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.
48
Solving the Painted Cube Cutting Problem
This problem involves a large cube that is painted and then cut into smaller cubes. We need to determine the number of smaller cubes that have exactly two faces painted.
Understanding the Cube Cutting Process
When a large cube is cut into smaller identical cubes, the original faces, edges, and corners of the large cube determine how many faces of the smaller cubes are painted.
The large cube has a side length of 18 cm, and the smaller cubes have a side length of 3 cm. To find out how many small cubes fit along one edge of the large cube, we divide the side length of the large cube by the side length of a small cube:
\( n = \frac{\text{Side of large cube}}{\text{Side of small cube}} \)
\( n = \frac{18 \text{ cm}}{3 \text{ cm}} \)
\( n = 6 \)
This means the large cube is divided into \(6 \times 6 \times 6\) smaller cubes, totaling \(6^3 = 216\) smaller cubes.
Identifying Cubes with Two Painted Faces
Consider the smaller cubes based on their original position in the large cube:
- Corner cubes: These were at the 8 corners of the large cube. They have 3 faces painted.
- Edge cubes: These were along the edges of the large cube, excluding the corner cubes. They have 2 faces painted.
- Face cubes: These were on the faces of the large cube, excluding the edge and corner cubes. They have 1 face painted.
- Inner cubes: These were in the interior of the large cube. They have 0 faces painted.
We are interested in the cubes with exactly two faces painted. These are the cubes located along the edges of the large cube.
Calculating the Number of Two-Faced Cubes
A cube has 12 edges. Each edge of the large cube contains \(n\) small cubes. However, the two cubes at the very ends of each edge are corner cubes (with 3 painted faces) and are not counted as two-faced cubes.
So, the number of small cubes with exactly two painted faces along each edge is \(n - 2\).
Number of two-faced cubes per edge \( = n - 2 = 6 - 2 = 4 \).
Since there are 12 edges in a cube, the total number of small cubes with exactly two faces painted is:
\( \text{Total number of two-faced cubes} = \text{Number of edges} \times (n - 2) \)
\( \text{Total number of two-faced cubes} = 12 \times (6 - 2) \)
\( \text{Total number of two-faced cubes} = 12 \times 4 \)
\( \text{Total number of two-faced cubes} = 48 \)
Therefore, there are 48 smaller cubes that have only two faces painted.
Summary of Cube Types by Painted Faces
We can summarize the different types of smaller cubes based on the number of painted faces:
| Type of Cube | Location in Large Cube | Number of Painted Faces | Formula (n = side large / side small) | Calculation (n=6) | Number of Cubes |
|---|---|---|---|---|---|
| Corner cubes | At the corners | 3 | 8 | 8 | 8 |
| Edge cubes | Along the edges (not corners) | 2 | \(12 \times (n-2)\) | \(12 \times (6-2)\) | 48 |
| Face cubes | On the faces (not edges/corners) | 1 | \(6 \times (n-2)^2\) | \(6 \times (6-2)^2 = 6 \times 4^2 = 6 \times 16\) | 96 |
| Inner cubes | In the interior | 0 | \((n-2)^3\) | \((6-2)^3 = 4^3\) | 64 |
| Total | \(n^3\) | \(6^3\) | 216 |
The calculation confirms that the number of cubes with exactly two faces painted is 48.
Revision Table: Cube Cutting Formulas
| Number of Painted Faces | Formula (n = large side / small side) |
|---|---|
| 3 Faces | 8 |
| 2 Faces | \(12 \times (n-2)\) |
| 1 Face | \(6 \times (n-2)^2\) |
| 0 Faces | \((n-2)^3\) |
| Total Cubes | \(n^3\) |
Additional Information: Spatial Reasoning Problems
Cube cutting problems like this are common in spatial reasoning and quantitative aptitude tests. They require visualizing how a 3D object is divided and how different parts of the original object correspond to parts of the smaller pieces. Understanding the formulas for different types of painted faces can quickly solve these problems. The key is to determine the value of 'n', which is the number of smaller divisions along each edge of the larger object.
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 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?
Six numbers, 1, 2, 3, 4, 5, and 6, are written on the different faces of a dice. Three different positions of the same dice are shown (Figures 1-3). Find the number on the face opposite to the face showing ‘3’.
