By: David Bressoud @dbressoud

I do not remember when or how I first discovered the MAA’s film of George Pólya in “Let Use Teach Guessing.” It was on a VHS tape owned by Macalester. I was immediately entranced. It would become the source for the second of the projects I always assigned in Discrete Mathematics, a powerful lesson in induction. CLICK HERE to access it on VIMEO.

Problem: If we cut space by five random planes, how many parts or regions will there be?

“Let Us Teach Guessing” was number 1 in a series of films called MAA Video Classics made in the early 1960s. It was filmed at Stanford. From someone who was there at the time, I learned that after setting up a classroom for the filming, they simply put a couple people in the hallway as classes changed and asked the passing students if they wanted to be in a film. This random assortment of students worked well. On the surface, the purpose of this film was to illustrate the importance of guessing in mathematics. In fact, this is part of a deeper thrust behind Pólya’s pedagogy, the importance of creating what I talked about last month, Guershon Harel’s concept of intellectual need.

Before delving into the problem presented in this video, I need to acknowledge what an inspiration George Pólya has been to me, both directly through his writing, and indirectly through the people he influenced. Chief among these is Imre Lakatos, about whom I will say much more when I talk about the genesis of Proofs and Confirmations. Lakatos’s doctoral thesis, the core of his ultimate masterpiece Proofs and Refutations, was inspired and informed by his exchanges with Pólya. Like “Let Us Teach Guessing,” Proofs and Refutations illustrates the importance of mathematical surprise as a source of intellectual need.

Pólya starts by encouraging random guesses for the number of regions. In my class, I would pause the video here and gather comparable guesses. Like the Stanford students, mine were a bit hesitant before starting to come up with a wide range of possible values. Almost always, one of them was 32. As Pólya says in the video, “Oh, you have something behind your guess.”

He now starts building up to five planes, starting with simpler cases and asking students to imagine a big block of cheese that is being sliced. No planes yield 1 region, one plane produces 2. With two planes there are 4. Three planes cut space into 8 regions. Now he (and I) ask for a guess with four planes. Everyone is confident the answer is 16.

To check this answer, he again goes back to a simpler case, the number of regions in a plane cut by lines. No lines yield 1 region, one line yields 2, two lines yield 4, three lines…? When you count the regions, you get 7, not 8. Something unexpected is happening. Now he actually counts the regions formed by four planes and leads the students to the discovery that there are only 15 regions, not the expected 16. Now he (and I) have their full attention.

Pólya again backs up to simpler cases, starting with the number of regions in a line cut by points, then a plane cut by lines, then space cut by planes. Before we get up n = 5, students have realized that the right-most column is simply n + 1, and each of the other numbers is the sum of two numbers from the previous row: the one directly above and the one to its right. They are able to generate the table in Figure 1.

This is where we end the first class. When students come back for the next class, their challenge is to find an explicit formula for the number of parts or regions in k-dimensional space cut by n k-1 dimensional objects for k up to 4, and then to prove their formula.

Obviously, this takes a fair amount of scaffolding, beginning by presenting a table of binomial coefficients and asking them to guess how the numbers they have found can be expressed in terms of these coefficients. It may take a while, but they all eventually see the need to add consecutive binomial coefficients. The proof requires double induction, inducting on the dimension and then on the number of k-1 dimensional objects. Students do not need to be taught how induction works in order to use it instinctively here. Once they have employed it, we can talk about how useful it is and what to watch out for (especially that the initial case holds). Again, this is part of creating intellectual need. We do not discuss induction until students need it and see the need for it. CLICK HERE for a pdf of the worksheet, and CLICK HERE for the LaTeX file.

It is clear by the end of this that these students can produce formulas for any positive dimension. This often elicits questions about why bother with more three dimensions. Of course, the simple fact that we can answer the question of the number of regions in any dimension is interesting in and of itself. If time permits, I will explain the connection to linear inequalities. Here there is no natural limit to the number of variables, which corresponds to the dimension. The number of regions represents the maximum number of combinations of linear inequalities that are actually possible.

There are many directions in which this can and has been further developed: How many finite regions? What if the planes are not random but two are parallel, three share a common line, or four share a common point? What numbers of regions are possible?

Most of all, students emerge with a real sense of accomplishment. They have discovered and proven a significant and challenging mathematical result.

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