Saturday, December 15, 2012
My favorite classroom puzzle
When I was teaching I would always keep some form of this grid around, either printed out on sheets of paper or in a PowerPoint or some other digital form. Whenever I had to keep a room full of students out of trouble I'd put this up on the screen or pass it out and ask them to count the squares, keeping in mind that a square could be one by one, two by two, all the way up to twelve by twelve.
Take a minute and give it a try. I'll wait...
I like this problem for a number of reasons. It's a problem that's extraordinarily difficult unless you hit upon the underlying pattern at which point it can be solved in about a minute. The insight does not require any background other than fifth grade math. The solution is simple and elegant and only relies on sixth grade math but it often eludes people with considerable mathematics background.
Like many problems this is easier if you pick the right side to start from. Most people will start with the one by ones by multiplying 12 by 12, then will move on to the two by twos. You're more likely to succeed if you start with the twelve by twelve then try the eleven by elevens (or better yet, start with the one by ones then jump to the the twelve by twelve then the eleven by elevens) then the ten by tens. Counting the bigger squares is simpler and the pattern will probably jump out quicker this way.
There is, of course, one twelve by twelve square in the picture, and it shouldn't be difficult to see that there are four eleven by elevens and nine ten by tens. So we want to find the sum of a sequence of twelve numbers that starts with one, four, nine and ends with one hundred and forty-four. Once you have the pattern in mind, you can start asking yourself why it holds. With a little thought you can probably get to this insight: for an n by n square, the bottom left corner has to fall in a square with sides twelve minus n plus one.
Like most good math puzzles, this is very Polya -- you use inductive reasoning to formulate a hypothesis then find a proof that shows the hypothesis must be true -- but that's a topic for a lot of future posts