4.1: Adding Odd Numbers

Have you ever worked through a proof, understood and confirmed each step, yet still not believed the theorem? You realize that the theorem is true, but not why it is true.

To see the same contrast in a familiar example, imagine learning that your child has a fever and hearing the temperature in Fahrenheit or Celsius degrees, whichever is less familiar. In my everyday experience, temperatures are mostly in Fahrenheit. When I hear about a temperature of \(40^{◦}C\), I therefore react in two stages:

1. I convert \(40^{◦}C\) to Fahrenheit: \(40 \times 1.8 + 32 = 104.\)
2. I react: “Wow, \(104^{◦}F\). That’s dangerous! Get thee to a doctor!”

The Celsius temperature, although symbolically equivalent to the Fahrenheit temperature, elicits no reaction. My danger sense activates only after the temperature conversion connects the temperature to my experience.

A symbolic description, whether a proof or an unfamiliar temperature, is unconvincing compared to an argument that speaks to our perceptual system. The reason lies in how our brains acquired the capacity for symbolic reasoning. (See Evolving Brains [2] for an illustrated, scholarly history of the brain.) Symbolic, sequential reasoning requires language, which has evolved for only \(10^{5}\) yr. Although \(10^{5}\) yr spans many human lifetimes, it is an evolutionary eyeblink. In particular, it is short compared to the time span over which our perceptual hardware has evolved: For several hundred million years, organisms have refined their capacities for hearing, smelling, tasting, touching, and seeing.

Evolution has worked 1000 times longer on our perceptual abilities than on our symbolic-reasoning abilities. Compared to our perceptual hardware, our symbolic, sequential hardware is an ill-developed latecomer. Not surprisingly, our perceptual abilities far surpass our symbolic abilities. Even an apparently high-level symbolic activity such as playing grand master chess uses mostly perceptual hardware [16]. Seeing an idea conveys to us a depth of understanding that a symbolic description of it cannot easily match.

Multiple problems

Problem 4.1 Computers versus people

At tasks like expanding \((x + 2y)^{50}\), computers are much faster than people. At tasks like recognizing faces or smells, even young children are much faster than current computers. How do you explain these contrasts?

Problem 4.2 Linguistic evidence for the importance of perception

In your favorite language(s), think of the many sensory synonyms for under- standing (for example, grasping).

Adding odd numbers

To illustrate the value of pictures, let’s find the sum of the first \(n\) odd numbers (also the subject of Problem 2.25):

\[S_{n} = \underbrace{1 + 3 + 5 + ... + (2n - 1).}_{n \text{terms}} \label{4.1} \]

Easy cases such as \(n\) = 1, 2, or 3 lead to the conjecture that \(S_{n} = n^{2}\). But how can the conjecture be proved? The standard symbolic method is proof by induction:

1. Verify that \(S_{n}\) = \(n^{2}\) for the base case \(n = 1\). In that case, \(S_{1}\) is 1, as is \(n_{2}\), so the base case is verified.

2. Make the induction hypothesis: Assume that \(S_{m}\) = \(m^{2}\) for m less than or equal to a maximum value n. For this proof, the following, weaker induction hypothesis is sufficient:

\[\sum_{1}^{n} (2k - 1) = n^{2} \label{4.2} \]

In other words, we assume the theorem only in the case that \(m = n\).

3. Perform the induction step: Use the induction hypothesis to show that \(S_{n+1}\) = (n + 1)\(^{2}\). The sum \(S_{n+1}\) splits into two pieces:

\[S_{n+1}=\sum_{1}^{n+1}(2 k-1)=(2 n+1)+\sum_{1}^{n}(2 k-1)\label{4.3} \]

Thanks to the induction hypothesis, the sum on the right is \(n^{2}\). Thus

\[S_{n+1} = (2n + 1) + n^{2}, \label{4.4} \]

which is \((n + 1)^{2}\); and the theorem is proved.

Although these steps prove the theorem, why the sum \(S_{n}\) ends up as \(n^{2}\) still feels elusive.

That missing understanding the kind of gestalt insight described by Wertheimer [48] requires a pictorial proof. Start by drawing each odd number as an L-shaped puzzle piece:

\[\label{4.5} \]

Question

How do these pieces fit together?

Then compute \(S_{n}\) by fitting together the puzzle pieces as follows:

\[label{4.6} \nonumber \]

Each successive odd number each piece extends the square by 1 unit in height and width, so the \(n\) terms build an \(n \times n\) square. [Or is it an \((n − 1) \times (n − 1)\) square?] Therefore, their sum is \(n^{2}\). After grasping this pictorial proof, you cannot forget why adding up the first n odd numbers produces \(n^{2}\).

Multiple problems

Problem 4.3 Triangular numbers

Draw a picture or pictures to show that

\[1 + 2 + 3 + ··· + n + ··· + 3 + 2 + 1 = n^{2}. \label{4.7} \]

Then show that

\[1 + 2 + 3 + ··· + n = \frac{n(n+1)}{2}.\label{4.8} \]

Problem 4.4 Three dimensions

Draw a picture to show that

\[\sum_{0}^{n} (3k^{2} + 3k + 1) = (n + 1)^{3}. \label{4.9} \]

Give pictorial explanations for the 1 in the summand \(3k^{2} + 3k + 1\); for the 3 and the \(k^{2}\) in \(3k^{2}\); and for the 3 and the k in 3k.