1. Katya is 6 times younger than her great-grandfather; if a 0 is placed between the digits of her age, the age of her great-grandfather is obtained. How old is she
1. The first way to solve the problem. From the problem statement, it follows that Katya's age is a two-digit number, and her great-grandfather's age is a three-digit number. The number of units in Katya's age, when multiplied by 6, gives a number in which the number of units will be the same as in Katya's age. Let's find such numbers:

$1 \times 6=6-$ does not fit;

$2 \times 6=12$ - does not fit;

$3 \times 6=18-$ does not fit;

$4 \times 6=24$ - fits;

$5 \times 6=30$ - does not fit;

$6 \times 6=36$ - fits;

$7 \times 6=42$ - does not fit;

$8 \times 6=48$ - fits;

$9 \times 6=42$ - does not fit.

Thus, Katya's age ends in 4, 6, or 8. The digit in the units place of Katya's age cannot be greater than one, as in this case, a number greater than one hundred would result, which does not contain a 0 in the tens place. Therefore, we need to check the numbers $14, 16, 18$. The numbers 14 and 16 do not fit because when multiplied by 6, they result in two-digit numbers. Only one number remains - 18.

The second way to solve the problem (Lida Frolova, a student at Gymnasium No. 24).

If we place a 0 between the digits of Katya's age, we get her great-grandfather's age. This means Katya is a two-digit number of years old, and her great-grandfather is a three-digit number of years old with a 0 in the middle.

The numbers $10, 11, 12, 13, 14, 15, 16$ - do not fit, as when each of them is multiplied by 6, they result in two-digit numbers. Only $17, 18, 19$ remain to be checked:

$17 \times 6=102$ - does not fit;

$18 \times 6=108-$ fits;

$19 \times 6=114$ - does not fit.

Answer: Katya is 18 years old.