A diamond expert cuts a huge cubical diamond into 960 identical diamond pieces in minimum number of 'n' cuts. If he wants to maximize the number of identical diamond pieces making same number of n cuts to it. So the maximum number of such diamond pieces are :

Options

A. a.1000
B. b. 1331
C. c. 1200
D. d. none of (a),(b),(c)

Answer

To solve the problem of maximizing the number of identical diamond pieces that can be cut from a cubical diamond using the same number of cuts as required to make 960 pieces, we can follow these steps:

### Step 1: Understand the Problem
We need to find the maximum number of identical pieces that can be obtained from a cube with the same number of cuts used to get 960 pieces.

### Step 2: Determine the Number of Cuts
To find out how many cuts are needed to create 960 pieces from a cube, we can use the formula for the number of pieces created by making cuts in a cube. If we make \( n \) cuts along each dimension (length, width, height), the number of pieces formed is given by:
\[ \text{Number of pieces} = (n + 1)^3 \]
We need to find \( n \) such that:
\[ (n + 1)^3 = 960 \]

### Step 3: Calculate \( n \)
To find \( n \), we first take the cube root of 960:
\[ n + 1 = \sqrt[3]{960} \approx 9.8 \]
This means \( n + 1 \) must be an integer, so we round down to 9:
\[ n + 1 = 9 \implies n = 8 \]
Thus, \( n = 8 \) cuts are needed to create 729 pieces, which is the closest perfect cube less than 960.

### Step 4: Find the Maximum Identical Pieces
Now, we want to maximize the number of identical pieces with the same number of cuts. We can use \( n + 1 = 10 \) (which means \( n = 9 \)):
\[ (n + 1)^3 = 10^3 = 1000 \]
This means that with 9 cuts, we can create a maximum of 1000 identical pieces.

### Conclusion
The maximum number of identical diamond pieces that can be obtained with the same number of cuts (9 cuts) is:
\[ \boxed{1000} \]