Problems on the Board II: Simplify!

I asked a pair of girls (age 8 and 9) I tutor to ask questions about the chessboard, and got another really great lesson out of it, this one highlighting the importance of making things simple.

With a little prompting, we came up with a nice list of questions:

  • How many squares are there on the chess board?
  • How many squares of any size?
  • How many rectangles?
  • How many chessboards would it take to fill the space needle? To reach the moon? To cover the Earth 10 times?
  • Can you bend a chessboard into a sphere?
  • How many corners, or vertices are there on the board?
  • How many edges?

When they get interested in a problem (and these kids, like most, were interested in their own problems right away) the tendency is to try to solve them right away. I thought it might be nice to work on the problem of how many edges there are, but one of the girls suggested working on how many rectangles there are. Take a look at a chessboard.

There are a lot of rectangles on this thing. So many, I’d say, that if you try to start counting them, you’re setting yourself up for a frustrated end, especially if you’re a kid. Here’s the dialogue between us:

Me: Do you know what a mathematician does when they face a problem that’s too hard to solve right away? They make it simple. (I fold the board in half.)

Them: Oh, we could start on a four by four board!  (They immediately start to count rectangles.)

Me: Mathematicians really prefer to start with really easy problems. (I fold the board in half again)

Them: We could start on a two by two board!

Me: Sure. Though a mathematician would really like to start with something even easier. Like a one by one board. In fact, a mathematician would probably start with a zero by zero board. How many rectangles are there on that board?

Them: (laughing) Zero!

Me: See? We’ve already solved a problem! How many rectangles are there on the one by one board?

Them: One.

Me: All right! We’ve solved two problems; let’s write down what we’ve got so far (we do). What about the two by two board?

Them: (counting): 1, 2, 3, 4, … 5. No. There are 1, 2, 3, 4, 5, 6, 7, 8, 9. Nine.

By this point, we’re all involved, and we’ve got real momentum. One of the girls starts to think of possible patterns in the numbers 0, 1, 9, but can’t think of what might come next (another great reason to start on the zero by zero board: we get an extra piece of data, and can look for patterns before we’ve even started the hard work). So, inspired, they dive in to the three by three board.

It’s complicated enough to take some doing, and after they had tried to count them all, missed some, tried again, got closer, tried again, and got the right answer, I suggested writing down their process instead of doing everything in their heads. We ended up with the following table:

The numbers in each box represent the numbers of each type of rectangle. For example, there are 4 two by two rectangles (i.e., squares) on the 3 by 3 chessboard, and there are 6 two by one rectangles, and 6 one by two rectangles (which look the same, but one is vertical, one horizontal). Adding up all the kinds of rectangles, this confirmed the next number in our pattern.

0, 1, 9, 36…

Do you see what should come next? We didn’t yet, but our momentum was better than ever. One of the girls jumped in to counting all the rectangles on the four by four; the other, daunted by the magnitude of that effort, started looking for patterns in the table we’d made. By the end, both had solved the four by four board, and one, having found a pattern in the table, had conjectured the number of rectangles in the five by five board. Our pattern now looks like this:

0, 1, 9, 36, 100, 225,…

Do you see a pattern now?

One of the nice things about this lesson was that, in addition to the deeper skills of analyzing patterns, simplifying, organizing, etc., there was a need to employ some basic technical skills: both girls ended up motivated to add up long columns of numbers. So this ended up giving us motivation to practice complicated addition problems that, without this framework, would have been boring. One of the girls, faced with the task of adding up the numbers on the table of all rectangles representing rectangles on the four by four board, found that adding the numbers in columns first makes it easier.

For anyone who wants to dive in at home, here’s a little extension: if you start with the number of squares on the chessboard, there are two natural generalizations: find the number of rectangles on the chessboard, or the three dimensional variation, find the number of cubes in a bigger cube (i.e., how many cubes are there in a rubik’s cube?). These questions seem totally different, but I if you start to write your answers down for each, you’ll notice a similarity between them that bespeaks a connection between these questions and the structures.