Let `f(n)` denotes the square of the sum of the digits of natural number n, where `f^2(n)` denotes `f(f(n)).f^3(n)` denote `f(f(f(n)))` and so on.the value of `(f^2011(2011)-f^2010 (2011))/(f^2013 (2011)-f^2012 (2011))` is....

Answer

To solve the problem, we need to calculate the values of \( f^{2011}(2011) \), \( f^{2010}(2011) \), \( f^{2013}(2011) \), and \( f^{2012}(2011) \) using the function \( f(n) \), which is defined as the square of the sum of the digits of \( n \).

### Step-by-step Solution:

1. **Calculate \( f(2011) \)**:
- The digits of 2011 are 2, 0, 1, and 1.
- Sum of the digits: \( 2 + 0 + 1 + 1 = 4 \).
- Therefore, \( f(2011) = 4^2 = 16 \).

2. **Calculate \( f^{2}(2011) = f(f(2011)) = f(16) \)**:
- The digits of 16 are 1 and 6.
- Sum of the digits: \( 1 + 6 = 7 \).
- Therefore, \( f(16) = 7^2 = 49 \).

3. **Calculate \( f^{3}(2011) = f(f^{2}(2011)) = f(49) \)**:
- The digits of 49 are 4 and 9.
- Sum of the digits: \( 4 + 9 = 13 \).
- Therefore, \( f(49) = 13^2 = 169 \).

4. **Calculate \( f^{4}(2011) = f(f^{3}(2011)) = f(169) \)**:
- The digits of 169 are 1, 6, and 9.
- Sum of the digits: \( 1 + 6 + 9 = 16 \).
- Therefore, \( f(169) = 16^2 = 256 \).

5. **Calculate \( f^{5}(2011) = f(f^{4}(2011)) = f(256) \)**:
- The digits of 256 are 2, 5, and 6.
- Sum of the digits: \( 2 + 5 + 6 = 13 \).
- Therefore, \( f(256) = 13^2 = 169 \).

6. **Calculate \( f^{6}(2011) = f(f^{5}(2011)) = f(169) \)**:
- From the previous calculation, we know \( f(169) = 256 \).

7. **Identify the pattern**:
- We see that:
- \( f^{1}(2011) = 16 \)
- \( f^{2}(2011) = 49 \)
- \( f^{3}(2011) = 169 \)
- \( f^{4}(2011) = 256 \)
- \( f^{5}(2011) = 169 \)
- \( f^{6}(2011) = 256 \)
- The values alternate between 169 and 256 for odd and even powers greater than 3.

8. **Calculate the required expression**:
- We need to find:
\[ \frac{f^{2011}(2011) - f^{2010}(2011)}{f^{2013}(2011) - f^{2012}(2011)} \]
- Since \( 2011 \) is odd, \( f^{2011}(2011) = 169 \) and \( f^{2010}(2011) = 256 \).
- Since \( 2013 \) is odd, \( f^{2013}(2011) = 169 \) and \( f^{2012}(2011) = 256 \).
- Therefore, the expression becomes:
\[ \frac{169 - 256}{169 - 256} = \frac{-87}{-87} = 1 \]

### Final Answer:
The value of \( \frac{f^{2011}(2011) - f^{2010}(2011)}{f^{2013}(2011) - f^{2012}(2011)} \) is \( 1 \).