Lets take any integer, $z=abc\cdots$, form the sum of its digits, $a+b+c+\cdots$, subtract this from $z$, cross out any one digit from the result, and denote the sum of the remaining digits by $w$. From a knowledge of $w$ alone, can a rule be found for determining the value of the digit crossed out?
Problem source: What Is Mathematics? An Elementary Approach to Ideas and Methods
Almost -- you can determine the digit except you can't know whether it was a $0$ or a $9$.
The remainder of $z-(a+b+c+\dotso)$ $\bmod9$ is $0$, and so is the remainder of the sum of its digits. If you leave out one of the digits $1$ through $8$, the effect will be to make the remainder of the rest come out as one of the remainders $8$ through $1$, respectively. However, if you leave out a $0$ or a $9$, the remainder will be $0$ in either case, and you can't tell which one was left out.
For instance, if you start with $9090$ and subtract $18$, you have $9072$; now if you cross out the $9$ you get $w=9$. On the other hand, if you start with $9018$ and subtract $18$, you have $9000$; now if you cross out a $0$ you also get $w=9$. Thus the same value of $w$ can occur whether a $0$ or a $9$ has been crossed out.