Tribble Math

Quote of the Day

Politics is ethics done in public.

— Bernard Crick

Introduction

Figure 1: Captain Kirk with Tribbles. (Source)

Over the 2016 New Year's holiday, a number of television stations were showing the remastered original Star Trek episodes. While you generally do not hear much math in a Star Trek episode, the "Trouble with Tribbles" actually had two statements to which a bit math can be applied. You can search around the Internet and find solutions for one of the statements, but I wanted to work through the math myself. It is interesting that a script writer must have actually worked through the details during time before calculators.

My Mathcad worksheet and a PDF are included here.

Background

Math Statement #1

Here is a quote from the episode (Source) that sets up the math problem.

KIRK: Here. Let me try it.
(But he can't open it either, so he tried one of the overhead doors instead. That does open, and a whole load of tribbles fall out, burying the gallant Captain up to his shoulders. Spock examines one.)
SPOCK: They seem to be gorged.
BARIS: Gorged? On my grain? Kirk, I am going to hold you responsible. There must be thousands of them.
KIRK: Hundreds of thousands.
SPOCK: One million seven hundred seventy one thousand five hundred sixty one. That's assuming one tribble, multiplying with an average litter of ten, producing a new generation every twelve hours over a period of three days.
KIRK: That's assuming they got here three days ago.
SPOCK: And allowing for the amount of grain consumed and the volume of the storage compartment.

My plan here is to duplicate the Spock's math here.

Math Statement #2

The other math statement, which left much information out, was how long it would take Cyrano Jones to cleanup all the dead tribbles – 17.9 years.

KIRK: There is one thing you can do.
JONES: Yes?
KIRK: Pick up every tribble on the space station. If you do that, I'll speak to Mister Lurry about returning your spaceship.
JONES: It would take years.
SPOCK: Seventeen point nine, to be exact.
JONES: Seventeen point nine years.
KIRK: Consider it Job security.
JONES: Captain, you're a hard man. All right! All right!

I am going to treat this statement as a Fermi problem and estimate the average tribble cleanup rate that Cyrano Jones will need to maintain to clean up all the tribbles from Math Problem #1 within 17.9 years.

Solution of a First-Order Difference Equation

Math problem #1 can be expressed as a first-order difference equation, which has the solution shown in Figure 2.

Figure 2: Solution of a First-Order Differential Equation. (Source)

Analysis

Math Statement #1 Solution

Figure 3 shows my solution for the number of tribbles after 3 days.

Figure 3: Math Problem #1 Solution.

Math Statement #2 Solution

I will estimate the tribble cleanup rate that Cyrano Jones will need to maintain of 17.9 years in order to clean up all the tribbles from problem #1. Based on the assumptions laid out in Figure 4, Cyrano Jones will need to average picking up 35 tribble per hour over the 17.9 years in order to clean all the tribbles up.

Figure 4: Math Problem #2 Solution.

Conclusion

This was good problem that left me pleased knowing the episode author, David Gerrold, must know a bit of math.

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