Imagine discovering an alien monolith on a distant planet called RH-1729. Scientists worldwide are eager to uncover its secrets. To study this mysterious object, your engineering team has created a high-tech probe made up of 27 cube modules. These modules can perform all the necessary scientific tests and can assemble into a large 3x3x3 cube. Each module can be placed anywhere in the cube and oriented in any direction. They can also disassemble and reassemble into different shapes.
Your mission is to apply a special protective coating to the probe for the extreme environments it will face. The red coating will protect it from the cold of deep space, the purple coating will shield it from the intense heat of entering RH-1729’s atmosphere, and the green coating will guard against the planet’s electric storms. You can paint each face of the 27 cube modules in any way you choose, but each face can only have one color. Your task is to figure out how to apply the colors so the cubes can reassemble to show only red, then purple, then green.
Begin by painting the outside of the complete cube red. This is necessary no matter what. Once painted, break the cube into 27 pieces and examine what you have. There are 8 corner cubes, each with three red faces, 12 edge cubes with two red faces, 6 face cubes with one red face, and a single center cube with no red faces. You’ve painted a total of 54 faces red, so you’ll need to do the same for the green and purple cubes. This means painting 54 faces red, 54 faces green, and 54 faces purple, totaling 162 faces, which matches the number of faces on all the cubes, leaving no room for waste.
To achieve this, symmetry is key. Look at the center cube and paint it half green and half purple, allowing it to serve as a corner for each of those cubes without wasting any faces. You’ll also need center cubes with no green and no purple. Take 2 corner cubes from the red cube and paint the 3 blank faces of one purple and the 3 blank faces of the other green. Now you have 6 face cubes, each with 1 face painted red, leaving 5 empty faces on each.
Divide these face cubes into two groups. In the first group, paint 3 faces green and 2 faces purple; in the second group, paint 3 faces purple and 2 faces green. Using symmetry, replicate these groups again with the colors rearranged. This gives you 6 cubes with 1 green face, 6 with 1 red face, and 6 with 1 purple face.
After counting, you have 8 corner cubes in each color, 6 edge cubes in each color, 6 face cubes in each color, and 1 center cube. You need 6 more edge cubes in green and purple. There are exactly 6 cubes left, each with 4 empty faces. Paint 2 faces of each green and 2 faces of each purple.
Now, you have a perfectly painted cube ready for an incredible journey. It rearranges itself to be red in deep space, purple as it enters RH-1729’s atmosphere, and green when it flies through the electric storms. As it reaches the monolith, you realize you’ve achieved something humans have dreamed of for ages: alien contact.
Recreate the 3x3x3 cube using 27 smaller cubes. Paint each cube face according to the article’s instructions. Once completed, disassemble and reassemble the cube to show only one color at a time. This hands-on activity will help you understand the concept of symmetry and spatial arrangement.
Work in groups to devise a strategy for painting the cubes. Discuss how to efficiently use the 162 faces to ensure each color can be displayed. Present your plan to the class, explaining your reasoning and the challenges you anticipate. This will enhance your problem-solving and teamwork skills.
Using graph paper, draw the cube in different orientations. Color each face according to the article’s guidelines. This visual exercise will help you grasp the concept of symmetry and how it applies to the cube’s assembly and reassembly.
Calculate the number of faces needed for each color and verify the total matches the article’s description. Discuss how mathematical principles apply to the problem and explore any patterns or formulas that emerge. This will deepen your understanding of mathematical reasoning.
Write a short story or create a comic strip about the probe’s journey to RH-1729. Include the challenges it faces and how the color-changing ability helps it succeed. This creative exercise will allow you to explore the narrative aspect of the problem and express your understanding in a unique way.
The discovery of an alien monolith on planet RH-1729 has scientists around the world racing to unlock its mysteries. Your engineering team has developed a sophisticated probe to study it. The probe consists of 27 cube modules capable of conducting all the necessary scientific tests to analyze the monolith. The modules can self-assemble into a large 3x3x3 cube, with each individual module placed anywhere in the cube and in any orientation. It can also disassemble and reassemble into different configurations.
Now comes your task. The probe will require a special protective coating for each of the extreme environments it will encounter. The red coating will protect it against the cold of deep space, the purple coating will shield it from the intense heat as it enters the atmosphere of RH-1729, and the green coating will guard it from the planet’s electric storms. You can apply the coatings to each face of all 27 cubic modules in any way you choose, but each face can only have a single color coating. You need to determine how to apply the colors so that the cubes can reassemble themselves to display only red, then purple, then green.
You can start by painting the outside of the complete cube red, as that will be necessary regardless. Then you can break it into 27 pieces and analyze what you have. There are 8 corner cubes, each with three red faces, 12 edge cubes with two red faces, 6 face cubes with one red face, and a single center cube with no red faces. At this point, you’ve painted a total of 54 faces red, so you’ll need the same number of faces for the green and purple cubes as well. When you’re finished, you’ll have painted 54 faces red, 54 faces green, and 54 faces purple. That totals 162 faces, which is exactly how many the cubes have in total, leaving no room for waste.
If there’s a way to accomplish this, it will likely be highly symmetrical. You examine the center cube and decide to paint it half green and half purple, allowing it to serve as a corner for each of those cubes without wasting any faces. You’ll also need center cubes with no green and no purple. So, you take 2 corner cubes from the red cube and paint the 3 blank faces of one purple and the 3 blank faces of the other green. Now you have the 6 face cubes, each with 1 face painted red, leaving 5 empty faces on each.
You can split these into two groups. In the first group, paint 3 faces green and 2 faces purple; in the second group, paint 3 faces purple and 2 faces green. Using symmetry, you replicate these groups again with the colors rearranged. This gives you 6 cubes with 1 green face, 6 with 1 red face, and 6 with 1 purple face.
Counting what you’ve painted, you see 8 corner cubes in each color, 6 edge cubes in each color, 6 face cubes in each color, and 1 center cube. This means you need 6 more edge cubes in green and purple. There are exactly 6 cubes left, each with 4 empty faces. You paint 2 faces of each green and 2 faces of each purple.
Now, you have a cube that is perfectly painted for an incredible journey. It rearranges itself to be red in deep space, purple as it enters RH-1729’s atmosphere, and green when it flies through the electric storms. As it reaches the monolith, you realize you’ve achieved something humans have dreamed of for ages: alien contact.
Probe – A device used to explore or measure physical properties in engineering and physics. – Engineers used a thermal probe to measure the temperature distribution across the surface of the spacecraft.
Module – A self-contained unit or component that can be combined with others to form a larger system. – The space station’s life-support module is crucial for maintaining a habitable environment for astronauts.
Symmetry – The property by which a system or object remains unchanged under certain transformations, such as reflection or rotation. – The symmetry of the bridge design ensures that it can withstand equal loads from all directions.
Cube – A three-dimensional geometric shape with six equal square faces. – In physics, a cube is often used to simplify calculations involving volume and surface area.
Paint – A liquid substance applied to surfaces to protect them or to provide color and texture. – Engineers applied a special thermal paint to the satellite to protect it from extreme temperatures in space.
Color – The characteristic of visual perception described through color categories, with physical properties like wavelength. – The color of the laser light was adjusted to ensure maximum absorption by the target material.
Face – A flat surface on a three-dimensional shape. – The engineers calculated the stress distribution on each face of the structural beam to ensure safety and stability.
Corner – The point where two or more edges or lines meet. – The design of the machine included rounded corners to reduce stress concentrations and improve durability.
Edge – The line where two surfaces of a solid meet. – The precision of the cut along the metal edge was crucial for the assembly of the aerospace components.
Center – The point that is equidistant from all points on the surface of a shape or object. – The engineers calculated the center of mass of the satellite to ensure proper balance during launch.
