6. Vasya thought of a natural number $n \leq 2020$. Petya tries to guess it as follows: he names some natural number x and asks if his number is greater (is it true that $\mathrm{x}<\mathrm{n}$?), and Vasya answers him with "yes" or "no". Petya wins if he finds out the number, and loses if after receiving a "no" answer for the second time, he cannot name the guessed number. What is the smallest number of questions Petya needs to win?
Answer: 64.

Solution. First, note that if Petya receives the first "no" answer at some point, this means that at that moment he knows that the number $n$ is in some interval [ $n_{1}, n_{2}]$. If his next attempt is a number $x > n_{1}$, and he receives a "no" answer - he loses, because he is not allowed to ask more questions, and he knows that Vasya thought of one of the numbers $n_{1}, n_{1}+1, \ldots, x$. Therefore, Petya must sequentially ask about the numbers $n_{1}, n_{1}+1, \ldots, n_{2}-1$, and this is $n_{2}-n_{1}$ questions.:

Now we will show that Petya can always guess the number in 64 questions by asking Vasya about the numbers: $64, 64+63, 64+63+62, \ldots, 64+63+\cdots+13=2008$, until he gets a "no" answer. If this never happens, that is, if Vasya always answers "yes," then after the 52nd question, Petya establishes that Vasya's number is in the range $[2009,2020]$ and will guess the number in 11 more questions.

If he gets a "no" answer after the question with the number $63+62+\cdots+k$, then the number Vasya thought of is in the interval $[64+63+\cdots+(k+1)+1, 64+63+\cdots+(k+1)+k]$, and from the above, it follows that he needs no more than $(64-k+1)+k-1=64$ questions.

Suppose Petya has a strategy to find the number Vasya thought of in no more than 63 questions. If he gets a "no" answer on the first question, this means that Vasya thought of a number that does not exceed 63. If Petya gets a "yes" answer and on the second question he gets a "no" answer, then the second number cannot exceed $63+62=125$. Continuing the same reasoning, we conclude that the number after the 63rd question does not exceed $63+62+\cdots+1=2016<2020$, and with a "yes" answer to this question, Petya will not be able to determine Vasya's number, a contradiction.

Criteria: Proved that 63 questions are not enough: 3 points.

Shown how to find the number in 64 questions: 4 points.