An insurance agent walks into a bar…

This is a riddle forwarded to me from the BCAMT Math list-serve (originally by George PĆ³lya, according to someone on the list-serve):

A Life Insurance agent comes up to a ladies 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?

The puzzle itself is interesting, but what I found more interesting is what happened next. Another mathematics teacher from the list-serve gave the question to some of their students, and got this response back:

“Couldn’t sleep, wrote a script that not only solved your problem, but solves it with any number of children, and 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.”

Script: https://gist.github.com/yasyf/5299301

This is a nice bit of mathematical reasoning by this student. They took a problem, solved it, and then found a way to generalize the problem, and solved the more general problem. You can’t ask for much more than that!

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Open ended problem: Dots and lines

In this video James Grime examines the “challenging” math problem given in the movie Good Will Hunting and points out that it is not actually all that challenging. Unfortunately he is pressed by the person interviewing him to give all of the solutions to the dots and lines problem given.

This problem could easily be extended to be more open-ended simply by leaving the number of dots open. Are there any patterns when you generate diagrams with 2 dots, 3 dots, 4 dots, 5 dots, and so on? What kinds of diagrams are essentially the same (homeomorphic)? What kinds of diagrams cannot be made more simple without changing the character of the diagram (irreducible)?

Collaborative Mathematics project

The Collaborative Mathematics project, created by Jason Ermer, looks like another excellent source of rich mathematical tasks for students. I recommend following the Problem a Day blog. Jason encourages the problems to be done collaboratively, hence the name of the project.

Here is a sample:

Notice how Jason takes a closed form question (What are the four digit numbers that can be flipped when multiplied by four?) and converts it into a much more open-ended investigation simply by making the restraints less restrictive. This is a useful general strategy you can use to make closed-form problems more open-ended.

Thanks to the Math Munch for sharing this project.