Difficult Question

[Dice Man]

A good poker player should think about what their opponent is thinking. A great poker player should think about what what their opponent thinks they're thinking, and an excellent poker player should think about what their opponent thinks they think their opponent is thinking. You need to know what information your opponent has and what information they think you have, in order to make good decisions, and understand your opponent's decisions. I came across this question about a game which should test how well people can empathise with the information of others.

Ann and Bob are told that a positive integer, x, has been chosen. One of them has x written on their forehead and one has x+1 written on their forehead. Obviously they can see the number on their partner's head but not the number on their own head. Ann has 4 written on her forehead, and Bob has 5 written on his forehead.

They are asked in turn what the number written on their own forehead is. They can only answer with the correct answer (i.e. they know) or say that they don't know. (no guesses)

The dialogue goes like this:

Interviewer: Ann, what is your number?
Ann: I don't know.
Interviewer: Bob, what is your number?
Bob: I don't know.
Interviewer: Ann, what is your number?
Ann: I don't know.
Interviewer: Bob, what is your number?
Bob: My number is 5.

How did Bob know what his number was?

No stupid answers like him being telepathic, seeing the reflection in Ann's eyes, absorbing the ink into his brain, or calculating the weight of ink required to write his number in conjunction with seeing the arm movements of whoever wrote it on his head. He knows the answer purely from the information he has gained from Ann, the number on her forehead, knowledge of the rules and knowledge that Ann knows the rules.

It took me pretty long to get the answer, but it still feels pretty weird how he could know. The vast majority of people (myself included) would not be as rational as Ann and Bob. They are perfectly rational agents.

I'll give 5000 vChips to anybody who gets it.

[waltypies]

0 + 1 = 1
1 + 1 = 2
2 + 1 = 3
3 + 1 = 4

therfore 4 + 1 = 5

[Dice Man]

There's no real method to that answer is there? If it was just a case of adding up the times people had said they didn't know +1, why didn't he just say 2 after she said she didn't know the first time?

[The Professor]

Intresting, maybe Bob had worked out that ann (like me) hadn`t got a clue what a positive integer was so,whatever No she could see on his head would mean little or nothing to her or the number she could see on his head wasn`t a positive integer but, Bob new exactly what a positive integer was and he new that Ann had a positive integer the No 4 on her head so after she had ansewered several times I dont know he assumed the No on his own head may not be a positive integer therefore Ann had the positive integer on her head and he had the positive integer in this case 4 +1 on his head which even a maths Dodo like myself could work out was 5.
Sounds plausable to me after a few glasses of Vino any way.

[Raiser]

Ann sees 5: therefore her card is 4 or 6
Bob sees 4 therefore his card is 3 or 5

If ann thinks her card is a 4 then she thinks bob must think her card is a 3 or 5
If ann thinks her card is a 6 then she thinks bob must think her card is a 5 or 7

So the common denominator in anns thinking what he is thinking is the number 5 there for bob must have a 5 which he does!!

If bob thinks his card is a 3 then he thinks that ann must think her card is a 4 or 2
If bob thinks his card is a 5 then he thinks that ann must think her card is a 6 or 4

So the common denominator in bobs thinking what she is thinking is the number 4 there for bob must have a 5!!

Do you know what i cant be arsed any more..... i was sat happily working it out and now i need slepp - so i will post this and finish it tomorrow!!

[Dice Man]

OK here's the first step of the process. If Bob can see that Ann has 4 on her head, then he knows that he either has 3 or 5 on his head right? Because he doesn't know if he has the x or the x+1 card.

[Dice Man]

Ann sees 5: therefore her card is 4 or 6
Bob sees 4 therefore his card is 3 or 5

If ann thinks her card is a 4 then she thinks bob must think her card is a 3 or 5
If ann thinks her card is a 6 then she thinks bob must think her card is a 5 or 7

So the common denominator in anns thinking what he is thinking is the number 5 there for bob must have a 5 which he does!!

If bob thinks his card is a 3 then he thinks that ann must think her card is a 4 or 2
If bob thinks his card is a 5 then he thinks that ann must think her card is a 6 or 4

So the common denominator in bobs thinking what she is thinking is the number 4 there for bob must have a 5!!

Do you know what i cant be arsed any more..... i was sat happily working it out and now i need slepp - so i will post this and finish it tomorrow!!

That is a rational approach to it. Following your first point I have to ask why would Ann care what Bob thinks Ann's number is? (4 or 2) Bob can see Ann's number! He doesn't have to guess it. You've made the mistake of mixing up the information each player has into one all-knowing observer who actually gave you a weaker conclusion than what was given in the premises!

Same with your theory about Bob's number. But you got pretty close to sussing out some stuff.

[Bizzare]

Ok, in final stage in Bob's opinion x=3 or 4. If x=3, then Ann would think that x=2 or 3. But Ann can't think that x=2 as if she did then she would think that Ben thought that x=1 or 2. Here x=1 implies that Ann thinks that her number is 2, but that isnt the case. x=2 implies that Ben thinks he has a solution but he hasn't offered one yet. So if x=3 then Ann would think that x=3 but she hasn't offered a solution yet. So x=4 and his own number is 5.

[waltypies]

I'll give 5000 vChips to anybody who gets it.

I will give 10,000 vchips to anyone who deletes this nonsense!!

[sl0ggs]

It's obvious, and I think Diceman is taking the piss... and you are all falling for it. (Nice one Diceman! )

They have been told that they each have a number on their head, one of them has a positive integer and the other has the positive integer +1 on theirs. Therefore, one look at each others head should be enough to know the answer, Bob can see a 4, which is a positive integer, therefore he must have 5 on his head because 4+1=5, she can see a 5 on his head, which is not positive, therefore her number must be 5-1=4.

Ann is a dumb ass, when asked first she should have wrapped it up straight away based on that. Bob shouldn't of even had a chance to answer. The reason Bob got it was because he was taking the piss out of the stupid woman, Ann didn't get it because she is the stupid woman.

How could Ann think her number is a 4 or a 6, or Bob think his is a 3 or 5. The sum is 'positive integer + 1'. It isn't rocket science! They can only have one number on their heads after seeing the number on each others!

Now, put the 5000 vchips in my account and stop asking stupid questions and winding people up...