I’m often captured by random ideas. This is one that came to me the other night, when I should have been sleeping.

In the Lord of the Rings series, there are a bunch of Elves floating around. Not literally, but, allow me a bit of artistic license here, please.

One of the main Elvish dudes is Elrond. Elrond is supposedly about 3,000 years old. OK, he probably has a really good 401(k). But, more than that, I’m intrigued with why there are so few Elves. I mean, that’s a long time to have kids.

If Elrond had a kid every 100 years, he’d have 30 kids. But what if the kids breed too?

If each of them breeds 100 years after their birth, and then every 100 years thereafter, the oldest kid would have 29 kids.

But, oops, those kids all breed too.

Trade in these wings for some wheels
—Bruce Springsteen, “Thunder Road”

Hmm. Let’s make this problem simpler.

Assume there are 1 Elf, he’s 2 years old, and he breeds every year after his first year. So, he’s got 1 kid, who hasn’t bred yet. So, in 2 years, he’s doubled the Elf ranks.

Now, make that dude 1 year older. Now he’s 3, he’s got 2 kids (one 2 and the other 1 year old). The 2-year-old has one kid. In 3 years, the Elves have quadrupled (from poor lonely Elrond to Elrond, older kid, younger kid, and grandkid)

// Side note: I have to introduce some nomenclature here to make this work. Kid#1 is Elrond’s first kid. Any kids that kid#1 has will be marked as kid#1-kid#<n>. So kid#1-kid#2 is the 2nd kid of the oldest kid. //

4 years yields Elrond, kid#1, kid#2, kid#3, kid#1-kid#1, kid#1-kid#2, kid#2-kid#1, or 7 Elves.

By 5 years, the forest is getting crowded. You have:
Elrond
Kid#1 (who now has kid#1-kid#1, kid#1-kid#2, kid#1-kid#3)
Kid#2 (who now has kid#2-kid#1, kid#2-kid#2)
Kid#3 (who now has kid#3-kid#1)
Kid#4

That’s a total of 11 Elves.

Let’s do the math, a bit. For every new year you add, you get one more Elrond kid. You also create the opportunity for one more kid to have a kid. But it takes too long to do this work by hand.

This is a series expansion that is a variant of a Triangle number.

// Side note: Talking to others is good. I exploited my brother’s big math brain a lot for this post. All screwups to me, all smartness to him. //

He's got the saints and the sinners
Coming up from behind
-Eurythmics

It works out to adding up the sum of all numbers less than the generational age and correcting for the extra generational count. For 3, for example, it is 1 (Elrond) plus (2 + 1) = 1 + 3 = 4. For 5, it is 1 (Elrond) plus (4 + 3 + 2 + 1).

This ultimately becomes 1/2 * (generation^2 - generation) + 1.

So, back to our buddy Elrond. If he is 3,000 years old, and gets to breed once per century, he had 30 generations. Which, plugging 30 into our handy-dandy formula, yields 436.

That's a lot of Elves.

And what if there are a lot of Elronds? Like, say 10,000. That is 4.4M Elves.

But wait! There were supposedly “great cities” full of Elves, which sounds like millions of elves to start with. Let's say there were 1M Elves; now Elrond's 30 generations would have made 400M Elves, which is more than the total population of the US.

So, my question… why aren’t Elves like cockroaches, sitting around everywhere, generally in the way.

It’s stuff like this that keeps me up nights.