You're a biologist on a mission to keep the rare honeybee Apis Trifecta from going extinct. The last 40 bees of the species are in your terrarium. You've already constructed wire frames of the appropriate size and shape. Your frame is made by equalateral hexagons, and it's shape is like an equalateral triangle with 9 hex on each side of your frame.
Now you need to turn them into working beehives by filling every hex with wax. There are 2 ways to fill a given hexes:
1. Place a bee into the cell. On placed, you can't remove it without killing it.
2. If any unfill cell has 3 or more neighboring wax-filled hexes, the bees already in the hive will move in and transform it. Once the bees have transfomed every hex in the hive, you can add an aditional bee and it'll specialies into the queen. The hive, if been well cared for, will eventually produce new bees and continue the species.
If there are no hexes with 3 or more transform neighbors, the bees will just sit and wait. And once a bee transform a hex, it will never become a queen.
You could put 40 bees in one wire hive, wait till they transform all the hexes, and then create a queen. But then just one collapses will end the species.
Question: At most how many queen bees you can make with 40 bees?
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