Suppose you have a complex $0 \to C^1 \to C^2 \to ... \to C^n \to 0$ which can be included in a commutative diagram of the following type (see image) where all rows and columns (except the complex itself) are exact and all rows sufficiently high and sufficiently low consist only of zeros.
The question is, for which $i$ the cohomology group $H^i(C)$ can be non-zero.
This was an exam question I was unable to solve.
The only thing that came to my mind while trying to solve this is the following: if there is an example for small $n$, say $n=3$, then we can construct an example for big $n$, just adding zeros to the complex from the example. That would give the answer for almost all $i$.
But I am not able to find such an example.