How Many Tribbles Were There?

Question

In the Star Trek episode "The Trouble with Tribbles," when Kirk opens the door and all the tribbles fall on him, Spock says that there are 1,771,561 tribbles in the wheat compartment, basing that calculation on one tribble entering 3 days ago, producing an average litter of 10 every 12 hours (and also factors in space and amount of wheat).

Is this based on any math or has David Gerrold ever said he did any math on this?

I started playing around and found the first litter, in 12 hours, would be 10 tribbles, and 10 tribbles having another litter in 12 hours would mean an additional 100, then an additional 1,000 12 hours later.

Then I realized I was leaving out the existing tribbles when counting each generation and tried to keep track in different columns, but made some place value mistakes. Still, it looked like that number, which I thought absurd at first, might be reasonable.

Has anyone ever done the actual math on this to see if 1,771,561 tribbles is reasonable under the circumstances?

Answer 1

Every 12 hours over 3 days is 6 litters. Start with one tribble. Every 12 hours, you multiply the number of tribbles by 11 (10 for children + 1 for self). So 6 litters would be 11^6 = 1,771,561. It appears I made the right assumptions and did the math right, because that's the number Spock got too.

Answer 2

Spock hedged his bets.

Spock's math is correct, if a tribble's life span is greater than 3 days. If 1 tribble produced a litter of 10 to 11 tribble offspring. Using the maximum number of 11 to the 6th power = 1,771,561.

A multiple of 11 every 12 hours for 3 days.

Knowing Spock he would round up making the maximum number possible the highest possible outcome. The actual number might have been smaller if some of the tribbles did not survive the entire three day period.

Answer 3

To add a little . . . the number given is 11^6. Now the question is how does Mr. Spock arrive at that in his head. Evaluate 11^6 as the binomial expansion of (1+10)^6. Recalling that the binomial coefficients can be taken from the seventh row of Pascal's triangle, you have: 1 * 10^6 + 6 * 10^5 + 15 * 10^4 + 20 * 10^3 + 15 * 10^2 + 6 * 10 + 1 = 1771561 The final addition is the only brain taxing part. It took me a couple minutes of staring into space, so no Vulcan Academy, here. But since the episode was broadcast in 1967 (before inexpensive pocket calculators), this trick would have been known to any high school student.

Answer 4

Starting with the original tribble and its 10 offspring at:

12 hours you have (1 + 10)=11
24 hrs you then have (11^2)=121,
36 hrs (11^3) = 1331,
48 hrs (11^4) = 14641,
60 hrs (11^5) = 161051,
72 hrs (11^6) = 17715561,

So Spock's calculations should be right by adding the 1 that produced the 10 offspring gives you 11 instead of 10 for each new litter.

Answer 5

See the other answers for the math, 1,771,561 is the highest possible number. It should be noted, however, that Sisko and Dax found many of the Tribbles in the food processor had died. As there is no way to know how long they were dead, this number may be lower. I'd guesstimate around 1.2-1.5 million.

Answer 6

1,771,561 is the correct answer. You need to account for the compounding interest.

You have to think that the population INCREASES by 1,000% (10 offspring) every generation. The first tribble has 10 offspring plus the mother for a total of 11 which is 11^1. Those 11 have 110 offspring plus their 11 mothers totals 121 which is 11^2.

Answer 7

But if you multiply 11^6, then aren't you counting the newly birthed tribble in the next iteration as well as the one in which it was just born? And that's assuming that you started with 10 tribbles already rather than just one and that you start the clock at the birth of the first set of litters rather than 12 hours from then.

Look at it this way, you can't just calculate 10^6, because after the first litter is born from 10 starting tribbles you have 110 tribbles (the 10 new tribbles in a new litter times ten litters, plus the original 10 who birthed them). You do not end up with 100 (10^2) or 121 (11^2).

No matter how you slice it, Spock and Dax had faulty logic when they made their calculations. Based on an average litter of 10, starting with 10 tribbles who are about to give birth to their first litter, you would have only 1,100,000 tribbles after 72 hours!

Answer 8

I did some math to calculate approximately how large a storage compartment would need to be to hold 1,771,561 tribbles. Considering that a tribble is mostly fur, you could probably compress one down to a three inch square, and in the storage compartment they would be stacked on top of each other. So, based on those assumptions, 64 of them could fit in a volume of 1 cubic foot. Now, let's assume that the floor of the storage compartment was 12 feet by 12 feet. That means that each foot of height in the compartment has 144 cubic feet of area. If each cubic foot of area could contain 64 tribbles, then each one foot layer of the compartment could hold 144 X 64 = 9,216 tribbles. How many one foot layers would be needed for 1,771,561 tribbles? 1,771,561 ÷ 9,216 = 192.2. How large would the storage compartment need to be? 12 feet by 12 feet, and 192.2 feet high. Captain Kirk would not survive the onslaught.