Fall 2008 Problem 4: Is that enough information?
Mathematicians Alice and Bob were having a discussion.
Alice: Did you know I have four daughters?
Bob: No, I didn't. What are their ages?
Alice: Well, the product of their ages is 72 and the sum of their ages equals the age of your oldest daughter.
Bob (thinking for a minute): No, that's not enough information for me to figure out their ages. Can you tell me something more?
Alice: My oldest daughter carries around my youngest daughter all the time.
Bob: Oh great —- now I know their ages!
What are the ages of the four daughters?
Solution
Brent Krise got the hat trick (correct answer, best explanation, and first correct answer)! That gave him five points. Here's his explanation:
The four daughters are 1, 3, 3, and 8. For this problem we need to know all the factors of 72, and subsets with cardinality four with products equal to 72. So I started to list subgroups of length 4 whose product was 72; {1,2,3,12},{1,3,3,8},{1,1,4,18}... and so on.
Now since Bob knows his own daughter's age and the fact that the four daughters product is 72 but still does not know the ages implies that there are at least two subsets summing to his own daughter's age. I found two whose sums were equal{1,3,3,8} and {2,2,2,9}, with sum equal to 15. Thus since the “oldest” carries around the “youngest” they must be 1, 3, 3 and 8.
Correct solutions from non-Centenary students came from Chris Evert and Micah Strange.