All questions of Cubes and Dice for Class 8 Exam

Understanding Cube Cuts
To determine the maximum number of identical pieces a cube can be divided into by 7 cuts, we can follow a systematic approach. Each cut can increase the number of pieces exponentially, depending on how they intersect.
Maximum Pieces Formula
The formula to calculate the maximum number of pieces (P) created by n cuts is:
P = (n^3 + 5n + 6) / 6
This formula arises from the general principle of how each new cut can potentially intersect all previous cuts.
Calculating for 7 Cuts
Now, substituting n = 7 into the formula:
- P = (7^3 + 5*7 + 6) / 6
- P = (343 + 35 + 6) / 6
- P = 384 / 6
- P = 64
However, this result is for the maximum number of pieces resulting from 7 cuts, leading us to a misunderstanding.
Correct Approach for Identical Pieces
For identical pieces, we must consider how cuts can be made uniformly. To achieve the maximum number of identical pieces, we can divide the cube into smaller cubical units.
- Each cut can effectively divide the cube into smaller cubes.
- The best configuration involves making equal cuts along each dimension (length, width, height).
Identical Pieces Calculation
1. Making 3 Cuts in One Direction: This divides the cube into 4 pieces along that dimension.
2. Making 2 Cuts in the Second Direction: This divides the remaining pieces further, doubling the count.
3. Making 2 Cuts in the Third Direction: This again doubles the count.
Thus, the total number of pieces can be calculated as:
- 4 pieces (from first cuts) x 3 pieces (from second cuts) x 3 pieces (from third cuts) = 36 identical pieces.
Conclusion
Therefore, the correct answer for the maximum number of identical pieces a cube can be cut into by 7 cuts is 36, which corresponds to option 'B'.

Understanding the Problem
To cut a cube into 24 identical pieces, we need to determine the least number of cuts required. A cube has three dimensions, and we can effectively utilize cuts along these dimensions.
Strategic Cutting Plan
1. First Cut:
- Make the first cut horizontally through the center of the cube.
- This divides the cube into 2 equal halves.
2. Second Cut:
- Make the second cut vertically along the first dimension (length).
- This will result in 4 equal pieces.
3. Third Cut:
- Make the third cut vertically along the second dimension (width).
- This gives us 8 pieces.
4. Fourth Cut:
- Now, take one of the 8 pieces and make a horizontal cut through the middle of this piece.
- This doubles the number of pieces to 16.
5. Fifth Cut:
- Finally, make one more cut, either vertically in length or width, through the center of the remaining pieces.
- This will result in a total of 24 identical pieces.
Conclusion
- Thus, we can efficiently cut a cube into 24 identical pieces using just 5 cuts.
- The strategy involves cutting through the middle of the cube and then making additional cuts to divide the segments further.
In summary, the least number of cuts required to achieve 24 identical pieces from a cube is 5.