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. In your lab there are 36 habitat cells, arranged in an inverted pyramid, with 8 cells in the top row. The first has one rabbit, the second has two, and so on, with eight rabbits in the last one. The other rows of cells are empty… for now. 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. 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. Each cell can hold 10^80 nano-rabbits – that’s a 1 followed by 80 zeros – before they break free  and overrun the world. Your calculations have given you a  46-digit number for the count of rabbits  in the bottom cell– plenty of room to spare. But just as you pull the lever  to start the experiment, your assistant runs in with terrible news. A rival lab has sabotaged your code so that all the zeros at the end  of your results got cut off. 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! To make matters worse, your devices and calculators  are all malfunctioning, so you only have a few minutes  to work it out by hand. How many trailing zeros should there be at the end of the count of rabbits  in the bottom habitat? And do you need to pull the emergency  shut-down lever? Pause the video now if you want  to figure it out for yourself. 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. The good news is we don’t need to. All we need to figure out is how many trailing zeros it has. But how can we know how many trailing  zeros a number has without calculating the number itself? What we do know is that we arrive at the number of rabbits in the bottom cell through a process of multiplication – literally. The number of rabbits in each cell is the product of the number of rabbits  in each of the two cells above it. 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. Let’s calculate the number of rabbits  in the second row and see what patterns emerge. Two of the numbers have trailing zeros – 20 rabbits in the fourth cell  and 30 in the fifth cell. But there are no numbers ending in 5. 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. That means we only need to worry about the numbers that have  trailing zeros themselves. 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. So let’s take the numbers in the fourth  and fifth cells and multiply down from there. 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. When we continue all the way down, we end up with 35 zeros  in the bottom cell. 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. Adding those 35 zeros to the  46 digit number we had before yields an 81 digit number – too big for the habitat to contain! You rush over and pull  the emergency switch just as the seventh generation of rabbits  was about to mature – hare-raisingly close to disaster.