Two Mathematicians: Factoring Logic

Date: 03/24/2003 at 15:38:30
From: Raphi
Subject: Two mathematicians

Two mathematicians are each assigned a positive integer. They are told 
that the product of the two numbers is either 8 or 16. Neither knows 
the other's number. This is their conversation:

First mathematician:  "I don't know your number."                    
Second mathematician: "I don't know your number."
First mathematician:  "Give me a hint."
Second mathematician: "No, you give me a hint."

At this point, one of the mathematicians knows the other's number. 
Assuming that they always tell the truth and do not guess, what is the 
number and who has it? 

Date: 03/25/2003 at 03:20:14
From: Doctor Jeremiah
Subject: Re: Two mathematicians

Hi Raphi,

Both of them know the product is 8 or 16 and that their numbers are 
positive integers. The only possible numbers the two mathematicians 
can have are 1, 2, 4, 8, and 16, because those are the only numbers 
whose product is either 8 or 16.

First mathematician: "I don't know your number."                    
This implies that the first mathematician cannot have a 16 because if 
he did, he would know that the second had a 1. So the first 
mathematician can only have a 1, 2, 4, or 8.

Second mathematician: "I don't know your number."
This implies that the second mathematician does not have a 16 or a 1 
because if he did, he would know that the first had a 1 or an 8. So 
the second mathematician can only have a 2, 4, or 8.

First mathematician: "Give me a hint."
This implies that the first mathematician doesn't have a 1 or an 8 
because if he did, he would know the second had the 8 or the 2 and he 
wouldn't have to ask for a hint. So the first mathematician can only 
have a 2 or a 4.

Second mathematician: "No, you give me a hint."
This implies that the second mathematician doesn't have an 8 or a 2 
because if he did, he would know the first had the 2 or the 4 and he 
wouldn't have to ask for a hint. So the second mathematician must 
have a 4.

The second mathematician asks for a hint because he doesn't know what 
the other has but that means the first mathematician knows that the 
second has a 4.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/13/2003 at 18:48:28
From: Raphi
Subject: Thank you (Two mathematicians)

Thank you, Dr. Jeremiah, and the rest of the Ask Dr. Math staff. 
I e-mailed you all the way from Jamaica. I am a high school 
mathematics teacher and this question caused quite a stir in the Math 
Department. I am a bit embarrassed, though, since none of us could get 
the answer. I have not yet revealed the source of my answer.

Date: 04/14/2003 at 14:36:17
From: Doctor Jeremiah
Subject: Re: Thank you (Two mathematicians)

Hi Raphi,

If you liked that one, see this one in the archives:

   Integer Logic Puzzle
   http://mathforum.org/library/drmath/view/51609.html  

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/