Can you solve the honeybee riddle? - Dan Finkel

You’re a biologist on a mission

to keep the rare honeybee Apis
Trifecta from going extinct.

The last 60 bees of the species
are in your terrarium.

You’ve already constructed wire frames
of the appropriate size and shape.

Now you need to turn them into working
beehives

by helping the bees fill
every hex with wax.

There are two ways to fill a given hex.

The first is to place a bee into it.

Once placed, a bee cannot be
removed without killing it.

The second:

if at any point an unfilled hex has three
or more neighboring wax-filled hexes,

the bees already in the hive will
move in and transform it.

Once the bees have transformed every hex
in a hive,

you can place an additional bee inside
and it’ll specialize into a queen.

The hive, if well cared for, will
eventually produce new bees

and continue the species.

If there are no hexes with three or more
transformed neighbors,

the bees will just sit and wait.

And once a bee transforms a hex,
it can never become a queen.

You could put 59 bees in one wire hive,
wait till they transform all the hexes,

and then create a queen.

But then just one collapse would end
the species.

The more viable hives you can
make now, the better.

So how many can you make with 60 bees?

Pause the video to figure
it out yourself

Answer in 3

Answer in 2

Answer in 1

Answer in 0

What you're looking for here is some
kind of self-sustaining chain reaction,

where a small number of bees will
transform an entire hive.

The lower the number of bees needed,
the better.

So how low can we go, and how can
we engineer a chain reaction?

Let’s start with the first question.

There's a really clever approach to this,

which involves counting the sides of the
filled-in hexes,

and examining their total perimeter.

Let’s suppose we put bees
in these three hexes.

The total transformed
perimeter has 18 sides.

But the middle hex has three
transformed neighbors,

so the bees will transform it too.

What happens to the perimeter?

It’s still 18!

And even after the bees transform the next
sets of hexes with three neighbors,

it still won’t change.

What’s going on here?

Each hex that has at least three sides
touching the bee-friendly space

will remove those sides from the perimeter
when it transforms.

Then it adds at most three new sides
to the perimeter.

So the perimeter of the transformed hexes
will either stay the same or shrink.

The final perimeter of
the entire hive is 54,

so the total perimeter of the hexes we
place bees in at the start

must be at least 54 as well.

Dividing that 54 by the six sides on each
non-adjacent hex

tells us it’ll take at least 9 bees to
transform the entire hive.

That’s a great start,

but we still have the tough question
of where the nine bees should go,

and if we’ll need more.

Let’s think smaller.

We already know that three bees could
completely transform a hive this big.

What about a slightly bigger one?

The perimeter of this hive is 30,

which means we’ll need at
least 5 bees to fill it in.

With 6 it’d be easy.

Placing them like this would fill out the
whole hive in just three steps.

But we can do better! We don’t actually
need to place a bee on this hex,

since the other bees will transform
that spot on their own.

It looks like we have the
beginning of a pattern.

Can we extend it to our full hive?

That would mean placing
our 9 bees like so.

Once they get to work, they’ll create a
chain reaction

that fills in the center of the hive and
extend it to its edges.

Add a 10th bee to the completed hive
and it becomes a queen.

Repeat that process five more times

and you’ve helped the last 60 members
of Apis trifecta

create 6 producing hives.

All in all, it’s a pretty
good bee-ginning.