >= 24, then A would be able to conclude a. This is the motor for the recursive conclusion.
Now, from the first “No” of B and from (3) and (5) follows (6) a > 3.
And so on.
A: “No” => b < 21.
B: “No” => a > 6.
A: “No” => b < 18.
B: “No” => a > 9.
A: “No” => b < 15.
B: “Yes”. Because together with the information of (2) there remains only one possibility.