I pulled the Parade magazine out of the Jan 13 Sunday paper, and, as if the Universe were grinning (cue the Twilight Zone music), I swear it fell open to the Marilyn vos Savant column. I glanced at it – then burst into laughter!
This is just too much fun to bypass! And, since it does lead to some important educational observations, I’m going to share. Pull up a chair.
Over twenty years ago, one of Marilyn’s columns contained this question, presumably from a reader: “Can you put 50 coins into 10 envelopes such that each envelope contains a different number of coins?”
Marilyn’s answer back then went something like this: “It’s impossible. Envision putting one coin in one envelope, two coins in another and so forth. By the time you put nine coins in the ninth envelope, you would have used 45 of the 50 coins. So, you wouldn’t have enough coins to put a different number in the 10th envelope.”
Because I had seen a discussion of this problem before, I knew that it wasn’t impossible if one was willing to think outside the box. If all the envelopes are as stated above, then inside the last (tenth) envelope, one can put two things: 1) the five remaining coins and 2) Envelope 5 (with its 5 pennies), say. (Other envelopes work too). Now Envelope 10 has 10 coins total, and all ten envelopes contain a different number of coins. Yes, some will obviously call that a ‘trick’, but it meets the conditions of the problem, though perhaps not the assumed/intended conditions, of course.
So, for fun, I sent that general solution to Marilyn via Parade magazine, and promptly forgot about it. Before long, an envelope from Parade magazine arrived. Inside was an official letter, with her signature, containing only one word: “Boo!” Since I didn’t think she was trying to scare me by mail, I could only assume she had seen the solution I sent and wasn’t crazy about it. Back then, I decided to take it as a compliment.
Back to 2019: Marilyn’s latest column repeated the exact same question, this time from a Bryan James in Dallas, TX. Marilyn’s answer was identical to what is given above, except that this time, there was another short paragraph added:
“Note: Solutions that cheat are possible. For example, you could put one envelope inside another envelope, thereby using the same coins more than once.”
I couldn’t help but laugh out loud! For several reasons, really.
Perspective here. I’m not about to nit-pick with Marilyn. But, educationally, there’s a very fine line we’re walking, so it’s worth discussing. On the one hand, I understand that one of the reasons students dislike math is they perceive there’s a ‘trick’ hiding around every corner. It’s one of the irrational fears that math teachers work hard to eliminate. And teachers never want to ‘trick’ students.
At the same time – and this applies beyond math, of course – one of the things we want to encourage and develop in all students is the ability to think creatively, and to solve problems in non-rote fashion, whether they be problems in math, on the job, or, in life itself. How can we do that if we don’t allow – indeed encourage, or even reward – outside-the-box solutions? Are all such solutions ‘cheating’?
Part of being a good teacher is knowing how to navigate the thin line above most efficiently with students. Most teachers do it expertly. It’s just another one of the unseen/unsung miracles they perform daily.
So now I wonder. Shall I send this column to Marilyn?