Alex Gendler: Can you solve the multiplying rabbits riddle?

Recorded atJanuary 10, 2019
Duration (min:sec)04:26
Video TypeTED-Ed Original
Words per minute180.64 medium
Readability (FK)58.58 easy
SpeakerAlex Gendler

Official TED page for this talk


After years of experiments, you've finally created the pets of the future – nano-rabbits! They're tiny, they're fuzzy ... and they multiply faster than the eye can see. But a rival lab has sabotaged you, threatening the survival of your new friends. Can you figure out how to avert this hare-raising catastrophe? Alex Gendler shows how. [Directed Artrake Studio, narrated by Addison Anderson].

Text Highlight (experimental)
100:06 After years of experiments, you’ve finally created the pets of the future– nano-rabbits!
200:12 They’re tiny, they’re fuzzy… and they multiply faster than the eye can see.
300:18 In your lab there are 36 habitat cells, arranged in an inverted pyramid, with 8 cells in the top row.
400:26 The first has one rabbit, the second has two, and so on, with eight rabbits in the last one.
500:33 The other rows of cells are empty… for now.
600:37 The rabbits are hermaphroditic, and each rabbit in a given cell will breed once with every rabbit in the horizontally adjacent cells, producing exactly one offspring each time.
700:49 The newborn rabbits will drop into the cell directly below the two cells of its parents, and within minutes will mature and reproduce in turn.
800:59 Each cell can hold 10^80 nano-rabbits – that’s a 1 followed by 80 zeros – before they break free and overrun the world.
901:10 Your calculations have given you a 46-digit number for the count of rabbits in the bottom cellplenty of room to spare.
1001:19 But just as you pull the lever to start the experiment, your assistant runs in with terrible news.
1101:25 A rival lab has sabotaged your code so that all the zeros at the end of your results got cut off.
1201:32 That means you don’t actually know if the bottom cell will be able to hold all the rabbitsand the reproduction is already underway!
1301:40 To make matters worse, your devices and calculators are all malfunctioning, so you only have a few minutes to work it out by hand.
1401:49 How many trailing zeros should there be at the end of the count of rabbits in the bottom habitat?
1501:55 And do you need to pull the emergency shut-down lever?
1601:59 Pause the video now if you want to figure it out for yourself.
1702:02 Answer in 3 Answer in 2 Answer in 1 There isn’t enough time to calculate the exact number of rabbits in the final cell.
1802:13 The good news is we don’t need to.
1902:15 All we need to figure out is how many trailing zeros it has.
2002:19 But how can we know how many trailing zeros a number has without calculating the number itself?
2102:25 What we do know is that we arrive at the number of rabbits in the bottom cell through a process of multiplicationliterally.
2202:33 The number of rabbits in each cell is the product of the number of rabbits in each of the two cells above it.
2302:38 And there are only two ways to get numbers with trailing zeros through multiplication: either multiplying a number ending in 5 by any even number, or by multiplying numbers that have trailing zeroes themselves.
2402:52 Let’s calculate the number of rabbits in the second row and see what patterns emerge.
2502:57 Two of the numbers have trailing zeros – 20 rabbits in the fourth cell and 30 in the fifth cell.
2603:03 But there are no numbers ending in 5.
2703:05 And since the only way to get a number ending in 5 through multiplication is by starting with a number ending in 5, there won’t be any more down the line either.
2803:15 That means we only need to worry about the numbers that have trailing zeros themselves.
2903:20 And a neat trick to figure out the amount of trailing zeros in a product is to count and add the trailing zeros in each of the factors – for example, 10 x 100 = 1,000.
3003:32 So let’s take the numbers in the fourth and fifth cells and multiply down from there.
3103:38 20 and 30 each have one zero, so the product of both cells will have two trailing zeros, while the product of either cell and an adjacent non-zero-ending cell will have only one.
3203:50 When we continue all the way down, we end up with 35 zeros in the bottom cell.
3303:56 And if you’re not too stressed about the potential nano-rabbit apocalypse, you might notice that counting the zeros this way forms part of Pascal’s triangle.
3404:06 Adding those 35 zeros to the 46 digit number we had before yields an 81 digit number – too big for the habitat to contain!
3504:16 You rush over and pull the emergency switch just as the seventh generation of rabbits was about to mature – hare-raisingly close to disaster.