Problem 2. Gosha entered a natural number into the calculator. Then he performed the following operation, consisting of two actions, three times: first, he extracted the square root, and then took the integer part of the obtained number. In the end, he got the number 1. What is the largest number that Gosha could have initially entered?

Let's recall, the integer part of a number is the largest integer not exceeding the given number.
Answer: 255.

Solution. Suppose he entered a number not less than 256. Then after the first operation, he would get a number not less than $[\sqrt{256}]=16$, after the second - not less than $[\sqrt{16}]=4$, after the third - not less than $[\sqrt{4}]=2$, which is a contradiction.

Let's assume Gosha entered the number 255. Then after the first operation, he would get $[\sqrt{255}]=15$, after the second $-[\sqrt{15}]=3$, after the third $-[\sqrt{3}]=1$. Therefore, the answer to the problem is the number 255.

## Criteria

## The following criteria are cumulative:

4 p. It is proven that the number entered by Gosha is not greater than 255.

3 p. It is proven that the number 255 satisfies the condition of the problem.