Alice and Bob play the following game: Alice thinks of a number from 0 to 2000 which Bob must guess with the least number of tries. To do so, each time he writes in a piece of paper one question which Alice must reply by using only “>”, “<” or “=”. If the question set from Bob does not precisely accept one of the 3 above signs, then Bob loses this challenge and has to ask a new question. What is the least number of tries (questions) required so that Bob guesses the number, no matter how unlucky he is, and what are the questions he must ask?
I believe he must ask the below questions: How is the number related to 1000? If the answer is =,we are done. If the answer is >, then he asks How is the number related to 1500? Otherwise, How is the number related to 500? Then we continue after 1500-1750-1875-1938-1969 and so on… or respectively, according to the reply given. Does this work? Any better ideas?
Within $k$ questions, there will only be $3^k$ possible known outcomes so Bob can not determine which of more then $3^6 = 729$ options is correct in $6$ or fewer questions. So no strategy can guarentee success in fewer than $7$ questions.
But the following strategy will guarantee success in exactly $7$ questions.
"If we wrote this number in base $3$ it would be a $7$ digit number $a_6a_5a_4a_3a_2a_1a_0$ somewhere between $0000001_3 = 1$ and $2202001=2000$.
"My $7$ question are going to be:
"Question no. $i$. Is $a_{i- 1}$ less than, equal to, or more than $1$?"
That way within $7$ questions Bob will know the value of all the digits of the number in base $3$ and thus Bob will know the number.