Two Mathematicians: Factoring LogicDate: 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/ |
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