a cube of 6 cm is painted of all its 6 faces with red colour . it is then broken up into 216 smaller identical cubes .what is the number of smaller cubes with no coloured surface ,number of smaller cubes with 1 red face and number cubes with 2 red faces .

We have a cube of 6 cm broken into 216 cubes (of 1cm cubed), meaning each face is divided into 6x6=36 small squares.

The interior cube that has no paint therefore has a size of 4 cm cubed, or 64 cubic cm.
The cubes that have paint on two sides would be the middle four along each of the 12 edges (i.e. excluding the corners).
The cubes which have paint on one side would be the middle cubes of each face (16 cubes per face) There are six faces.

All these should add up to 208 cubes. Together with the 8 corners (with 3 faces painted), there will be 216 cubes.

Why did the cube go to the paint store?

Because it wanted to be a colorful character!

Now, let's break down the numbers for you:

- The number of smaller cubes with no colored surface: Well, if we imagine the original cube, we can see that the innermost cubes won't have any painted surfaces. Since the original cube is made up of smaller cubes that have the same dimensions, each of these inner cubes has 1 cm dimensions. Therefore, the total number of smaller cubes with no colored surface is 4^3 = 64.

- The number of smaller cubes with 1 red face: Looking at the original cube, we can see that the smaller cubes that make up the outermost layer will have one painted face, as it's exposed to the paint. There are 4 sets of smaller cubes on each face, which means there are 4^2 = 16 smaller cubes on each face. Since there are 6 faces, the total number of smaller cubes with one red face is 6 x 16 = 96.

- The number of smaller cubes with 2 red faces: The smaller cubes with two red faces will be located at the corners of the original cube. There are 8 corners in total, and each corner consists of 3 smaller cubes. So, the total number of smaller cubes with two red faces is 8 x 3 = 24.

So, to summarize:
- Number of smaller cubes with no colored surface: 64.
- Number of smaller cubes with 1 red face: 96.
- Number of smaller cubes with 2 red faces: 24.

Hope this brings a colorful smile to your face!

To find the number of smaller cubes with no colored surface, we need to calculate the number of cubes that are completely inside the larger cube.

Since the larger cube has an edge length of 6 cm, it can be divided into smaller cubes with an edge length of 1 cm. So, there will be 6 x 6 x 6 = 216 identical smaller cubes.

Now let's calculate the number of smaller cubes with no colored surface:
Each small cube has 6 faces, and since all the faces of the larger cube are painted red, each small cube will have at least one colored face. Therefore, there will be no smaller cubes with no colored surface.

Next, let's find the number of smaller cubes with 1 red face:
The larger cube has 6 faces, and each face consists of a 6 x 6 grid of smaller cubes. So, each face will have 6 x 6 = 36 smaller cubes.
Since each face is fully painted, there will be no smaller cube with more than 1 red face.
So, there are a total of 6 faces x 36 smaller cubes per face = 216 smaller cubes with 1 red face.

Finally, let's find the number of smaller cubes with 2 red faces:
As calculated above, each face has 36 smaller cubes and there are 6 faces in total.
Out of these 36 cubes, the corner cubes have 3 faces exposed, the edge cubes have 2 faces exposed (excluding the corner faces), and the remaining interior cubes have only 1 face exposed.
Since there are 8 corner cubes, each with 3 faces exposed, the total number of faces exposed by the corner cubes is 8 x 3 = 24 faces.
Similarly, there are 12 edge cubes, each with 2 faces exposed, so the total number of faces exposed by the edge cubes is 12 x 2 = 24 faces.
The remaining interior cubes will have all 6 faces painted, and since each interior cube is surrounded by 26 other smaller cubes, there will be (216 - 8 - 12) x 6 = 1248 faces exposed by the interior cubes.

Hence, there will be a total of 24 faces + 24 faces + 1248 faces = 1296 faces exposed.
Since each exposed face belongs to 2 different smaller cubes, there will be a total of 1296 / 2 = 648 smaller cubes with 2 red faces.

To summarize:
- Number of smaller cubes with no colored surface: 0
- Number of smaller cubes with 1 red face: 216
- Number of smaller cubes with 2 red faces: 648

To find the number of smaller cubes with no colored surface, we need to determine the number of smaller cubes that do not touch any of the outer faces of the painted cube.

Since the original cube has sides measuring 6 cm, it can be divided into 6 smaller cubes along each face, resulting in a total of (6 x 6) = 36 smaller cubes that are on the outer layer.

Now, let's consider what's left after removing these outer cubes. We are left with a smaller cube inside, with each side measuring 4 cm (6 cm - 2 * 1 cm, taking into account the layer with colored cube faces).

This smaller inner cube can also be divided into smaller identical cubes using the same process. The sides of this inner cube measure 4 cm, so we can divide each side into (4 - 2) = 2 parts (taking into account the outer layer of the face cubes).

Therefore, the total number of smaller cubes without any colored surface would be (2 x 2 x 2) = 8 cubes.

Now, let's move on to finding the number of smaller cubes with one red face. Since each face of the original cube is painted, there are (6 x 6) = 36 smaller cubes that have one red face. This includes the smaller cubes from the outer layer and the inner cube.

Finally, let's find the number of smaller cubes with two red faces. Since each face of the original cube is painted, there are no smaller cubes with two red faces.

In summary:
- Number of smaller cubes with no colored surface: 8 cubes
- Number of smaller cubes with 1 red face: 36 cubes
- Number of smaller cubes with 2 red faces: 0 cubes