11. The members of a tribe have ten fingers on their hands and nine toes on their feet, and therefore count indifferently in base 10 or 19. In their mathematical culture, a positive integer is called "sacred" if it is written with the same two digits (between 1 and 9) in both bases. How many sacred numbers are there?
11. The answer is 4. Let $n$ be a sacred number and let $AB$ be its representation in base 10. Then, for its sacredness, in base 19 it is written as $AB$ or as $BA$. The first case does not yield solutions because $A$ and $B$ are different from 0, and therefore $AB$ in base 19 is greater than $AB$ in base 10 for any $A, B$. Therefore, we must have (writing in base 10) $10A + B = 19B + A$ or $9A = 18B$, from which $A = 2B$. Since $10 > A > 0, 10 > B > 0$, the allowed values for $B$ are then $1, 2, 3, 4$, which give rise to the numbers $21, 42, 63, 84$, which are all indeed sacred. There are therefore 4 sacred numbers.