Imagine you’ve created some amazing robo-ants for an experiment. The good news is they work! The bad news? They accidentally have laser-shooting abilities that you can’t turn off. You have just five minutes to stop them before they activate their lasers and escape into the world, causing chaos.
The robo-ants live in a habitat made of tubes, each 1 meter long. They walk at a speed of 1 meter per minute. If they bump into each other or reach a dead end, they turn around and walk back. When they reach an intersection, they randomly choose to go left, right, or forward, but never backward unless they hit something.
Your mission is to stop the robo-ants by using two emergency vacuum nozzles. These nozzles can be placed at any spot on the front of the habitat. Once placed, they can’t be moved without creating a hole for the ants to escape. The goal is to place the nozzles in the best spots to capture all the ants before they escape.
With hundreds of robo-ants moving around, it might seem impossible to catch them all. But here’s a trick: think about what happens when two ants meet. They simply swap directions, making it look like they passed through each other. This means you only need to focus on where a single ant could walk without interruption for less than five minutes.
Placing the nozzles at intersections where three or four tubes meet is a smart move. These are spots where ants might change directions and miss the nozzles. There are four intersections in total, but you need to pick the right two.
The top right intersection is crucial. If you don’t place a nozzle there, an ant could crawl down to a dead end and back, taking four minutes, and then choose any direction, taking at least another minute to escape.
Once you’ve placed a nozzle in the top right, the next best spot is the bottom left intersection. Imagine an ant starting anywhere else in the habitat. Even in the worst-case scenario, it would move for at most 4 meters before being captured by the vacuum. No other two intersections can guarantee capturing all the ants within five minutes.
By placing the nozzles at these strategic points, you’ve successfully captured all the robo-ants and prevented a disaster. Next time, before experimenting with robo-ants, maybe consider having a robo-anteater on standby. And if it could fly and breathe fire, that would be quite the spectacle! What could possibly go wrong?
Imagine you are a scientist tasked with designing a new habitat for robo-ants. Draw a diagram of your habitat, including tubes and intersections. Make sure to label each part and think about how the ants will move through it. Share your design with the class and explain your strategy for controlling the ants.
Create a simple simulation using paper or a digital tool to show how ants move through the habitat. Use markers or icons to represent ants and demonstrate how they change direction at intersections. Present your simulation to the class and discuss how different nozzle placements affect the ants’ paths.
Work in pairs to play a game where one student places robo-ants in a habitat, and the other places nozzles to capture them. Use a grid to represent the habitat and take turns placing ants and nozzles. The goal is to capture all ants within five minutes. Discuss the strategies used and what worked best.
Using the information about ant speed and tube length, calculate the maximum distance an ant can travel in five minutes. Create different scenarios and calculate how long it would take for ants to reach various intersections. Share your calculations and reasoning with the class.
Write a short story about what might happen if the robo-ants escaped. Use your imagination to describe the chaos they could cause and how you would solve the problem. Share your story with the class and discuss the potential consequences of not capturing the ants in time.
The good news is that your experimental robo-ants are a success! The bad news is that you accidentally gave them the ability to shoot lasers, and you can’t turn it off. You have five minutes to stop them before the lasers go online. Until then, all of your robo-ants will walk inside their habitat at a speed of exactly 1 meter per minute. If they bump into each other or hit a dead end, they’ll instantly turn around and walk back the way they came. When five minutes are up, they’ll turn on their lasers, break free, and stream out into the world, causing chaos as they go.
Your one chance to stop them is to insert two emergency vacuum nozzles into the habitat and suck the ants up before they break free. The nozzles can be pressed into any one location in the habitat through a membrane covering its front side, and any ants that walk past will be deactivated. You can’t move the nozzles once they’re placed without leaving a hole that the robo-ants could escape from, so choosing the right spots will be key.
The habitat is made out of meter-long tubes. When the robots reach an intersection, they will randomly choose to go left, right, or forward. They only go backward if they hit another robo-ant or a dead end. Unfortunately, there are hundreds of them inside the habitat, and if even one escapes, it could cause significant damage.
With just under five minutes remaining, where should you place the two vacuum nozzles to capture all the robo-ants?
Pause the video now if you want to figure it out for yourself.
With robo-ants moving all over the habitat, it might seem impossible to stop them before they break free. However, this situation is simpler than it seems.
Imagine just two robo-ants crawling toward each other. When they collide, they immediately reverse directions. If they crawled past each other instead, it would look exactly the same before and after their interaction, but with their positions swapped. This is true every time a pair of robo-ants meet. Because the identities of individual ants don’t matter, you just need to figure out where to place the nozzles to capture any single ant walking without interruption for less than five minutes, starting from any point in the habitat.
Placing the nozzles at intersections where three or four tubes meet seems like your best bet since that’s where the robo-ants might otherwise change directions and miss your nozzles. There are only four intersections; which two should you pick? The top right intersection has to be one of them. If it isn’t, an ant crawling down from this intersection toward the dead end would crawl for four minutes to get back to the intersection and then go in any of three directions, walking for at least another minute.
Once you’ve placed a nozzle in the top right, the only other choice that has a chance to work is the bottom left. To see that this works, imagine an ant anywhere else in the habitat. In the worst-case scenario, the ant would start right next to the vacuum nozzle, moving away from it. But in all those worst cases, the ant would move for at most 4 meters before being sucked up into the vacuum. No other choice of two intersection points is guaranteed to capture all the robo-ants within five minutes.
Having vacuumed them all up, you’ve averted a major crisis. Before you experiment with robo-ants again, you’ll want to have a robo-anteater ready. And wouldn’t it be interesting if it could fly and breathe fire? There’s no way that could go wrong!
Robo-ants – Small robotic devices designed to mimic the behavior of ants, often used in experiments to study algorithms and problem-solving strategies. – In our math class, we programmed robo-ants to find the shortest path through a maze, simulating how ants find food.
Habitat – The natural environment where an organism lives, which can be used as a model for mathematical simulations and problem-solving. – We created a mathematical model of a forest habitat to study how different species interact and share resources.
Speed – The rate at which an object moves, calculated as distance divided by time, often used in mathematical problems involving motion. – To solve the problem, we calculated the speed of the car by dividing the total distance by the time taken.
Nozzles – Devices used to control the direction or characteristics of a fluid flow, often used in mathematical modeling of fluid dynamics. – In our science project, we used different nozzles to study how the shape affects the speed and direction of water flow.
Capture – To record or take hold of data or information, often used in mathematics when collecting data for analysis. – We used sensors to capture the temperature data every hour for our math project on climate patterns.
Intersections – Points where two or more lines or paths cross each other, commonly used in geometry and graph theory. – The problem asked us to find the intersections of the two lines on the graph to determine where they meet.
Directions – Instructions or guidelines on how to proceed with a task, often used in problem-solving and mathematical procedures. – The teacher gave us clear directions on how to solve the quadratic equations step by step.
Escape – To break free from a constraint or problem, often used in mathematical puzzles and logic problems. – We had to find a way to escape the maze by solving a series of mathematical riddles.
Strategy – A plan of action designed to achieve a specific goal, often used in solving complex mathematical problems. – Our strategy for solving the math competition problems was to divide them among team members based on their strengths.
Experiment – A test or trial conducted to discover something unknown, often involving mathematical analysis of the results. – We conducted an experiment to see how different variables affect the growth of plants, and then analyzed the data using statistics.
