Calculating Cubes with Two Painted Sides
This problem involves visualizing a larger cube that is painted and then cut into smaller cubes. We need to find the number of smaller cubes that have paint on exactly two sides.
Understanding Cube Divisions
A large cube of size $N \times N \times N$ cut into $1 \times 1 \times 1$ smaller cubes results in $N^3$ smaller cubes.
- Corner Cubes: These are located at the 8 corners of the large cube and have 3 painted sides.
- Edge Cubes: These are located along the 12 edges (but not corners) and have 2 painted sides.
- Face Cubes: These are located on the 6 faces (but not edges or corners) and have 1 painted side.
- Interior Cubes: These are inside the cube with no paint.
Applying to the $3^{''} \times 3^{''} \times 3^{''}$ Cube
The large cube has dimensions $N=3$. It is cut into $1^{''} \times 1^{''} \times 1^{''}$ cubes, resulting in $3 \times 3 \times 3 = 27$ smaller cubes.
We are interested in the cubes painted on exactly two sides. These are the edge cubes (excluding the corners).
- The number of edges on a cube is always 12.
- For a cube of size $N \times N \times N$, the number of smaller cubes along each edge is $N$.
- The number of cubes on each edge that have exactly two painted sides is $(N-2)$, as the two corner cubes on each edge have three painted sides.
Calculation
In this case, $N=3$. The number of smaller cubes painted on exactly two sides is:
Number of 2-sided cubes = (Number of edges) $\times$ (Number of non-corner cubes per edge)
Number of 2-sided cubes = $12 \times (N-2)$
Number of 2-sided cubes = $12 \times (3-2)$
Number of 2-sided cubes = $12 \times 1$
Number of 2-sided cubes = $12$
Therefore, there are 12 smaller cubes painted on exactly two sides.
Similar Questions
Four different positions of the same dice are shown. Select the letter that will be on the face opposite to the one having the letter F.
Which cube given in the options CANNOT be made using the following sheet?
Two different positions of the same dice are shown. Select the number of dots that will be on the face opposite to the one with 2 dots.
Four different positions of the same dice are shown. Select the number that will be on the face opposite to the one having 5.
Two positions of a cubical block with faces A, B, C, D, E and F are shown below. When F is at the top which face will be at the bottom?
A cube is made by folding the given sheet. In the cube so formed, what would be the letter on the opposite side of the letter A?
Three different positions of the same dice are shown. Select the pattern that will be on the face opposite to the one with the triangle.
Three different positions of the same dice (figures 1, 2, and 3) are shown. Which digit is on the face opposite to the one having 5?
1)
2)
3)
The following are different views of the same wooden block.
What would be the number opposite 5? (Ignore the orientation of the numbers.)
The following paper cutting is folded along the lines to make a cubical box. Select the correct image of the resulting box from among the options.
Important Questions from Cube and Dice
Pairs of adjacent sides of a big cube are coloured with red, yellow and brown colour. Now this big cube is divided into 64 small equal cubes. How many small cubes will have 3 faces painted with different colours?
A cube of side 4 cm is painted green on all its faces and then divided into smaller cubes of side 2 cm each. How many small cubes have been obtained?
Two adjacent faces of a solid cube are painted with black colour. The faces opposite to the black faces is painted with blue colour while the remaining faces are painted with yellow colour. After painting, this cube has been divided into 125 equal cubes. How many cubes have no faces painted?
A cube, whose two adjacent faces are coloured, is cut into 64 identical small cubes. How many of these small cubes are not coloured at all?
Three different positions of the same dice are shown. Select the number that will be on the face opposite to the face having the number '3'?
