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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.

• 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?

• 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 fifty-four 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 to keep the rare honeybee, Apis Trifecta, from going extinct.

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# Can you solve the honeybee riddle? - Dan Finkel

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林宜悉 posted on 2023/01/23
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