Many challenges, mainly restricted-source, are impossible in some languages. Find an Illegal String is a good example—many languages without syntax errors, such as brainfuck (deleted answer), can simply be put in an infinite loop before the illegal string is reached within the code. The challenge even brings Jelly to its knees (deleted answer by Dennis himself), as you can simply put
¶1 at the end of any program (after closing any open strings) and the parser won't even look at the line with the syntax error.
Now, it's not always obvious whether a challenge like this is possible in a given language. Even if someone has a proof that there are no solutions, it's not currently clear whether or not it's legal to post this as an answer in itself. (This is the closest question I could find, but it asks specifically about challenges that aren't possible in general.)
So, what should be done when a language provably cannot answer a challenge? What about when it's almost certain, but not proven impossible?