[tab][Tab]it does’t matter that the other one doesn’t know that I know, so long as each of them knows that both are not 9[/tab]
phoneutria: Arminius:Phoneutria.
[tab]Both A and B have "12"s on their foreheads, and 12 + 12 = 24. So you should know from the premise (12 + 12) that the sum is 24, not 27. The sum must be 24. That is why your solution is false. The sum 27 is not possible because of the premise that both have "12"s on their foreheads.[/tab]
On their foreheads![tab]I know that, but they don’t. All they know is the other dude has a 12 and that the total is either 24 or 27.
A 1a: If B had a 9, I’d have a 15.
1b: B has a 12, therefore I don’t have a 9.
1c: 12+12=24 and 12+15=27, therefore the number on my forehead is either 12 or 15A answers no
B 1a: If A had a 9, I’d have a 15.
1b: A has a 12, therefore I don’t have a 9.
1c: 12+12=24 and 12+15=27, therefore the number on my forehead is either 12 or 15
1d: A answered no on the first round, so he doesn’t know whether the number on his forehead is 12 or 15 either.
1e: If the number he sees on my forehead was 15, he would know for sure that his is 12, since 15+15 is not a valid option.
1f: Since he does not know for sure he must see a 12 on my forehead.B answers that his number is 12
I change my answer to ONE"[/tab]
[tab]
phoneutria:I know that, but they don’t.
Phoneutria, my comment was addressed to you, not to A and B. You have to know that both have “12”'s on their foreheads (so that the sum must be 24 in your calculaltion). That was meant. This premise is given in the riddle.[/tab]
Good luck!
So your riddle is, there’s 2 guys with 12 on their foreheads. What’s on their foreheads?
… 12, I know because… it’s in the premise.
Are we having a natural language issue, robot?
[tab][Tab]it does’t matter that the other one doesn’t know that I know, so long as each of them knows that both are not 9[/tab]
Not so. Each depends upon what the other is thinking when they answer.
Can you show me how not knowing that prevents them from arriving at the answer after 1 no?
So your riddle is, there’s 2 guys with 12 on their foreheads. What’s on their foreheads?
… 12, I know because… it’s in the premise.
Are we having a natural language issue, robot?
Spi hider, … ahem, … hi spider.
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:
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.
You should go on with that. (7), (8), (9), … and so on. Do you understand? If yes: Can you do that?[/tab]
I did not give a sum as answer. I said that both of them know that they have 12 on their forehead after one no.
Can you show me how not knowing that prevents them from arriving at the answer after 1 no?
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”.
I did not give a sum as answer. I said that both of them know that they have 12 on their forehead after one no.
Why did you stop at 15 and 12 then?
[tab]After 9 was eliminated, they know that 12 and 15 are the only valid options.
They can’t both be 15.
If I see a 15 I would know that my number is 12, however I see a 12, so I have to answer no.
The other one must realize that the situation above ensued and therefore be must see a 12 on my forehead.[/tab]
Why did you not go on?
[tab]Remember that five "no"s are already given:
Perfect Logicians.
Players A and B both have got the number 12 written on her forehead. Everyone sees the number on the front of the other but does not know the own number. The game master tells them that the sum of their numbers is either 24 or 27 and that this numbers are positive integers (thus also no zero).
Then the game master asks repeatedly A and B alternately, if they can determine the number on her forehead.
A: "No". B: "No". A: "No". B: "No". A: "No". ....
After how many "no"s does the game end, if at all?
[/tab]
phoneutria:Can you show me how not knowing that prevents them from arriving at the answer after 1 no?
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”.
Both of them know that both of them don’t have 9. So it is not necessary for one to know that the other knows.
phoneutria:I did not give a sum as answer. I said that both of them know that they have 12 on their forehead after one no.
Why did you stop at 15 and 12 then?
phoneutria:[tab]After 9 was eliminated, they know that 12 and 15 are the only valid options.
They can’t both be 15.
If I see a 15 I would know that my number is 12, however I see a 12, so I have to answer no.
The other one must realize that the situation above ensued and therefore be must see a 12 on my forehead.[/tab]
Why did you not go on?
[tab]Remember that five "no"s are already given:
Arminius:Perfect Logicians.
Players A and B both have got the number 12 written on her forehead. Everyone sees the number on the front of the other but does not know the own number. The game master tells them that the sum of their numbers is either 24 or 27 and that this numbers are positive integers (thus also no zero).
Then the game master asks repeatedly A and B alternately, if they can determine the number on her forehead.
A: "No". B: "No". A: "No". B: "No". A: "No". ....
After how many "no"s does the game end, if at all?
[/tab]
I stopped because I provided what the problem asked.
Can we get carleas in here?
Arminius: phoneutria:I did not give a sum as answer. I said that both of them know that they have 12 on their forehead after one no.
Why did you stop at 15 and 12 then?
phoneutria:[tab]After 9 was eliminated, they know that 12 and 15 are the only valid options.
They can’t both be 15.
If I see a 15 I would know that my number is 12, however I see a 12, so I have to answer no.
The other one must realize that the situation above ensued and therefore be must see a 12 on my forehead.[/tab]
Why did you not go on?
[tab]Remember that five "no"s are already given:
Arminius:Perfect Logicians.
Players A and B both have got the number 12 written on her forehead. Everyone sees the number on the front of the other but does not know the own number. The game master tells them that the sum of their numbers is either 24 or 27 and that this numbers are positive integers (thus also no zero).
Then the game master asks repeatedly A and B alternately, if they can determine the number on her forehead.
A: "No". B: "No". A: "No". B: "No". A: "No". ....
After how many "no"s does the game end, if at all?
[/tab]
I stopped because I provided what the problem asked.
Can we get carleas in here?
No, spider. We are alone here. Show your weapons!
Carleas is observing the precesses in this thread from outside anyway, but currently he has no chance to get in.
[tab]Okay, I will give you the next step.
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.
Next step:
A: “No” => b < 21.
B: “No” => a > 6.
And so on.
By this I have given you almost the whole solution. [size=85](Now, hurry up, because the others are coming soon.)[/size]
Good luck![/tab]
No I don’t want to play with a recursive solution until you acknowledge that my inductive solution is sound.
No I don’t want to play with a recursive solution until you acknowledge that my inductive solution is sound.
It isn’t “sound” because
Both of them know that both of them don’t have 9. So it is not necessary for one to know that the other knows.
…that isn’t true.
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.
If anything, that’ll add a couple of nos.
If anything, that’ll add a couple of nos.
Then just count them.
Like at the first no I know I don’t have a 9, then at the second no you know you don’t have a 9 and I know that you know.
phoneutria:No I don’t want to play with a recursive solution until you acknowledge that my inductive solution is sound.
It isn’t “sound” because
phoneutria:Both of them know that both of them don’t have 9. So it is not necessary for one to know that the other knows.
…that isn’t true.
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.
Exactly.
Like at the first no I know I don’t have a 9, then at the second no you know you don’t have a 9 and I know that you know.
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.