After a rather relaxing spring break, I returned to this lovely message waiting in my school email inbox:
Welcome back to Brentwood. I hope you enjoyed your break. I have a math question for you to get your brain back into gear. Bring your answers with you to class.
A Life Insurance agent comes up to a lady’s door to sell her a policy. He asks her how many children does she have and what are their ages. She replies, I have three children, but I will not even listen to your sales pitch if you cannot guess their ages. (Assume all ages are whole numbers) Here is clue number 1: The product of their ages is equal to 36.
Salesman jots down some figures and replies, “I do not have enough information to answer that yet”.
The lady replies, “Okay, here is clue number 2: If you go the house next door (points to it), the sum of their ages is the same as the house number”.
Salesman goes next door, looks at the number, jots down a few more things and then returns to say,” I still don’t have enough information”.
The lady then replies, “Okay, your third and final clue: My eldest is learning to play the piano!!”
The salesman, replied with the answer, it was correct, he gave his sales pitch, she bought a policy.
What were their ages?
Well with that challenge at hand, there was no way I was resting easy on my first night back. The result of an hour-long collaboration with a peer, and my houseparent, resulted in the logical explanation that, of all the possible combinations of ages (which all had to be factors of 36), there must have been at least two that added to the same sum, thus giving the ambiguous nature to the second clue of the riddle, and therefore one of these combinations-that-resulted-in-a-repeated-sum must be the answer. When combined with the third clue, which clearly only serves to show that there is only one eldest child, and that we can therefore remove any possible answers with the two eldest children being the same age, we get the answer of the daughters being two years old, two years old, and nine years old.
Of course, I couldn’t stop there, so the remainder of my night consisted of coding a Python script (embedded below) which not only solves the problem, but solves it for any number of children, and for any product of their ages, if possible. It also will generate numbers that will give you valid variations of this question, given the number of answers that you want. The script can be found here, and embedded below. Night well spent!