Back in 2004, I saw this Slashdot comment:

Every hard drive I’ve ever bought has been larger than all my previous hard drives combined. And this is without even trying.

– blancolioni

Reflecting on this, I realized that this was also true of all the hard drives I had ever owned, beginning with the 20 MB one in the 80s.

I resolved to make this a target, instead of mere happenstance. I refuse to buy a new hard drive until I can afford one that would maintain this invariant.

To achieve this, I noted down the sizes of all the hard drives I’d owned, and every time I bought a new one, I would add to this list.

It took a few years for my brain to realize a simple mathematical fact: If I simply doubled the size of my current hard drive, the invariant would be satisfied.

Here’s why:

Let \(a_{0},a_{1},\dots,a_{n}\) be the sizes of all the hard drives I have bought.

Now the invariant dictates that:

\begin{equation*} a_{n}>a_{0}+\cdots+a_{n-1} \end{equation*}

If I add \(a_{n}\) to both sides:

\begin{equation*} 2a_{n}=a_{n}+a_{n}>a_{0}+\cdots+a_{n-1}+a_{n} \end{equation*}

To satisfy the invariant, simply ensure that \(a_{n+1}\ge2a_{n}\).

Saying: “I always buy a hard drive at least double the size of my existing one” doesn’t sound as cool as “Every hard drive I’ve ever bought has been larger than all my previous ones combined.”

I no longer maintain that list.

This was a rare occasion where mathematics made my life more dull.