Imagine discovering an ancient alien space station filled with amazing technology. Every species in the galaxy wants to get there first, but there’s a problem. Your spaceship uses a special drive that needs 1 unit of fuel for every parsec it travels. Unfortunately, your ship can only carry 15 units of fuel at a time, and the space station is 23 parsecs away with nothing but empty space in between.
Luckily, dark matter fuel is stable in deep space. This means you can leave some fuel behind in space and come back to pick it up later. You start with 45 units of fuel at your current location, so with some clever planning, you might just make it to the space station.
To reach the alien space station, you need to figure out where to leave fuel along the way. You can solve this puzzle with as few as two fuel cache points, although there are other ways to do it with more. The trick is to decide where to place these caches.
Let’s think backwards from the space station. To reach 23 parsecs, you need to leave the 8-parsec mark with a full tank of fuel. But the 8-parsec point is too far to use as a cache right away. You could jump there, but you wouldn’t have enough fuel to return to the start or store any for later. So, you need to find a cache point between the start and 8 parsecs.
There’s a helpful pattern to consider. At the start, you have enough fuel for 3 full tanks. At 8 parsecs, you need just 1 tank. Is there a point, let’s call it point X, where you could have exactly 2 tanks of fuel? This would be perfect because you could refuel there twice, using your fuel efficiently.
From point X, you would jump forward twice: once to leave fuel at the 8-parsec cache point, and again for the final jump. You’d make three trips between X and 8 parsecs. You’d have 2 tanks of fuel at point X and need 1 left at the 8-parsec cache point, allowing you to use one tank—15 units—for the round trips. Since 15 units divided by 3 trips is 5, you can place these two cache points 5 parsecs apart. Any farther, and you wouldn’t have enough fuel to reach the space station.
The earliest you can place point X is at the 3-parsec mark. Can you transport 30 units of fuel there? Let’s see. You start with a full tank of 15 units. You jump 3 parsecs, drop off 9 units at the cache point, and then jump back home, arriving with an empty tank. Doing this twice gets you 18 units of fuel at the cache point, and one more jump gives you a total of 30 units of fuel at the 3-parsec cache.
Next, you jump to the 8-parsec mark, drop off 5 units of fuel, and jump back to the 3-parsec mark. You fill up your tank and jump forward again, arriving with 10 units of fuel in your tank. Now, you’re almost there. You beam the 5 units of fuel from deep space to fill your tank to capacity and input the coordinates of your final destination. A 15-parsec jump leaves you with just enough fuel to dock with the alien space station. It’s time to use this incredible technology to make life better for everyone in the galaxy!
Imagine you are the captain of the spaceship. Use a grid or board to simulate the journey to the space station. Each square represents a parsec. Use tokens to represent fuel units and practice placing them strategically to reach the space station with the given constraints. This will help you understand the importance of planning and resource management.
Work in groups to solve math puzzles related to the fuel riddle. Calculate the number of trips needed to place fuel caches at different points. Discuss the mathematical patterns and strategies that can be used to solve the problem efficiently. This will enhance your problem-solving and critical thinking skills.
Write a short story about your adventure to the alien space station. Include details about the challenges you faced and how you overcame them using the fuel cache strategy. Share your story with the class to explore different perspectives and creative solutions.
Conduct a simple experiment to understand fuel efficiency. Use small objects to represent fuel and measure how far they can be moved with a limited amount of energy (e.g., blowing air or using a rubber band). Relate your findings to the spaceship’s journey and discuss how efficiency plays a role in space travel.
Create a poster that illustrates the journey to the alien space station. Include the path taken, fuel cache points, and any obstacles encountered. Use visuals to explain the strategy used to solve the dark matter fuel riddle. Present your poster to the class to demonstrate your understanding of the concept.
Here’s a sanitized version of the provided YouTube transcript:
—
It’s an incredible discovery: an ancient, abandoned alien space station filled with advanced technology. Now, every species in the galaxy is racing to get there first and claim it for themselves. However, there’s a challenge. Your ship’s faster-than-light jump drive consumes 1 unit of fuel for every parsec of distance, and your ship holds only 15 units of fuel. The space station is 23 parsecs away, with only empty space in between.
Fortunately, dark matter fuel is stable in deep space, allowing you to vent a cache of it from your fuel chamber and return to pick it up later. Although your ship holds only 15 units of fuel, you have access to 45 units in your current location. With some strategic fuel caching along the way, you might be able to reach all 23 parsecs.
So, how can you reach the alien space station? The solution can be achieved with as few as two cache points, though there are valid solutions that use more. The key is determining where to cache fuel along your route.
Let’s work backwards from the alien space station. To reach 23 parsecs, you’ll need to leave the 8-parsec mark with a full tank of fuel. The 8-parsec point is too far from the start to use as a cache immediately; you could jump there but wouldn’t have enough fuel to return to the start, let alone store any for later. Therefore, you’ll need to find a cache somewhere between the start and 8 parsecs.
There’s an interesting pattern that can help. At the start, you have exactly 3 tanks’ worth of fuel. At 8 parsecs, you need exactly 1. Is there a point, which we can call point X, where you could have exactly 2? This would be useful, as you could refuel there twice, maximizing your storage capacity without waste.
Wherever point X is, you’ll jump forward from it twice: once to deposit fuel at the 8-parsec cache point, and a second time for the final jump. You’ll jump the distance between X and 8 parsecs three times in total. You’d have 2 tanks of fuel at point X and need 1 left at the 8-parsec cache point, allowing you to spend one tank—15 units—going back and forth. Since 15 units divided by 3 trips is 5, you can place these two cache points 5 parsecs apart. Any farther, and you wouldn’t have enough fuel to reach the alien space station.
The earliest you can place point X is at the 3-parsec mark. Is it possible to transport 30 units of fuel there? Let’s try. You set out with a full tank of 15 units. You jump 3 parsecs, drop off 9 units at the cache point, and then jump back home, arriving with an empty tank. Repeating this process gets you 18 units of fuel at the cache point, and one more jump puts you at the 3-parsec cache with a total of 30 units of fuel.
Next, you jump to the 8-parsec mark, drop off 5 units of fuel, and jump back to the 3-parsec mark. You fill up your tank and jump forward again, arriving with 10 units of fuel in your tank. Now, the end is in sight. You beam the 5 units of fuel from deep space to fill your tank to capacity and input the coordinates of your final destination. A 15-parsec jump leaves you with minimal fuel, but you’re ready to dock with the precursor space station. It’s time to put this alien technology to work and improve life for everyone in the galaxy.
—
This version maintains the core content while removing any informal language and ensuring clarity.
Fuel – A substance that is used to produce energy, especially in engines or reactors. – The rocket needed a large amount of fuel to reach the outer layers of the Earth’s atmosphere.
Parsec – A unit of distance used in astronomy, equal to about 3.26 light-years. – The nearest star to our solar system, Proxima Centauri, is about 1.3 parsecs away.
Cache – A storage location for frequently accessed data to speed up retrieval. – The computer’s cache helps to quickly load the mathematical software used in class.
Jump – A sudden increase or change in position, often used to describe a shift in data or values. – The graph showed a jump in temperature readings during the experiment.
Units – Standard quantities used to specify measurements. – In physics, it’s important to use the correct units when calculating force, such as newtons.
Space – The boundless three-dimensional extent in which objects and events occur and have relative position and direction. – Astronomers study the vastness of space to understand the universe better.
Tank – A large container for storing liquid or gas. – The water tank was used to demonstrate principles of pressure in the physics lab.
Mark – A point or line used as a reference in measurement or calculation. – The teacher asked the students to mark the midpoint of the line segment on their geometry worksheets.
Trip – A journey or excursion, often used in physics to describe the path taken by an object. – The physics problem involved calculating the total distance of the trip made by the car.
Dark – Lacking or having very little light, often used to describe areas in space with minimal visible light. – Scientists study dark matter to understand its effects on the universe’s structure.
