A 3 cm cube is painted on all its faces.It is then cut into 1 cm cubes.How many cubes will have paint on exactly two faces ?
The middle edge cubes on each side will have two sides painted. That is 4 on each side and 2 more each on the front and back.
4 + 4 + 2 + 2 = ?
There will be 12 cubes with paint on exactly two faces.
There will be 27 smaller cubes in total, 9 on the 'top', 9 in the 'middle' and 9 on the 'bottom. The top and bottom are equivalent.
Consider the top 9 cubes. The cube in the center only has one painted side.
The corners each have three painted sides.
This leaves 4 cubes left, which are around the perimeter of the top, and are in the middle (in between each corner).
Since the top and bottom are equivalent, the bottom must also have 4 cubes with exactly two sides that are painted.
This leaves the middle section. The center cube has zero painted faces. The four corners each have two painted faces, while the cubes in between each corner only have one painted face.
So there are four cubes on each the 'top,' 'middle' and 'bottom' that have exactly two painted sides.
To find out how many 1 cm cubes will have paint on exactly two faces, we first need to determine how many 1 cm cubes are there in the 3 cm cube.
We know that a 3 cm cube has sides measuring 3 cm each. The total number of 1 cm cubes along one side is obtained by dividing the side length of the larger cube (3 cm) by the side length of the smaller cube (1 cm), which gives us 3.
Since the cube has three dimensions (length, width, and height), we need to multiply the number of 1 cm cubes along each side to find the total number of 1 cm cubes in the larger cube. Therefore, the total number of 1 cm cubes in the 3 cm cube is 3 x 3 x 3 = 27.
Now, let's consider the cubes that have paint on exactly two faces.
For a cube to have paint on exactly two faces, it means that it must be located on the edge or corner of the larger cube.
On each edge of the larger cube, there are 4 1 cm cubes with paint on exactly two faces. Since there are 12 edges on a cube (each having 4 cubes), there are a total of 12 x 4 = 48 cubes with paint on exactly two faces along the edges.
In each corner of the larger cube, there are 8 1 cm cubes with paint on exactly two faces. Since there are 8 corners on a cube (each having 8 cubes), there are a total of 8 x 8 = 64 cubes with paint on exactly two faces at the corners.
Therefore, the total number of cubes with paint on exactly two faces is 48 + 64 = 112 cubes.