Сonstructive problems and optimization

first digit and the square of the second digit.
6. In three digits, they crossed out the second digit, as a result of which it decreased 6 times. Find the original number.
7. A three-digit number crossed out the number of hundreds, as a result, the number decreased by 7 times. Find the original number.
8. Find two two-digit numbers, the cube of one of which is equal to the square of the other.
9. The sum of several numbers is 1, and each of them is less than 1. Can the sum of their cubes be more than 1?

first digit and the square of the second digit.
11. In three digits, they crossed out the average number, as a result of which it decreased 6 times. Find the original number.
12. A three-digit number crossed out the number of hundreds, as a result, the number decreased by 7 times. Find the original number.
13. Find two two-digit numbers, the cube of one of which is equal to the square of the other.
14. The sum of several numbers is 1, and each of them is less than 1. Can the sum of their cubes be more than 1?

these numbers was reduced by one, while their product has not changed. Give an example of such numbers.
16. The snail crawled for 9 minutes of time, rested for a while. For each 2 minutes taken separately, she crawled 20 cm. Does it follow from this that in 9 minutes she crawled 90 centimeters?
17. Is it possible to mark the centers of 16 cells of an 8 × 8 chessboard so that no three centers were on one straight line?
18. Is it possible to cross out all 13 dots in the figure with five lines, without removing the pencil from the paper and not drawing any line twice?

It is known that their sum equals 500. Peter can ask him to name the sum of numbers in any cell and all its neighbors. Can a few of such questions Peter find out what number is written in the central cell?.
20. Find all pairs of primes p and q with the following property:7p + 1 is divisible by q, and 7q + 1 is divisible by p.
21. An inquisitive tourist wants to walk along the streets of the Old Town from the train station (point A on the plan) to his hotel (point B). The tourist wants his route to be as long as possible, but he is not interested in being twice at the same intersection, and he does not do that. Draw on the plan the longest possible route and prove that there is no longer.