Riddles

[Arminius]

Phoneutria.

untab

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.

On their foreheads!

[phoneutria]

Phoneutria.

untab

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.

On their foreheads!

untab

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 15

A 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]

[Arminius]

Phoneutria.

untab

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.

On their foreheads!

untab

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 15

A 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"

No. That is false. I am sorry.

untab

A 1a: If B had a 9, I'd have a 15.

So you are A. Okay.

A has a 12, therefore I don't have a 9.

Now you are B? Hey?

1c: 12+12=24 and 12+15=27, therefore the number on my forehead is either 12 or 15

Yes, regardless whether you are A or B. Okay.

A answers no

B 1a: If A had a 9, I'd have a 15.

So you are B again. Okay.

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.

But does he see a 15?

1f: Since he does not know for sure he must see a 12 on my forehead.

What?
It is clear, because of the premise of the riddle, that he sees a 12.

B answers that his number is 12

No, that is not allwoed because of the premise of the riddle.

I change my answer to ONE

Please read the task again:
- viewtopic.php?f=4&t=188593&start=150#p2580041 .

Remember: Both are PERFECT logicians. So they knew, for example, mathematics too.

And read also the following posts again:
- viewtopic.php?f=4&t=188593&start=175#p2598528
- viewtopic.php?f=4&t=188593&start=175#p2598586 .

[James S Saint]

untab

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 15

A 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 doesn't know that A has disqualified a 9. They each know that they don't have a 9, but they don't know that the other knows that. If B knows that B has either 12 or 15, he also knows that A is seeing 12 or 15 and A might be thinking that himself might have 12, 15, or 9, even though we know that A has disqualified 9.

[phoneutria]

James

untab

if I know that I don't have a 9 without even seeing my card, certainly the other logicians also knows that I don't have a 9. Since we are both perfect logicians, we both know that both of us don't have 9s.

[Arminius]

Maybe it is easier to look for a formula.

[James S Saint]

James

untab

if I know that I don't have a 9 without even seeing my card, certainly the other logicians also knows that I don't have a 9. Since we are both perfect logicians, we both know that both of us don't have 9s.

The other doesn't know that you know.

[Arminius]

Phoneutria.

untab

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.

On their foreheads!

untab

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 15

A 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"

untab

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.

Good luck!

[phoneutria]

James

untab

if I know that I don't have a 9 without even seeing my card, certainly the other logicians also knows that I don't have a 9. Since we are both perfect logicians, we both know that both of us don't have 9s.

The other doesn't know that you know.

untab

[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

[phoneutria]

Phoneutria.

untab

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.

On their foreheads!

untab

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 15

A 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"

untab

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.

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?

[James S Saint]

untab

[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

Not so. Each depends upon what the other is thinking when they answer.

[phoneutria]

Can you show me how not knowing that prevents them from arriving at the answer after 1 no?

[Arminius]

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.

untab

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?

[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.

[James S Saint]

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".

[Arminius]

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?

untab

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.

Why did you not go on?

untab

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?

[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?

untab

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.

Why did you not go on?

untab

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?

I stopped because I provided what the problem asked.

Can we get carleas in here?

[Arminius]

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?

untab

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.

Why did you not go on?

untab

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?

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.

[Arminius]

untab

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. (Now, hurry up, because the others are coming soon.)

Good luck!

[phoneutria]

No I don't want to play with a recursive solution until you acknowledge that my inductive solution is sound.

[James S Saint]

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.

[phoneutria]

If anything, that'll add a couple of nos.

[James S Saint]

If anything, that'll add a couple of nos.

Then just count them.

[phoneutria]

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.