Is it possible to design a programming task that is unsolvable?
A simplified example can be the following:

$$T( n ) = \begin{cases}
\max \{ M(n) \mid |M| \leq n \} + 1 &amp; \text{if such max exists}\\
0 &amp;\text{otherwise.}
\end{cases}$$

In other words the "unprogrammable" (uncomputable) task&nbsp;$T$ (or better uncomputable function&nbsp;$T$) simply picks all programs&nbsp;$M$ of size at most&nbsp;$n$ as "input" and outputs the maximum value that they output on input&nbsp;$n$, plus 1.

Given a program $M_i$ that satisfies your assumption ( ... which produce an answer in finite time for a (countably) infinite number of inputs ...), there will always be an $x_0$ such that $|M_i| &lt; x_0$ and $M_i$ on $x_0$ halts, and we have:

$$M_i(x_0) \leq \max \{ M(x_0) \mid |M| \leq x_0 \} &lt; T(x_0)\,.$$