## That and Why

When I was a kid, my parents had two ways of justifying rules. In the first class there was a pretty understandable reason:

Me: Why do I have to brush my teeth?
Mom: Because it will give you a beautiful smile.
Mom: Oh, please don’t tell him things like that, honey.
Dad: Don’t tell me what to do, woman.
Mom: Are you off your meds again?
Dad: What did I just say? Everyone’s a critic. [to the pet turtle] What are you staring at you retractable hockey puck?
Mom: Mark, dear, see, this is what happens. Your father didn’t brush his teeth when he was a little boy.
Dad: That turtle is a demon. Somebody get me a soldering iron and some holy water.

These days I have some doubts as to the authenticity of those little performances, but they were certainly effective. On the other hand, sometimes my parents’ justifications could be a little obscure:

Me: Why do I have to take out the trash?
Mom: Because I say so.
Dad: And because if you don’t, I will tell you in detail what sex really is, and remember in my Navy days I did two tours on a submarine.

Both are devastatingly convincing – either way I am completely sure I need to do my chores. But in only one case do I feel like there’s a real reason why.

I recently saw this mathematical relation somewhere (I forget where, but it’s pretty well-known):

$1^3 + 2^3 + \ldots + n^3 = (1 + 2 + \ldots + n)^2$

For example, if n = 5, then

$1^3 + 2^3 + 3^3 + 4^3 + 5^3 = 1 + 8 + 27 + 64 + 125 = 225$

and

$(1 + 2 + 3 + 4 + 5)^2 = 15^2 = 225$

This will be true no matter how big $n$ gets. Obviously no one has checked all the way up to $n = 935467568777043682111$, for example. Even with a computer it would not be possible, and if you did check up to that number, how do you know it would still work for that, plus one? We’ll come back to this.

A simpler example is this one:

$2 + 4 + 6 + \ldots + 2n = n(n+1)$.

Again you can check it out for as many numbers as you want. For $n = 7$ it says

$2 + 4 + 6 + 8 + 10 + 12 + 14 = 7*(7+1)$

And that’s right. The left hand side adds up to 56 and the right hand side is 7*8 = 56.

The idea in math, though, is to show that it’s always true, even for $n$ equal to the number of stars in the universe.

Here is proof based on dominoes with dots on them. We’ll lay the dominoes out on a table so that both the sum $2 + 4 + \ldots + n$ and the multiplication problem $n(n+1)$ count the number of dots on the dominoes.

Our dominoes will have two dots, one on each half, like this:

One domino. Two dots. This will be the 2 from 2 + 4 + ... + 2n

That domino represents 2, the first number in our sum $2 + 4 + \ldots + 2n$.

Next we add 4, which means 2 more dominoes:

Two more dominoes with four more dots. Now we have 2 + 4 dots.

But we can rearrange the dominoes however we want and still keep the number of dots the same, so for the hell of it, let’s put them like this:

The same dominoes rearranged.

Now we add three more dominoes to get this:

Adding three more dominoes. Now we have 2 + 4 + 6 dots.

Can you see the pattern? Each time we want to add a another number in the series, we add another layer of dominoes around the edge of the rectangle. It’ll be more obvious if we gray out the middle layer a little.

Same image with the layers highlighted. Each layer is one number from the sum (2 + 4 + ... + 2n) and n is the number of layers.

We can keep doing that until we have $n$ layers of dominoes. So, for $n=5$, our finished picture would look like this:

For n = 5, we have (2 + 4 + 6 + 8 + 10) dots.

We know that we’ve added the numbers $2 + 4 + 6 + 8 + 10$ because each layer has one more domino, hence two more dots, than the previous one. But we can also use a shortcut to count the dots. Ignoring the edges of the dominoes and just focusing on the dots, we see that the dots form a rectangle. The rectangle is 5 dots high and 6 dots long, so the total number of dots is 5*6 = 30. This checks out. 2 + 4 + 6 + 8 + 10 = 30.

The same pattern will hold for as many layer as we please. It’s clear from the picture that each new layer adds two dots, so $n$ layers will have $2 + 4 + \ldots + n$ dots. But it’s also clear that each new layer makes the rectangle one dot higher and one dot longer, so that in all the rectangle is $n$ dots high and $n+1$ dots long. That means the total number of dots can be counted two ways, and since the number of dots is the same either way,

$2 + 4 + \ldots + 2n = n(n+1)$

That’s one sort of proof. We might say, after seeing this proof, that now we understand not only that the equation

$2 + 4 + \ldots + 2n = n(n+1)$

is true, but also why it is true.

This is a subjective thing. This particular proof makes a lot of sense to me, but to someone else it might not. The proof is very informal. What if there’s an error I just don’t see?

Let’s look at a different type of proof – a more formal type based on symbols rather than pictures. This time we’ll prove the more difficult equality

$1^3 + 2^3 + \ldots + n^3 = (1 + 2 + \ldots + n)^2$

from the beginning of the post.

$2 + 4 + \ldots + 2n = n(n+1)$

so dividing both sides by $2$

$1 + 2 + \ldots + n = \frac{n(n+1)}{2}$

Squaring,

$(1 + 2 + \ldots + n)^2 = \frac{n^2(n+1)^2}{4} = \frac{n^4 + 2n^3 + n^2}{4}$

That’s the right hand side. Now for the left. By algebra

$k^4 - (k-1)^4 = 4k^3 - 6k^2 + 4k - 1$

so

$\sum_{k=1}^n k^4 - (k-1)^4 = 4\sum_{k=1}^n k^3 - 6\sum_{k=1}^n k^2 + 4\sum_{k=1}^n k - \sum_{k=1}^n 1$

But also

$\sum_{k=1}^n k^4 - (k-1)^4 = \sum_{k=1}^n k^4 - \sum_{k = 1}^n (k-1)^4 = \sum_{k=1}^n k^4 - \sum_{k=0}^{n-1} k^4 = n^4$

So

$n^4 = 4\sum_{k=1}^n k^3 - 6\sum_{k=1}^n k^2 + 4\sum_{k=1}^n k - \sum_{k=1}^n 1$

or

$\sum_1^n k^3 = \frac{n^4 + 6\sum_1^n k^2 + 4 \sum_1^n k - \sum_1^n 1}{4}$.

In general, we can find the sum $\sum_{k=1}^n k^p$ for any $p$ based on the binomial coefficients and the sums for lesser powers. Simplifying out the algebra in this case gives

$\sum_{k=1}^n k^3 = \frac{n^4 + 2n^3 + n^2}{4}$,

which is the same as the result from before, so

$1^3 + 2^3 + \ldots + n^3 = (1 + 2 + \ldots + n)^2$.

This proof is pretty much solid. You could make it formal and rigorous if you wanted to. But unlike the first proof, I don’t get from it much sense of the “why”. Sometimes I feel like numbers are just telling me what to do and threatening me with horrible consequences if I don’t.