I am thinking of two different positive integers between 1 and 100 (both inclusive). At most how many questions do you need to ask to find my two numbers if I will answer your questions truthfully and honestly?
My answers can only be "Yes", "No", or "I don't know". (I am a fairly good mathematician and equipped with a calculator.)
How many questions would you need if I a have three numbers in mind rather than two?
I think I got an idea of doing this, regardless of whether you are a "fairly good mathematician equipped with a calculator" or not. Thus I'm not sure whether it's the intended answer.
I take a piece of paper and start to write down all possible options on the paper. In this case, we have originally $\binom{100}2$ options.
I write the first $\frac 1 3$ options with a red pen. I write the second $\frac 1 3$ options with a blue pen. For the last $\frac 1 3$ options, I write each one with either red or blue pen. These are all done outside of your sight.
After writing down all options, I cover the last $\frac 1 3$ options with another piece of paper, so that they cannot be seen. I then show this piece of paper to you, tell you what I have done, and ask you this question:
"Is the correct answer written in red?"
According to whether the correct answer lies in the first, second or last $\frac 1 3$, your answer should be "Yes", "No" or "I don't know", respectively. This effectively reduces the number of possible options by a factor of $3$.
I then repeat the process until there is only one option left. The total number of questions needed is $\lceil \log_3 N\rceil$ if there are originally $N$ options.
This can be shown to be optimal, under certain restrictions on the type of questions that can be asked. Note that there must be some restrictions, otherwise questions such as "Will you answer this question with "no"?" are unanswerable.
