A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by making the minimum number of cuts. Which of the following statements is/are correct?
I. The minimum number of cuts is 9.
II. The number of smaller pieces which are not painted on any face is 6.
Both I and II
Understanding Cube Cutting and Painted Faces
This problem involves a solid cube that is painted on all its faces and then cut into smaller, equal-sized pieces. We need to determine the minimum number of cuts required to get a specific number of pieces (60 in this case) and analyze the number of pieces with zero painted faces based on this cutting configuration.
Minimizing Cuts for a Given Number of Pieces
When a cube is cut, cuts are typically made parallel to the faces. If you make \((N_x-1)\) cuts parallel to one pair of faces, \((N_y-1)\) cuts parallel to another pair, and \((N_z-1)\) cuts parallel to the third pair, you will get a total of \(N_x \times N_y \times N_z\) smaller pieces.
The total number of cuts is the sum of the cuts made in each direction: \((N_x-1) + (N_y-1) + (N_z-1)\). To get a specific total number of pieces, say \(N = N_x \times N_y \times N_z\), the minimum number of cuts is achieved when the values of \(N_x, N_y, N_z\) are as close to each other as possible. This is because for a fixed product, the sum of factors tends to be minimized when the factors are close together.
Applying to 60 Pieces
We need to find three integers \(N_x, N_y, N_z\) such that their product \(N_x \times N_y \times N_z = 60\) and the sum \((N_x-1) + (N_y-1) + (N_z-1)\) is minimized. Let's look at possible factorizations of 60 into three integers and calculate the corresponding number of cuts:
| Factorization (\(N_x \times N_y \times N_z\)) | Number of pieces in each direction (\(N_x, N_y, N_z\)) | Cuts in each direction (\(N_x-1, N_y-1, N_z-1\)) | Total Cuts (\((N_x-1)+(N_y-1)+(N_z-1)\)) |
|---|---|---|---|
| \(1 \times 1 \times 60\) | 1, 1, 60 | 0, 0, 59 | 59 |
| \(1 \times 2 \times 30\) | 1, 2, 30 | 0, 1, 29 | 30 |
| \(1 \times 3 \times 20\) | 1, 3, 20 | 0, 2, 19 | 21 |
| \(1 \times 4 \times 15\) | 1, 4, 15 | 0, 3, 14 | 17 |
| \(1 \times 5 \times 12\) | 1, 5, 12 | 0, 4, 11 | 15 |
| \(1 \times 6 \times 10\) | 1, 6, 10 | 0, 5, 9 | 14 |
| \(2 \times 2 \times 15\) | 2, 2, 15 | 1, 1, 14 | 16 |
| \(2 \times 3 \times 10\) | 2, 3, 10 | 1, 2, 9 | 12 |
| \(2 \times 5 \times 6\) | 2, 5, 6 | 1, 4, 5 | 10 |
| \(3 \times 4 \times 5\) | 3, 4, 5 | 2, 3, 4 | 9 |
The factorization \(3 \times 4 \times 5\) gives the minimum number of cuts, which is \(2+3+4=9\). Therefore, the minimum number of cuts required to get 60 smaller pieces is 9. This means Statement I is correct.
Counting Pieces with Zero Painted Faces
When a solid cube is cut, the pieces with zero painted faces are those that were located entirely in the interior of the original cube. They do not touch any of the original outer faces.
In a cube cut into \(N_x \times N_y \times N_z\) pieces, the pieces that have no painted faces are those that are removed by 1 layer from each side in all three directions. This results in an inner cuboid of size \((N_x-2) \times (N_y-2) \times (N_z-2)\) pieces.
Applying to the Minimum Cut Configuration (3x4x5)
For the minimum number of cuts (9), the cube is cut into \(3 \times 4 \times 5\) pieces. Using the formula for pieces with zero painted faces:
Number of unpainted pieces = \((N_x-2) \times (N_y-2) \times (N_z-2)\)
Substituting \(N_x=3, N_y=4, N_z=5\):
Number of unpainted pieces = \((3-2) \times (4-2) \times (5-2)\)
Number of unpainted pieces = \(1 \times 2 \times 3\)
Number of unpainted pieces = 6
Therefore, the number of smaller pieces which are not painted on any face is 6. This means Statement II is correct.
Conclusion
Based on our analysis:
- Statement I: The minimum number of cuts is 9. This is correct.
- Statement II: The number of smaller pieces which are not painted on any face is 6. This is also correct.
Since both statements I and II are correct, the option stating "Both I and II" is the correct answer.
Revision Table: Cube Cutting Concepts
| Concept | Explanation | Formula (for \(N_x \times N_y \times N_z\) pieces) |
|---|---|---|
| Total Pieces | Product of the number of pieces in each dimension after cuts. | \(N_x \times N_y \times N_z\) |
| Total Cuts | Sum of cuts made parallel to each pair of faces. | \((N_x-1) + (N_y-1) + (N_z-1)\) |
| Minimum Cuts | Achieved when \(N_x, N_y, N_z\) are factors of the total number of pieces and are as close as possible. | Minimize \((N_x-1) + (N_y-1) + (N_z-1)\) subject to \(N_x \times N_y \times N_z = \text{Total Pieces}\) |
| Pieces with 0 Painted Faces (Inner Pieces) | Pieces not touching any original face. | \((N_x-2) \times (N_y-2) \times (N_z-2)\) |
| Pieces with 1 Painted Face | Pieces on the faces but not edges or corners. | \(2[(N_x-2)(N_y-2) + (N_y-2)(N_z-2) + (N_z-2)(N_x-2)]\) |
| Pieces with 2 Painted Faces (Edge Pieces, excluding corners) | Pieces on the edges but not corners. | \(4[(N_x-2) + (N_y-2) + (N_z-2)]\) |
| Pieces with 3 Painted Faces (Corner Pieces) | Pieces at the corners. A cube always has 8 corners. | 8 (if \(N_x, N_y, N_z \ge 1\)) |
Additional Information on Painted Cube Problems
Problems involving cutting painted cubes are common in logical reasoning and spatial visualization sections of competitive exams. They test your ability to relate the overall structure of the cube to the properties of the smaller pieces.
Key points to remember:
- The total number of smaller pieces is the product of the number of subdivisions along each dimension.
- The number of cuts along a dimension is one less than the number of pieces created along that dimension.
- Minimizing cuts for a fixed number of pieces involves making the divisions in each dimension as equal as possible.
- The number of painted faces on a smaller piece depends on its position within the original cube (corner, edge, face center, or interior).
- Understanding the formulas for different types of pieces (0, 1, 2, or 3 painted faces) can quickly solve such problems. Note that the sum of pieces with 0, 1, 2, and 3 painted faces must equal the total number of small pieces.
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