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.
