The number of real quadruples $(x,y,z,w)$ satisfying $x^3+2=3y, y^3+2=3z, z^3+2=3w, w^3+2=3x$ is

$ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \text{None}$
To find the number of real quadruples \((x, y, z, w)\) that satisfy the system of equations:
\[
\begin{cases}
x^3 + 2 = 3y \\
y^3 + 2 = 3z \\
z^3 + 2 = 3w \\
w^3 + 2 = 3x
\end{cases}
\]
we start by analyzing the equations.

1. **Express each variable in terms of the next:**
   \[
   y = \frac{x^3 + 2}{3}, \quad z = \frac{y^3 + 2}{3}, \quad w = \frac{z^3 + 2}{3}, \quad x = \frac{w^3 + 2}{3}
   \]

2. **Substitute \(y\) into the second equation:**
   \[
   z = \frac{\left(\frac{x^3 + 2}{3}\right)^3 + 2}{3}
   \]

3. **Substitute \(z\) into the third equation:**
   \[
   w = \frac{\left(\frac{\left(\frac{x^3 + 2}{3}\right)^3 + 2}{3}\right)^3 + 2}{3}
   \]

4. **Substitute \(w\) into the fourth equation:**
   \[
   x = \frac{\left(\frac{\left(\frac{\left(\frac{x^3 + 2}{3}\right)^3 + 2}{3}\right)^3 + 2}{3}\right)^3 + 2}{3}
   \]

5. **Assume \(x = y = z = w\):**
   If \(x = y = z = w\), then each equation simplifies to:
   \[
   x^3 + 2 = 3x
   \]
   This simplifies to:
   \[
   x^3 - 3x + 2 = 0
   \]

6. **Solve the cubic equation \(x^3 - 3x + 2 = 0\):**
   Factor the cubic equation:
   \[
   x^3 - 3x + 2 = (x - 1)^2 (x + 2) = 0
   \]
   The solutions are:
   \[
   x = 1 \quad \text{(with multiplicity 2)}, \quad x = -2
   \]

7. **Verify the solutions:**
   - For \(x = 1\):
     \[
     y = \frac{1^3 + 2}{3} = 1, \quad z = \frac{1^3 + 2}{3} = 1, \quad w = \frac{1^3 + 2}{3} = 1
     \]
     Thus, \((1, 1, 1, 1)\) is a solution.
   - For \(x = -2\):
     \[
     y = \frac{(-2)^3 + 2}{3} = -2, \quad z = \frac{(-2)^3 + 2}{3} = -2, \quad w = \frac{(-2)^3 + 2}{3} = -2
     \]
     Thus, \((-2, -2, -2, -2)\) is a solution.

8. **Conclusion:**
   There are exactly two real quadruples \((x, y, z, w)\) that satisfy the given system of equations: \((1, 1, 1, 1)\) and \((-2, -2, -2, -2)\).

The final answer is \(\boxed{2}\).