Problem 189

1. Notice that the two given equations have a similar structure, so we can consider constructing a function.
2. Transform the given equations into $(a-1)^{3}+2(a-1)=-2$ and $(b-1)^{3}+2(b-1)=2$.
3. Construct the function $f(x)=x^{3}+2x$.
4. Observe that $f(-x)=-f(x)$, so $f(x)$ is an odd function.
5. Also, $f′(x)=3x^{2}+2 > 0$, so $f(x)$ is strictly increasing.
6. Since $f(a-1)=-2$ and $f(b-1)=2$, we have $f(a-1)=-f(b-1)=f(1-b)$.
7. Therefore, $a-1=1-b$, which implies $a+b=2$.
8. Thus, the answer is $\boxed{2}$.

Since the structures of the two given equations are similar, we can consider constructing a function.
Transform the given equations into $(a-1)^3 + 2(a-1) = -2$ and $(b-1)^3 + 2(b-1) = 2$,
Construct the function $f(x) = x^3 + 2x$,
Since $f(-x) = -f(x)$,
Therefore, $f(x)$ is an odd function.
Since $f'(x) = 3x^2 + 2 > 0$,
Therefore, $f(x)$ is monotonically increasing.
Thus, $f(x)$ is a monotonically increasing odd function.
Because $f(a-1) = -2$ and $f(b-1) = 2$,
So $f(a-1) = -f(b-1) = f(1-b)$,
Hence, we have $a-1 = 1-b$, which leads to $a+b = 2$.
Therefore, the answer is $\boxed{2}$.