Let $ a$, $ b$, $ c$, and $ d$ be real numbers with $ |a\minus{}b|\equal{}2$, $ |b\minus{}c|\equal{}3$, and $ |c\minus{}d|\equal{}4$. What is the sum of all possible values of $ |a\minus{}d|$?

$ \textbf{(A)}\ 9 \qquad
\textbf{(B)}\ 12 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 18 \qquad
\textbf{(E)}\ 24$
1. Given the absolute value equations:
   \[
   |a - b| = 2 \implies a - b = 2 \text{ or } a - b = -2
   \]
   \[
   |b - c| = 3 \implies b - c = 3 \text{ or } b - c = -3
   \]
   \[
   |c - d| = 4 \implies c - d = 4 \text{ or } c - d = -4
   \]

2. We need to find all possible values of \( |a - d| \). We will consider all combinations of the equations above.

3. Case 1: \( a - b = 2 \), \( b - c = 3 \), \( c - d = 4 \)
   \[
   a - b = 2 \implies a = b + 2
   \]
   \[
   b - c = 3 \implies c = b - 3
   \]
   \[
   c - d = 4 \implies d = c - 4 = (b - 3) - 4 = b - 7
   \]
   \[
   a - d = (b + 2) - (b - 7) = 2 + 7 = 9
   \]

4. Case 2: \( a - b = 2 \), \( b - c = 3 \), \( c - d = -4 \)
   \[
   a - b = 2 \implies a = b + 2
   \]
   \[
   b - c = 3 \implies c = b - 3
   \]
   \[
   c - d = -4 \implies d = c + 4 = (b - 3) + 4 = b + 1
   \]
   \[
   a - d = (b + 2) - (b + 1) = 2 - 1 = 1
   \]

5. Case 3: \( a - b = 2 \), \( b - c = -3 \), \( c - d = 4 \)
   \[
   a - b = 2 \implies a = b + 2
   \]
   \[
   b - c = -3 \implies c = b + 3
   \]
   \[
   c - d = 4 \implies d = c - 4 = (b + 3) - 4 = b - 1
   \]
   \[
   a - d = (b + 2) - (b - 1) = 2 + 1 = 3
   \]

6. Case 4: \( a - b = 2 \), \( b - c = -3 \), \( c - d = -4 \)
   \[
   a - b = 2 \implies a = b + 2
   \]
   \[
   b - c = -3 \implies c = b + 3
   \]
   \[
   c - d = -4 \implies d = c + 4 = (b + 3) + 4 = b + 7
   \]
   \[
   a - d = (b + 2) - (b + 7) = 2 - 7 = -5 \implies |a - d| = 5
   \]

7. Case 5: \( a - b = -2 \), \( b - c = 3 \), \( c - d = 4 \)
   \[
   a - b = -2 \implies a = b - 2
   \]
   \[
   b - c = 3 \implies c = b - 3
   \]
   \[
   c - d = 4 \implies d = c - 4 = (b - 3) - 4 = b - 7
   \]
   \[
   a - d = (b - 2) - (b - 7) = -2 + 7 = 5
   \]

8. Case 6: \( a - b = -2 \), \( b - c = 3 \), \( c - d = -4 \)
   \[
   a - b = -2 \implies a = b - 2
   \]
   \[
   b - c = 3 \implies c = b - 3
   \]
   \[
   c - d = -4 \implies d = c + 4 = (b - 3) + 4 = b + 1
   \]
   \[
   a - d = (b - 2) - (b + 1) = -2 - 1 = -3 \implies |a - d| = 3
   \]

9. Case 7: \( a - b = -2 \), \( b - c = -3 \), \( c - d = 4 \)
   \[
   a - b = -2 \implies a = b - 2
   \]
   \[
   b - c = -3 \implies c = b + 3
   \]
   \[
   c - d = 4 \implies d = c - 4 = (b + 3) - 4 = b - 1
   \]
   \[
   a - d = (b - 2) - (b - 1) = -2 + 1 = -1 \implies |a - d| = 1
   \]

10. Case 8: \( a - b = -2 \), \( b - c = -3 \), \( c - d = -4 \)
    \[
    a - b = -2 \implies a = b - 2
    \]
    \[
    b - c = -3 \implies c = b + 3
    \]
    \[
    c - d = -4 \implies d = c + 4 = (b + 3) + 4 = b + 7
    \]
    \[
    a - d = (b - 2) - (b + 7) = -2 - 7 = -9 \implies |a - d| = 9
    \]

11. Summarizing all possible values of \( |a - d| \):
    \[
    |a - d| = 9, 1, 3, 5
    \]

12. The sum of all possible values is:
    \[
    9 + 1 + 3 + 5 = 18
    \]

The final answer is \(\boxed{18}\)