Sophie's favorite (positive) number is a two-digit number. If she reverses the digits, the result is $36$ less than her favorite number. Also, one digit is $3$ less than double the other digit. What is Sophie's favorite number?
Let the tens digit be $t$ and the units digit be $u$. We are told that the number is $\underline{10t+u}$.
If Sophie's favorite number is reversed, the number becomes $10u + t$.
We are given that $10u + t = 10t + u - 36$. Simplifying both sides gives $9u = 9t - 36$. Thus $u = t - 4$.
We are also given that one digit is three less than double the other digit. Therefore, either $u = 2t - 3$ or $t = 2u - 3$. We note that these two equations are equivalent. Thus $u = t - 4 = 2t - 3$. Solving yields $t = 1$, so Sophie's favorite number is $10t + u = \boxed{14}$.