How many different surface areas are possible when 8 one inch cubes are arranged so that each has one or more faces in common (touching) with at least one of the other cubes?
We have to use a physical model. And I have no idea how to start this or even do it. Please be specific, like step by step, when explaining how to do it. Thanks
To determine the number of different surface areas possible when arranging 8 one-inch cubes, we can break down the problem into smaller parts. Here's a step-by-step approach:
Step 1: Identify the maximum number of faces each cube can contribute to the overall surface area.
- Each cube has 6 faces, but only 5 will be visible since one face is shared with another cube.
Step 2: Determine the minimum number of faces two cubes can share when touching.
- When two cubes touch, they can share at least one face, and at most two opposite faces.
Step 3: Identify the different ways in which cubes can be arranged.
- Start with one cube as the base and place the other 7 cubes around it.
- Each additional cube can be placed adjacent to one of the initial cubes, utilizing the shared faces.
Step 4: Calculate the number of different arrangements.
- Count the number of possibilities using different combinations of touching faces.
- Take into account rotations and reflections; only unique arrangements should be counted.
Step 5: Calculate the number of unique surface areas.
- Multiply the number of different arrangements by the maximum number of visible faces (5 faces per cube).
Let's go through these steps in detail:
Step 1: Each cube has 6 faces, but when positioned next to another cube, one face is shared, leaving 5 visible faces.
Step 2: When two cubes touch, they can share at least one face, and at most two opposite faces. For simplicity, assume that only one face is shared between two cubes.
Step 3: Start by placing one cube as the base. Then, place the other 7 cubes around it. Each additional cube can be positioned in several ways while sharing one face with another cube. Determine all the possible arrangements.
Step 4: Count the different arrangements:
- Place the first cube on the base: 1 arrangement.
- The second cube can be placed adjacent to the first cube: 1 arrangement.
- The third cube can be placed adjacent to either the first or second cube: 2 arrangements.
- The fourth cube can be placed adjacent to any of the previous three cubes: 3 arrangements.
- The fifth cube can be placed adjacent to any of the previous four cubes: 4 arrangements.
- The sixth cube can be placed adjacent to any of the previous five cubes: 5 arrangements.
- The seventh cube can be placed adjacent to any of the previous six cubes: 6 arrangements.
- The eighth cube has only one possible placement remaining.
Multiply the number of arrangements for each cube together: 1 x 1 x 2 x 3 x 4 x 5 x 6 x 1 = 720 possible arrangements.
Step 5: Calculate the number of unique surface areas:
- Each cube has 5 visible faces when positioned next to another cube.
- Multiply the number of different arrangements (720) by the maximum number of visible faces (5 per cube): 720 x 5 = 3600 different surface areas.
So, when arranging 8 one-inch cubes with each having one or more faces in common with others, there are 3600 different surface areas possible.
To find the number of different surface areas possible when 8 one-inch cubes are arranged with overlapping faces, we can use a physical model and follow these step-by-step instructions:
1. Start with 8 identical one-inch cubes.
2. Begin arranging the cubes in a way that each cube has at least one face touching another cube. You can do this by placing the cubes next to each other, making sure they share a face.
3. As you arrange the cubes, keep track of the different combinations and orientations. A combination is different if the arrangement has a unique set of faces exposed.
4. Continue arranging and joining cubes until you have explored all possible combinations and orientations.
5. Remember to rotate the cubes as needed to create different arrangements. A rotation is considered unique if the arrangement has a different set of faces exposed.
6. Count the number of distinct surface areas you have created.
Note: It might be useful to physically build different arrangements with actual cubes and note each distinct surface area. Alternatively, you can try sketching and visualizing the different arrangements on paper to keep track of the number of different surface areas.
Remember, the key is to explore all possible combinations and orientations while ensuring that each cube has at least one face touching another cube.