What I am trying to present to you today will not be a lecture. It will be rather a demonstration—or an experiment. And this experiment, or demonstration, or whatever you like to call it, has the aim of showing you my attitude toward teaching.
Introduction by Prof. R. A. Rosenbaum, Wesleyan University
The academic ideal in mathematics, as well as in other disciplines, is that of the scholar-teacher—the person who can contribute notably to the world store of knowledge and understanding, and who can also make others better through their teaching effectiveness. One man is acknowledged to be preeminent: the paradigm, the archetype and model for us all. This exemplar is George Pólya.
Much of Pólya’s academic life was spent in Switzerland after his education in Budapest. He came to the United States and joined the Stanford faculty in 1942. He holds a number of honorary degrees and has written many books and some 220 articles—most of them the fruits of his imaginative mathematical research. His contributions to the combined fields of mathematics and mathematics education are unequaled.
The demonstration you are about to see is truly unrehearsed and spontaneous. It illustrates his extraordinary, uncanny ability to stimulate a group to guess intelligently—to make reasonable conjectures, a process which is essential to mathematical discovery.
Prof. George Pólya:
Ladies and gentlemen, I don't wish to give you a lecture. I wish to teach you about guessing.
It's a very important part of life to be a good guesser, and a very important part of mathematics. You might wonder how—because mathematics is full of proofs, and only what is proven is valid in mathematics. So, where is the role of guessing?
Yes, mathematics, when it is finished and complete, consists of proofs. But when it is discovered, it always starts with a guess. I want to give you a real experience of this.
So instead of a lecture, we’ll play together a guessing game.
You should find out from your own experience what reasonable guessing is. Not wild guessing—anyone can do that. The less you know, the easier it is to make a wild guess. But educated, reasonable guessing—that’s something else. That should be learned, and a math class is a good place to learn it.
This game has two simple rules:
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If you already know the answer to my question—please don’t answer. That would spoil the fun.
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If you don’t know the answer—don’t hold back. Guess! Your guess may be wrong, but even a wrong guess helps lead to a better one, and eventually, to the truth.
Let’s begin.
The Problem:
I will give you a problem to guess.
It’s a problem of solid geometry. Not much to know there. For instance, what is a plane? A plane is very flat. The top of this desk, if it's well-made and smooth, approximates a mathematical plane. But the mathematical plane is ideal—flat and infinite in all directions.
Now imagine five such planes.
Imagine cutting a big block of cheese—Swiss cheese, green cheese, whatever you like—with five planes. One cut, two cuts, three, four, five. Each cut is a plane. These planes divide the space into several parts.
My question is: How many parts?
Who’s ready with a guess?
Student: 25?
Good! How did you get that?
Student: I did 5 × 5.
There’s an idea there. Great.
Anyone else?
Student: 32?
Oh! Interesting. You have something behind that.
Any more guesses?
Student: 10 spaces?
Okay, we’ll see who’s right.
But let’s pause. Guessing is just the beginning. Solving any real problem begins with a guess. A real problem always has some difficulty—otherwise, it wouldn't be a problem.
If you can’t solve it right away, what should you do? Wait for inspiration? No! The right thing is to imagine an easier problem to prepare you for the real one.
For example, I asked about five planes. Why not ask about fewer planes first?
So let’s go in order.
One plane:
Just one horizontal plane—like the surface of still water. It divides space into how many parts?
Student: Two.
Correct! Let's note that: one dividing plane → two parts.
Two planes:
Now imagine a second plane intersecting the first. You see two lines on the blackboard—those are intersections with the planes. Together, they divide space into how many parts?
Student: Four.
Right. Two planes → four parts.
Three planes:
Now, let’s imagine a third plane intersecting the other two.
Some parts are in front of the blackboard (the room), some behind. How many parts total?
Student: Eight.
Excellent.
Now let’s try a key idea in guessing: think of extreme cases.
What if there are zero dividing planes?
Then space is not divided—there’s just one region. So: zero planes → one part.
So far, we have this pattern:
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0 planes → 1 part
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1 plane → 2 parts
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2 planes → 4 parts
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3 planes → 8 parts
Now guess for four planes. Hands up!
Student: 16?
Who seconds that? Many of you? Good!
Yes, 16 is a reasonable guess. Why?
Because you observed the pattern: 1, 2, 4, 8… these are powers of 2. 2⁰, 2¹, 2², 2³.
You also had the courage to say: and so on. That’s an inductive generalization.
Is it proven?
Students: No.
Good. So let’s mark it with a question mark. It's a doubtful result—unproven.
Still, it’s a reasonable guess.
Now, how to test it?
Let’s actually try dividing space with four planes and count the resulting parts.
(He draws and demonstrates with tetrahedron.)
By analogy with dividing the plane by lines, if three lines in a plane create a triangle, four planes in space create a tetrahedron—a finite, enclosed solid.
We now count:
Add them: 1 + 4 + 6 + 4 = 15
But we guessed 16!
So what happened?
Answer: A very good guess—reasonable, inductive—but wrong. And that is the key experience.
Reasonable Guesses Can Be Wrong
That’s the lesson. A guess—even a respectable, inductive one—can be wrong. This is important in science and in life.
Now, let’s extend this analogy further.
We’ve looked at:
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Space divided by planes
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Planes divided by lines
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Lines divided by points
Let’s take lines divided by points.
So, we see a simple pattern here: number of parts = number of dividing elements + 1
Could we find a similar formula for space divided by planes?
This is an ambitious question—and it’s good to be ambitious in mathematics.
Return to the Original Question
So, if five planes divide space, how many parts result?
Look at our earlier numbers:
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0 planes → 1
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1 plane → 2
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2 planes → 4
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3 planes → 8
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4 planes → 15
Now what about 5 planes? What number should come next?
Students offer guesses: 21, 26, 28, 29...
One student suggests 26 and explains:
"I saw a pattern. In the columns of a triangular number table, if you add two previous values, you get the next."
This is another inductive guess. Is it proved?
Students: No.
Right. But do you believe it more now, since 11 (in a 2D analogy) was verified?
Students: More.
That’s reasonable. The verification of 11 gives us inductive evidence for the pattern.
Final Thoughts
So 26 is not proven. But it has more credibility now.
The key points in reasonable guessing are:
Let me close with a story about that distinction.
A member of the Royal Society in England once rushed into a meeting and gave his hat to the janitor without taking a ticket. After the meeting, the janitor handed it back to him.
The scientist said, a bit patronizingly, “How did you know it was my hat?”
The janitor replied sharply:
“Sir, I don’t know whether it’s your hat. It’s the hat you gave me.”
That’s the difference between a fact and a theory—a guess.
