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?
Arminius, I arrived at certainty that they both know they are 12 after 2 "no"s.
If you think that my last sentence does not provide certainty, can you please point out the flaw?
[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.