Are there problem instances which we know to be unsolvable?
Every problem with a finite number of inputs is computable.

Proof: Suppose that the inputs are $i_1,\ldots,i_n$ and that the desired outputs are $o_1,\ldots,o_n$. Then the following program solves the problem:


If the input is $i_1$, output $o_1$.
If the input is $i_2$, output $o_2$.
...
If the input is $i_{n-1}$, output $o_{n-1}$.
Output $o_n$.