A cube 4 cm on each side is painted blue and cut into 64 one cm cubes. How many one-cm cubes are painted blue on exactly two faces?
Those would be the cubes along the edges, but not the corners.
There are 12 edges, each with 2 non-corner cubes.
So, looks like 24 cubes with 2 blue sides.
To find out how many one-cm cubes are painted blue on exactly two faces, we first need to understand how the original cube was cut into smaller cubes.
The original cube had 4 cm on each side, which means it had a volume of 4 cm x 4 cm x 4 cm = 64 cubic cm. It was then cut into 64 one-cm cubes.
Each face of the original cube has dimensions of 4 cm x 4 cm, and there are six faces in total. So, when the cube is cut into smaller cubes, each face of the original cube is divided into 4 smaller faces on each of the resulting one-cm cubes.
Of these four smaller faces, only the outermost face on each side will be painted blue. Therefore, each cube will have either 0, 1, or 2 blue faces.
To count the one-cm cubes that have two blue faces, we need to consider the cubes located at the edges and corners of the original cube. These cubes have three sides exposed, and hence, two of these sides will be painted blue.
Let's count them:
- There are 12 edge cubes, with three exposed sides each. So, 12 x 3 = 36 cubes have two blue faces.
- There are 8 corner cubes, with three exposed sides each. So, 8 x 3 = 24 cubes have two blue faces.
Now, let's add up the number of cubes that have two blue faces:
- From the edges: 36 cubes
- From the corners: 24 cubes
Adding them together, there are a total of 36 + 24 = 60 one-cm cubes that are painted blue on exactly two faces.