I saw a problem where it asked: What question can you ask a knight (who always tells the truth) such that he cannot respond? For this, I came up with the question "Is the answer to this question No?" -- the knight can't say Yes (because he would be lying) and he also can't say no. Similarly for a knave, "Is the answer to this question Yes?" works.
My question is: is there a question that you can ask to both the knight and the knave that they both can't answer?
If you ever need to consult a knight or knave (call them $K$) to figure out whether something $P$ is the case or not, you can ask them the following:
"Would you say that $P$ is true?"
It turns out that the answer to this question will be "yes" if $P$ is true, and "no" if $P$ is false, no matter whether $K$ is a knight or knave.
Why? Well, note that with the above question, $K$ is asked considered the truth of the following claim:
"$K$ would say that $P$ is true"
Now, this claim is true if $K$ is a knight and $P$ is true, or if $K$ is a Knave and $P$ is false. If $K$ is a knave and $P$ is true, or $K$ is a knave and $P$ is false, then this claim is false.
So, if we use $K$ to represent the claim that $K$ is a knight, then the above claim is true iff $K \leftrightarrow P$.
But we also have that $K$ is a knight if and only if what $K$ answers to this question is correct.
So, if $K$ says "yes', then we have:
$$K \leftrightarrow (K \leftrightarrow P)$$
Since $\leftrightarrow$ is associative, we thus have:
$$(K \leftrightarrow K) \leftrightarrow P$$
and thus:
$$\top \leftrightarrow P$$
and thus:
$$P$$
On the other hand, if $K$ says "No", then we have:
$$K \leftrightarrow \neg (K \leftrightarrow P)$$
which works out to:
$$K \leftrightarrow (K \leftrightarrow \neg P)$$
$$(K \leftrightarrow K) \leftrightarrow \neg P$$
$$\top \leftrightarrow \neg P$$
$$\neg P$$
So there you have it: the answer to
"Would you say that $P$ is true?"
is "Yes" if $P$ is true, and "No" if $P$ is false
OK, so with that trick, we can address your puzzle: how to ask a question that no knight or knave can answer at all?
I propose to ask:
"Would you say that you do not say "Yes" in answer to this question?"
Given the above analysis, we know that no matter whether the person answering is a knight or knave, if they answer "Yes", then it must be true that they do not say "Yes" (which is a contradiction), while if they say "No", then it is not true that they do not say "Yes", i.e. they must be answering "Yes" (another contradiction). So, they cannot answer either "Yes" or "No".