Imagine you’re a biologist with an important mission: saving a rare honeybee species from extinction. You have the last 60 bees of this species in a special container called a terrarium. You’ve already built the wire frames that will become their new homes, but now you need to help the bees fill these frames with wax to create working beehives.
Each beehive is made up of hexagonal cells, or hexes. There are two ways to fill these hexes with wax:
Once all the hexes in a hive are filled, you can add an extra bee to become the queen. A well-cared-for hive will produce new bees, helping the species survive. If there aren’t enough transformed hexes, the bees will wait. Remember, once a bee transforms a hex, it can’t become a queen.
You could put 59 bees in one hive and wait for them to transform all the hexes, then create a queen. But if something goes wrong, the species could be in danger. The goal is to create as many hives as possible with the 60 bees you have. So, how many hives can you make?
The key is to create a chain reaction where a small number of bees can transform an entire hive. The fewer bees needed, the better. But how can we achieve this?
Let’s explore a clever approach. It involves counting the sides of the filled hexes and looking at their total perimeter. Suppose you place bees in three hexes. The total perimeter of these transformed hexes is 18 sides. The middle hex has three transformed neighbors, so it will also transform. Surprisingly, the perimeter remains 18!
Why does this happen? Each hex with at least three sides touching the bee-friendly space will remove those sides from the perimeter when it transforms, adding at most three new sides. This means the perimeter of transformed hexes stays the same or shrinks. The final perimeter of the entire hive is 54, so the starting hexes must also have a total perimeter of at least 54. Dividing 54 by the six sides of each non-adjacent hex means you’ll need at least 9 bees to transform the entire hive.
Now, where should these nine bees go? Let’s think smaller. We know three bees can transform a small hive. What about a slightly larger one? A hive with a perimeter of 30 needs at least 5 bees to fill it. With 6 bees, it would be easy, but we can do better!
You don’t need to place a bee on every hex, as some will transform on their own. This reveals a pattern. Can we apply this to the full hive? By placing 9 bees in a specific arrangement, they can create a chain reaction that fills the hive from the center to the edges. Add a 10th bee to the completed hive, and it becomes a queen. Repeat this process five more times, and you’ll have 6 productive hives from the 60 bees.
In the end, you’ve helped the last 60 bees create 6 thriving hives, giving the species a fighting chance for survival. It’s a great start to saving these precious honeybees!
Imagine you are a bee trying to fill a hive. Using hexagonal graph paper, draw a hive and place 9 bees strategically to create a chain reaction that fills the hive. Share your strategy with the class and discuss why your placement is effective.
In groups, role-play as bees and hexes. Decide who will be bees and who will be hexes. Bees will “move” to a hex and “transform” it. Hexes with three or more transformed neighbors will automatically transform. Reflect on how this activity helps you understand the chain reaction concept.
Calculate the perimeter of different hive configurations. Start with a small number of hexes and increase gradually. Determine how many bees are needed to transform the entire hive. Present your findings and explain the relationship between perimeter and bee placement.
Create a unique hive design using hexagons. Plan where to place the bees to achieve a full transformation with the least number of bees. Present your design to the class and explain your strategy for maximizing efficiency.
Research the importance of honeybees in ecosystems. Hold a class debate on the best strategies for conserving bee populations. Discuss how understanding hive dynamics can contribute to bee conservation efforts.
You’re a biologist on a mission to keep the rare honeybee species 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 harming it. The second method is that 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 will specialize into a queen. If well cared for, the hive will eventually produce new bees and continue the species. If there are no hexes with three or more transformed neighbors, the bees will just wait. Once a bee transforms a hex, it can never become a queen.
You could put 59 bees in one wire hive, wait until they transform all the hexes, and then create a queen. However, just one collapse could 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 reaction?
Let’s start with the first question. There’s a 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 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! 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 in a specific arrangement. Once they get to work, they’ll create a chain reaction that fills in the center of the hive and extends 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 the species create 6 producing hives. All in all, it’s a pretty good beginning.
Bees – Small, flying insects known for their role in pollination and producing honey. – Bees are essential for pollinating many of the crops that humans rely on for food.
Hive – A structure where bees live and work together as a colony. – The beekeeper carefully inspected the hive to ensure the bees were healthy and producing honey.
Hexes – Patterns or shapes with six sides, often seen in the structure of honeycombs. – The hexes in the honeycomb allow bees to efficiently store honey and pollen.
Transform – To change in form, appearance, or structure. – Caterpillars transform into butterflies through a process called metamorphosis.
Species – A group of living organisms consisting of similar individuals capable of exchanging genes or interbreeding. – The giant panda is a species that is native to China and is considered vulnerable due to habitat loss.
Survival – The ability to continue living or existing, especially under challenging conditions. – The survival of many animal species depends on the availability of clean water and food sources.
Perimeter – The continuous line forming the boundary of a closed geometric figure or area. – Scientists measured the perimeter of the forest to study the effects of deforestation on local wildlife.
Arrangement – The way in which things are organized or positioned. – The arrangement of leaves on a plant can affect how much sunlight each leaf receives for photosynthesis.
Reaction – A process in which substances interact to form new substances, often releasing or absorbing energy. – Photosynthesis is a chemical reaction that converts carbon dioxide and water into glucose and oxygen using sunlight.
Extinction – The state or process of a species, family, or larger group ceasing to exist. – The extinction of the dodo bird was primarily caused by human activities and introduced predators.
