Imagine one large cube made of 1000 white centimeter cubes. Suppose the outside of the large cube was painted black. How many of the small white cubes would have the following?
1) 0 faces painted black
2) 1 face painted black
3) 2 faces painted black
4) 3 faces painted black
1) 8*8*8 = 512 inside
2) 6*8*8 = 384 on faces
3) 12*8 = 96 on edges
4) 8 on corners
To solve this problem, we need to understand the structure of the large cube made up of 1000 white centimeter cubes. The large cube is made up of 10 smaller cubes on each edge, creating a 10x10x10 cube structure.
Now let's calculate the number of small white cubes that have a certain number of faces painted black.
1) 0 faces painted black:
Each small white cube has 6 faces, so if none of them are painted black, this means that all 6 faces are white. Therefore, all 1000 small white cubes will have 0 faces painted black.
2) 1 face painted black:
To calculate the number of small white cubes with 1 face painted black, we need to determine the number of cubes on the edge of the large cube.
Each edge of the large cube has 10 small white cubes. Since there are three edges in total (top, front, and side), the total number of small white cubes with 1 face painted black is 10 x 3 = 30.
3) 2 faces painted black:
To calculate the number of small white cubes with 2 faces painted black, we need to determine the number of cubes on the corners of the large cube.
Each corner of the large cube has 8 small white cubes. Since there are eight corners in total, the total number of small white cubes with 2 faces painted black is 8 x 8 = 64.
4) 3 faces painted black:
To calculate the number of small white cubes with 3 faces painted black, we need to determine the number of cubes on the edges of the large cube (excluding the corners).
Each edge (excluding the corners) of the large cube has 9 small white cubes. Since each edge has 12 cubes, there are (12 - 8) x 9 = 36 small white cubes with 3 faces painted black.
So, the number of small white cubes with each specific number of faces painted black is:
1) 0 faces painted black: 1000 cubes
2) 1 face painted black: 30 cubes
3) 2 faces painted black: 64 cubes
4) 3 faces painted black: 36 cubes