Can you solve the multiplying rabbits riddle? - Alex Gendler

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.