As is the case with all information puzzles, we have to find away to map each outcome (yes, no, I don't know) to a different number (1, 2, 3). To do this, we need some property that's unknown for exactly one of these three numbers.
There are multiple ways to do this. The most basic (and most reliable) way would be to introduce a number that only you know so I wouldn't know:
I'm thinking of either 1.5 or 2.5. Is your number greater than mine?
1 is smaller than either of these numbers, so the answer will always be no. 3 is greater, so yes. And 2 could be either greater or smaller, so I don't know.
Another way, which is less reliable, is to introduce an unknown is to use some unknown property in mathematics:
Is TREE(your number) divisible by your number?
The Kruskal tree theorem produces a sequence of numbers such that TREE(1) = 1 and TREE(2) = 3, but TREE(3) is so astronomically large that it makes Graham's number look like epsilon in comparison. Moreover, it's not even known how to calculate it, so we cannot even tell whether it's even or odd.
In this case, if I'm thinking of 1, the answer is yes, if I'm thinking of 2, the answer is no, and if I'm thinking of 3, I'm certainly not going to know whether TREE(3) is divisible by 3. The reason this one is less reliable is because if we ever do find a way to calculate TREE(3), the uncertainty will no longer be there.
Yet a third way, the least reliable of all, is to use some form of contingent unknown, like a form of obscure trivia or some event that has a relatively unpredictable or unknown occurrence.
Is the current world population greater than (your number + 5.3) billion?
Will Easter Sunday ten years from now happen less than (your number * 32) days after February 1?
Is the last bit of the SHA-256 hash of "entanglement" strictly smaller than (your number - 1)?
The current world population is currently between 7.2 and 7.3 billion. It reached 6.3 billion ten years ago, and won't reach 8.3 billion for another ten years, but somebody who didn't actually look up the population tables wouldn't know this fact for sure.
Easter Sunday is defined as the Sunday after the first full moon after the vernal equinox, so it can't happen before March 20 in any year (which is the earliest date the vernal equinox can happen), and it can't happen after April 25 (36 days after the vernal equinox) either. So it's not going to happen before March 3 or after May 8 either. But the date in the middle, April 5/6, is right in the middle of possible dates for Easter, so unless you've memorized the Computus table, it's not generally feasible to predict what day Easter is going to fall on.
The SHA-256 hash of "entanglement" is generally impossible to calculate in a few seconds in your head, unless you looked it up ahead of time, but you do know that its last bit (which is either 0 or 1) will always be less than (3 - 1), and never less than (1 - 1).
Although this method isn't completely reliable in that they might know that piece of trivia, the key is to provide two completely ridiculous values on each side and one that can be argued either way in the middle that the average person would answer "I don't know" to.
For the record:
The current world population at the time of posting is about 7.23 billion, which is less than 7.3 billion. This makes the answer "no" when my number is 2.
Easter Sunday will happen on March 31 in 2024, which makes the answer "no".
The SHA-256 hash of "entanglement" is
961B164F23EB33F8FDA12C95E8BD93F6 32A08A8B8A0A18B3DDE1CFE8926875FF, which has a last bit of
1. So the answer is "no".