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  • 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.

You're a biologist on a mission

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C2 hive hex perimeter bee transform transformed

Can you solve the honeybee riddle? - Dan Finkel

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    林宜悉 posted on 2020/10/24
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