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 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”.
Why did you stop at 15 and 12 then?
Why did you not go on?
[tab]Remember that five "no"s are already given:
[/tab]
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.
Any of them would immediately say yes if they could see a 9. This is obvious to both.
?? How so?
If you see a 9, how do you know whether you have a 18 (for 27) or a 15 (for 24)?
If both of them say no on the 1st round, then none have 9.
This is evident.
Even after the first round, 9 is still an option.
[tab]
Okay, I will give you the next step.
Arminius: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
Next step:
A: “No” => b < 18.
B: “No” => a > 9.
And so on.[/tab]
The whole solution (with the solution process):
[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.
Now add the „No“s!
The game ends after 7 „no“s.[/tab]
I remind you of the riddle I posted on 14 January 2016:
Six people in two groups.
There are six people A, B, C, D, E, F which are in each case either in group 1 or group 2. The following statements are given:
- Both A and B are in 1.
- F is in 2, and if E is in 2, then C is also in 2.
- D is in 1, and if F is in 2, then A is also in 2.
- A and E are both in 2.
- D is in 2, and E is in 1, and if C is in 2, then B is in 1.
- D and B are both in 2.
- The statements 1-6 are wrong.
Who is in which group?
Who is depicted here?
The whole solution (with the solution process):
[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.Now add the „No“s!
The game ends after 7 „no“s.[/tab]
Incorrect (for several reasons).
Sorry, try again.