Recorded at | January 10, 2019 |
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Event | TED-Ed |

Duration (min:sec) | 04:26 |

Video Type | TED-Ed Original |

Words per minute | 180.64 medium |

Readability (FK) | 58.58 easy |

Speaker | Alex Gendler |

Official TED page for this talk

**Synopsis**

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].

1 | 00:06 | After years of experiments, you’ve finally created the pets of the future– nano-rabbits! | ||

2 | 00:12 | They’re tiny, they’re fuzzy… and they multiply faster than the eye can see. | ||

3 | 00:18 | In your lab there are 36 habitat cells, arranged in an inverted pyramid, with 8 cells in the top row. | ||

4 | 00:26 | The first has one rabbit, the second has two, and so on, with eight rabbits in the last one. | ||

5 | 00:33 | The other rows of cells are empty… for now. | ||

6 | 00:37 | 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. | ||

7 | 00:49 | 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. | ||

8 | 00:59 | Each cell can hold 10^80 nano-rabbits – that’s a 1 followed by 80 zeros – before they break free and overrun the world. | ||

9 | 01:10 | Your calculations have given you a 46-digit number for the count of rabbits in the bottom cell– plenty of room to spare. | ||

10 | 01:19 | But just as you pull the lever to start the experiment, your assistant runs in with terrible news. | ||

11 | 01:25 | A rival lab has sabotaged your code so that all the zeros at the end of your results got cut off. | ||

12 | 01:32 | That means you don’t actually know if the bottom cell will be able to hold all the rabbits – and the reproduction is already underway! | ||

13 | 01:40 | To make matters worse, your devices and calculators are all malfunctioning, so you only have a few minutes to work it out by hand. | ||

14 | 01:49 | How many trailing zeros should there be at the end of the count of rabbits in the bottom habitat? | ||

15 | 01:55 | And do you need to pull the emergency shut-down lever? | ||

16 | 01:59 | Pause the video now if you want to figure it out for yourself. | ||

17 | 02:02 | 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. | ||

18 | 02:13 | The good news is we don’t need to. | ||

19 | 02:15 | All we need to figure out is how many trailing zeros it has. | ||

20 | 02:19 | But how can we know how many trailing zeros a number has without calculating the number itself? | ||

21 | 02:25 | What we do know is that we arrive at the number of rabbits in the bottom cell through a process of multiplication – literally. | ||

22 | 02:33 | The number of rabbits in each cell is the product of the number of rabbits in each of the two cells above it. | ||

23 | 02:38 | 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. | ||

24 | 02:52 | Let’s calculate the number of rabbits in the second row and see what patterns emerge. | ||

25 | 02:57 | Two of the numbers have trailing zeros – 20 rabbits in the fourth cell and 30 in the fifth cell. | ||

26 | 03:03 | But there are no numbers ending in 5. | ||

27 | 03:05 | 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. | ||

28 | 03:15 | That means we only need to worry about the numbers that have trailing zeros themselves. | ||

29 | 03:20 | 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. | ||

30 | 03:32 | So let’s take the numbers in the fourth and fifth cells and multiply down from there. | ||

31 | 03:38 | 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. | ||

32 | 03:50 | When we continue all the way down, we end up with 35 zeros in the bottom cell. | ||

33 | 03:56 | 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. | ||

34 | 04:06 | Adding those 35 zeros to the 46 digit number we had before yields an 81 digit number – too big for the habitat to contain! | ||

35 | 04:16 | You rush over and pull the emergency switch just as the seventh generation of rabbits was about to mature – hare-raisingly close to disaster. |