$f : \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $m+f(m+f(n+f(m))) = n + f(m)$ for every integers $m,n$. 
If $f(6) = 6$, then $f(2012) = ?$

$ \textbf{(A)}\ -2010 \qquad \textbf{(B)}\ -2000 \qquad \textbf{(C)}\ 2000 \qquad \textbf{(D)}\ 2010 \qquad \textbf{(E)}\ 2012$
1. **Substitute specific values into the functional equation:**
   Given the functional equation:
   \[
   m + f(m + f(n + f(m))) = n + f(m)
   \]
   We start by substituting \( m = 6 \) and \( n = 0 \):
   \[
   6 + f(6 + f(0 + f(6))) = 0 + f(6)
   \]
   Since \( f(6) = 6 \), this simplifies to:
   \[
   6 + f(6 + f(6)) = f(6)
   \]
   \[
   6 + f(6 + 6) = 6
   \]
   \[
   6 + f(12) = 6
   \]
   Therefore:
   \[
   f(12) = 0
   \]

2. **Substitute another set of values:**
   Next, we substitute \( m = 12 \) and \( n = 6 \):
   \[
   12 + f(12 + f(6 + f(12))) = 6 + f(12)
   \]
   Using \( f(6) = 6 \) and \( f(12) = 0 \), this simplifies to:
   \[
   12 + f(12 + f(6)) = 6 + 0
   \]
   \[
   12 + f(12 + 6) = 6
   \]
   \[
   12 + f(18) = 6
   \]
   Therefore:
   \[
   f(18) = -6
   \]

3. **Hypothesize a general form for \( f(x) \):**
   Based on the values we have found, we hypothesize that \( f(x) \) might be of the form:
   \[
   f(x) = -x + 12
   \]
   We will verify this by substituting \( f(x) = -x + 12 \) into the original functional equation.

4. **Verify the hypothesis:**
   Substitute \( f(x) = -x + 12 \) into the original equation:
   \[
   m + f(m + f(n + f(m))) = n + f(m)
   \]
   Substitute \( f(m) = -m + 12 \):
   \[
   m + f(m + f(n + (-m + 12))) = n + (-m + 12)
   \]
   Simplify the inner expression:
   \[
   f(n + (-m + 12)) = - (n - m + 12) + 12 = -n + m
   \]
   Substitute back:
   \[
   m + f(m + (-n + m)) = n + (-m + 12)
   \]
   Simplify:
   \[
   m + f(2m - n) = n - m + 12
   \]
   Substitute \( f(2m - n) = -(2m - n) + 12 \):
   \[
   m + (-(2m - n) + 12) = n - m + 12
   \]
   Simplify:
   \[
   m - 2m + n + 12 = n - m + 12
   \]
   \[
   -m + n + 12 = n - m + 12
   \]
   This is true, so our hypothesis \( f(x) = -x + 12 \) is correct.

5. **Find \( f(2012) \):**
   Using the verified form \( f(x) = -x + 12 \):
   \[
   f(2012) = -2012 + 12 = -2000
   \]

The final answer is \(\boxed{-2000}\).