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Answer the next 2 questions based on the information given below:
Certain number of cuts are made in a cube such that 8 smaller pieces are obtained.
What is the maximum number of cuts that could have been made?
Answer: 7
When a certain number of pieces are obtained by cutting a cube:
We would have maximum cuts when all the cuts are made along the same axis, and
Minimum cuts when cuts are made as equally as possible along the three axes.
Here, we have 8 smaller pieces,
∴ maximum cuts would be obtained when we have all cuts along same axis.
∴ maximum number of cuts will be (8 – 1) = 7 cuts.
Hence, 7.
What is the minimum number of cuts that could have been made?
Answer: 3
When a certain number of pieces are obtained by cutting a cube:
We would have maximum cuts when all the cuts are made along the same axis, and
Minimum cuts when cuts are made as equally as possible along the three axes.
Here, we have 8 smaller pieces,
∴ minimum cuts would be obtained when we have equal number of pieces along all 3 axes.
∴ 8 = 2 × 2 × 2.
⇒ 2 pieces along each axis.
⇒ 1 cut along each axis
∴ 3 cuts total.
Hence, 3.
Answer the next 6 questions based on the information given below:
A cube is cut into smaller pieces. Find the minimum number of cuts required to obtain
27 pieces
Answer: 6
Here, we have 27 smaller pieces,
∴ Minimum cuts would be obtained when we have equal number of pieces along all 3 axes.
∴ 27 = 3 × 3 × 3
⇒ 3 pieces along each axis.
⇒ 2 cuts along each axis
∴ 6 cuts total.
Hence, 6.
64 pieces
Answer: 9
Here, we have 64 smaller pieces,
∴ Minimum cuts would be obtained when we have equal number of pieces along all 3 axes.
∴ 64 = 4 × 4 × 4
⇒ 4 pieces along each axis.
⇒ 3 cuts along each axis
∴ 9 cuts total.
Hence, 9.
512 pieces
Answer: 21
Here, we have 512 smaller pieces,
∴ Minimum cuts would be obtained when we have equal number of pieces along all 3 axes.
∴ 512 = 8 × 8 × 8
⇒ 8 pieces along each axis.
⇒ 7 cuts along each axis
∴ 21 cuts total.
Hence, 21.
100 pieces
Answer: 11
Here, we have 100 smaller pieces,
∴ Minimum cuts would be obtained when we have pieces distributed as equally as possible along all 3 axes.
∴ 100 = 5 × 5 × 4
⇒ 5, 5, and 4 pieces along the three axes.
⇒ 4, 4 and 3 cuts along the three axes.
∴ 11 cuts total.
Hence, 11.
990 pieces
Answer: 27
Here, we have 990 smaller pieces,
∴ Minimum cuts would be obtained when we have pieces distributed as equally as possible along all 3 axes.
∴ 990 = 9 × 10 × 11
⇒ 9, 10, and 11 piece along the three axes.
⇒ 8, 9 and 10 cuts along the three axes.
∴ 27 cuts total.
Hence, 27.
73 pieces
Answer: 72
Here, we have 73 smaller pieces,
∴ Minimum cuts would be obtained when we have pieces distributed as equally as possible along all 3 axes.
∴ 73 = 73 × 1 × 1
⇒ 73, 1, and 1 piece along the three axes.
⇒ 72, 0 and 0 cuts along the three axes.
∴ 72 cuts total.
Note: There is only 1 way you can cut a cube to get 73 pieces, i.e., make 72 cuts along the same axis.
Hence, 72.
Answer the next 2 questions based on the information given below:
A cube is cut into smaller cubes. Find the minimum number of cuts that could have been made if
343 cubes are obtained?
Answer: 18
Here, we have 73 smaller cubes.
Since the smaller pieces are also cubes, this is possible only when equal number of pieces are obtained along each axis.
∴ 343 = 7 × 7 × 7
⇒ 7, 7, and 7 pieces along the three axes.
⇒ 6, 6 and 6 cuts along the three axes.
∴ 18 cuts total.
Hence, 18.
180 cubes are obtained?
Answer: D
Here, we have 180 smaller cubes.
Since the smaller pieces are also cubes, this is possible only when equal number of pieces are obtained along each axis.
∴ 180 is not a perfect cube hence, such a case is not possible.
Hence, option (d).
