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?
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 reactor? 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 Bs 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 nine Bs to transform the entire height. That's a great start, but we still have the tough question of where the nine B's 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 five bees to fill it in. With six, 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 nine 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 tenth 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 six producing hives. All in all, it's a pretty good beginning.