Problems

Using five sixes, arithmetic operations and exponentiation, form the numbers from 1 to 14.

Using five sevens, arithmetic operations and exponentiation, form the numbers from 1 to 22.

Using five eights, arithmetic operations and exponentiation, form the numbers from 1 to 20.

Using five nines, arithmetic operations and exponentiation, form the numbers from 1 to 13.

Try to make a square from a set of rods:

6 rods of length 1 cm, 3 rods of length 2 cm each, 6 rods of length 3 cm and 5 rods of length 4 cm. You are not able to break the rods or place them on top of one another.

In the equation $101-102=1$, move one digit in such a way that that it becomes true.

Try to get one billion $1000000000$ by multiplying two whole numbers, in each of which there cannot be a single zero.

On a table there are 2022 cards with the numbers 1, 2, 3, ..., 2022. Two players take one card in turn. After all the cards are taken, the winner is the one who has a greater last digit of the sum of the numbers on the cards taken. Find out which of the players can always win regardless of the opponent’s strategy, and also explain how he should go about playing.