### Learn Math With Troubling Tribbles!

#### By Kyle Hill on May 8, 2014

The original series of *Star Trek *didn’t just break the boundaries of race and expand our vision of the future — it even got its math right. There really is a huge trouble with tribbles, and it has everything to do with exponential growth.

In the classic 1967 episode, it only takes one tribble on the Enterprise and a few hours to produce waves of tribbles pouring out of the elevators and vents. Doctor “Bones” McCoy says that’s because the little furballs are basically “born pregnant.” Captain Kirk estimates that after three days on the ship, there could be hundreds of thousands of them squirming around. But Spock corrects him and announces that there are exactly 1,771,561 tribbles on board. He is exactly right because the writers of *Star Trek *knew how to work a calculator.

Like how bacteria grow or kaiju emerge, tribble reproduction follows an exponential equation that you can derive without much work. In fact, it’s easy enough to do so that I quickly found half a dozen math teachers and communicators that have used tribbles to teach students about the concept. The only logical thing to do is to try it out and come up with *the* tribble growth equation.

So, how did Spock know how many tribbles were on the Enterprise at the end of three days?

If you suspect that some population is growing at an exponential rate (e.g., it doubles itself pretty quickly), there is a certain way to write that mathematically. Take a look at my notepad:

Above, I started with what I knew. For P (population) at time 0 (when the tribbles first came aboard), there was 1 tribble. Easy enough. We also knew, thanks to Spock, that after 72 hours, P was equal to 1,771,561. Now it gets a little more complicated.

To write a useful equation, what we really want to know is what the population of tribbles will be at any one time. Therefore, I wrote **Population=P(t)**. But we know that tribbles grow exponentially (or we are checking that assumption), so we have to include that information. To find the *rate* of tribble growth, we take the mathematical derivative of Population=P(t) to get **P’(t)**. Putting those two together we get **P’(t)=kP(t)**. (This equation says that the rate of growth is equal to some constant **k** times the population at some time. **K **pops out of the normal process of taking the derivative of a function.)

So far so good, aspiring Vulcan?

Now, unless you took calculus, the next part might be a bit confusing. We have an equation that includes the tribble population at any one time and the growth rate. But we want an equation that pops out a number of tribbles, not just a growth rate, like Spock did. So, we have to turn **P’(t) **back into **P(t)**. To do that, you *integrate*. Integrating both sides of **P’(t)=kP(t) **produces something that we can finally work with: **P(t)=ce^ ^{kt}**. (You can learn more about basic integration here.)

**P(t)=ce^ ^{kt}** happens to be the very same equation that all biology students learn when studying basic population growth, and we basically just discovered it… how logical.

*[We are defining this equation based on the numbers Spock told us, but how do we know if Spock’s numbers are correct? There is an even simpler equation that you can follow. In the episode, we find out that one tribble has 10 babies every 12 hours. So, twelve hours after boarding, one tribble has 10 babies, meaning that there are 11 tribbles on board. At 24 hours, those 11 tribbles each have 10 babies, putting 121 total tribbles on the Enterprise. Follow this simple multiplication and addition process out to 72 hours, and you get 1,771,561 tribbles—exactly what Spock said!]*

Equation in hand, we can now plug in what we know about tribbles and get the final result. Looking back up at my notes, you see that I calculated that **c=1**. You get that from the initial conditions. If you plug **0 **in for **t **and **1 **in for **P(t)**, then **c=1**. All that’s left is to find **k**, or the rate of tribble growth.

If there are 1,771,561 tribbles on the Enterprise after 72 hours, our equation becomes **1771561=e^ ^{72k}**. To get that

**72k**down from that constant

**e**, you can do a little math trick by taking the natural logarithm—

*ln*—of both sides (a modern calculator makes this pretty easy). If you do, we get

**k=0.1998**, or about

**0.2**.

There you have it! The equation for tribble growth is **P(t)=e^ ^{0.2t}**. Now let’s mess around with it.

If you graph the tribble growth equation, you get an exponential curve that looks like this:

Look closely at the right side of the graph. Exponential growth means that you will be having serious trouble with millions of tribbles within a few days. In fact, if the tribbles stayed a bit longer on the Enterprise, they would have suffocated everyone with wriggling fuzz very quickly. The next graph shows the danger:

There was serious tribble trouble at 72 hours. At around 100 hours of growth, there would be a tribble occupying every cubic inch of volume on the Enterprise. Kirk and his crew would drown in pregnant, fuzzy aliens. And because the growth is exponential, it would get much worse very quickly. Less than 8 hours later, our equation tells us that Spock and Kirk would have to deal with *a thousand times* more tribbles — well over two billion. That’s enough to fill 1,200 Borg Cubes completely. Conclusions like that are worth all the math.

Left unchecked, the tribble birth rate equation shows an unstoppable expansion of exponential growth towards domination. That’s something the Klingons know all too well.

—

*Kyle Hill is the Chief Science Officer of the Nerdist enterprise. Follow the nerdery on Twitter @Sci_Phile.*

## 6 comments

[…] tribbles. Marvellous maths. Fantastic fun, by Kyle […]

This, of course, assumes that the tribbles had access to unlimited amounts of food, but IIRC, the tribbles in the holds had eaten all the wheat, and it was the access to such a large amount of food that caused their rapid population rise.

Considering that Gerrold lifted the basic math and problem from Heinlein’s “The Rolling Stones”, from a cahpter titled “Flat Cats Factorial”, the math ought to work.

Except the tribbles Spock counted weren’t on the Enterprise. They were on the space station.

Minus the ones that died from the poisoned quadrotriticale of course.

Thank you so much. I love this…