here's a puzzle:
My friend Gonzo is good with numbers. One day we had the following conversation:
Me: I'm thinking of two numbers. Each is greater than one. Their product is less than 20.
Gonzo: I can't figure out what your two numbers are. Instead of just telling me what their product is less than, could you tell me what their product is.
Me: Even if I told you that, you still wouldn't be able to figure out what my two numbers are.
Gonzo: Fair enough, but unless you just want me to start guessing, you're going to have to give me some more information.
Me: Their sum is equal to the age of your daughter Clara.
Gonzo: I still don't know what your two numbers are.
Me: Perhaps I should give you some more information?
Gonzo: Yes, please.
The conversation went on, but I won't tell you about the rest of it because you already know enough to answer the following question: How old is Clara?
Posted by mccartney at January 6, 2004 6:56 PMWas the additional information about whether or not your two numbers were identical? This would have allowed Gonzo to decide between a product of 12 or 16.
Posted by: Kevin at January 7, 2004 5:48 AMSo she's either 4, 19, or something between the two, as long as the two number don't have to be different. Refer to Kevin above if they do have to be different.
Posted by: mcgee at January 7, 2004 12:07 PMWait, never mind. Product goes with multiplication, right? I obviously haven't had a math class since my junior year of high school...
Now I'm just confused
Posted by: mcgee at January 7, 2004 12:20 PMWhat did Gonzo not know and when didn't he know it? He needs to guess two numbers that, together, are the factors forming a product less than 20. This leaves 1-19 as a choice of possible products. Each of these factors is greater than one. This eliminates all products that are prime numbers, which leaves 4,6,8,9,10,12,14,15,16,and 18 as possible products. Even if Gonzo were to be told what the product is, he would still have insufficient information to guess the factors. This eliminates all products that have only two factors besides themselves and one, which leaves 12 (2,3,4,6), 16 (2,4,8), and 18 (2,3,6,9). Now Gonzo is told that the right combination of factors, if added together, will equal the age of his daughter Clara. Gonzo knows how old Clara is. Since he still does not know what the product is, this must mean that the possible combination of addends in Clara's age can be multiplied together to get more than one of the remaining products. From this end of the question, which is to guess Clara's age, the possible ages are sums of the factors. If the product is 18, she is either 11 (2,9) or 9 (3,6). If 16, then 10 (2,8) or 8 (4,4). If 12, then 8 (2,6) or 7 (3,4). Clara's age is the only one that occurs twice. Gonzo still needs to know whether the two numbers he needs to guess are the same (4,4) or different (2,6).
Posted by: Kevin at January 7, 2004 7:15 PM