Monday Math Madness #12 is up at Blinkdagger, and features Marvin the Martian (who turned 60 years old this past July 24. Happy Birthday Marvin! And wasn’t it cool of NASA and friends to use Marvin in the patch for the Mars Exploration Rovers?)
At any rate, this week’s puzzle is particularly challenging. One person picks two whole numbers between 2 and 99, tells the sum to a second person and the product to a third person. The second person tells the third person they [Person #3] can’t possibly know the original numbers, and the third person realizes that that is enough information to figure it out. With that revelation, the second person is able to figure it out. Your job is to find the numbers.
Seriously, that’s all the information that you get, though it’s phrased perhaps a little more clearly at Blinkdagger. And at the moment I have little idea how to solve it, but I’m working on it. It did, however, remind me of one of my favorite problems that I occasionally given to non-majors in a “distribution requirement” math class. The problem involves a census taker who asks a parent the ages of the three children who live in the house. The ages (whole numbers) multiply to seventy-two, and add to the house number. The census-taker looks at the house number and says, “That’s not enough information.” The parent agrees, and comments that the oldest child has a pet rabbit, and that’s enough to solve the problem.
Like I said, I love this problem, but my students are often a little overwhelmed when I assign it. This led to one of my favorite ever teaching exchanges, which went something like this:
Student: Does it matter that it’s a rabbit?
Me: Not in particular. It could be a dog. Or a cow.
The student thought for a while, then:
Student: I got it! “Rabbit” in French is lapin, which has 5 letters. “Dog” in French is chien, which has 5 letters. And “Cow” in French is vache, which also has 5 letters. Am I on the right path?
One the one hand, I loved the student’s enthusiasm (which was not unusual for this student) and also the willingness to try new ways of thinking. And this student was no slouch mathematically, and was a joy to have in class. On the other hand, it really gave me insight into what word problems must seem like to a non-mathematician, if translating the words into a foreign language and then counting the letters seemed like a reasonable course of action. In problem solving, “easy”, “hard”, and “obvious” are in the eye of the beholder, not necessarily the eye of the author of the problem. [Which isn't me -- I've seen versions of this problem in several places.]
And in good news, my student did go on to solve the problem correctly.