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.
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Important Questions from Cube and Dice
A cube of side 80 cm is painted yellow on all the faces and then cut into smaller cubes of sides 8 cm each. Find the number of smaller cube having all the three faces painted.
A cube of side 49 cm is painted purple on all the faces and then cut into smaller cubes of sides 7 cm each. Find the number of smaller cubes having only one face printed.
A cube of side 18 cm is painted yellow on all the faces and then cut into smaller cubes of sides 3 cm each. Find the number of smaller cubes that have only two faces painted.
A cube of side 12 cm is painted green on all the faces and then cut into smaller cubes of sides 2 cm each. Find the number of smaller cubes that have only one face painted.
Four friends J, Q, B and Z rolled the dice in alphabetical order.
The scores were:
1st round: 3, 2, 5, 2
2nd round: 1, 5, 1, 4
3rd round: 4, 2, 1, 1
If each point on the dice would get 10 points, who won the maximum points after 3 rounds?
