To find the degree of the polynomial that produces the given successive results, we can observe the differences between the successive outputs.
Let's denote the outputs as \( f(1) = 24 \), \( f(2) = 0 \), \( f(3) = -24 \), \( f(4) = 0 \), \( f(5) = 144 \), and \( f(6) = 504 \).
Now, let's compute the first differences:
\[ \begin{align*} f(2) - f(1) & = 0 - 24 = -24 \ f(3) - f(2) & = -24 - 0 = -24 \ f(4) - f(3) & = 0 - (-24) = 24 \ f(5) - f(4) & = 144 - 0 = 144 \ f(6) - f(5) & = 504 - 144 = 360 \ \end{align*} \]
The first differences are: \(-24, -24, 24, 144, 360\).
Next, we compute the second differences:
\[ \begin{align*} -24 - (-24) & = 0 \ 24 - (-24) & = 48 \ 144 - 24 & = 120 \ 360 - 144 & = 216 \ \end{align*} \]
The second differences are: \(0, 48, 120, 216\).
Next, we compute the third differences:
\[ \begin{align*} 48 - 0 & = 48 \ 120 - 48 & = 72 \ 216 - 120 & = 96 \ \end{align*} \]
The third differences are: \(48, 72, 96\).
Next, we compute the fourth differences:
\[ \begin{align*} 72 - 48 & = 24 \ 96 - 72 & = 24 \ \end{align*} \]
The fourth differences are: \(24, 24\).
The differences stabilized (the same value) at the fourth difference level, which indicates that the original polynomial is of degree 4.
Thus, the degree of the polynomial is \( \boxed{4} \).