1. The operation of doubling a digit of a natural number consists in multiplying this digit by 2 (if the product turns out to be a two-digit number, the digit in the next place of the number is increased by 1, as in column addition). For example, from the number 9817, by doubling the digits 7, 1, 8, and 9, one can obtain the numbers 9824, 9827, 10617, and 18817, respectively. Can one obtain the number $22 . . .22$ (21 twos) from the number 22.. 22 (20 twos) with several such operations? (N. Agakhanov)
Answer. Possible. Solution. Triple the first digit of the number 22... 22 (20 twos). We get the number 1622... 22 (21 digits). Now double the digit 6 and we get the desired number $22 . .22$ (21 twos). Note. There are other ways as well.