One day, back in the time before computers, little John was scribing the answer his arithmetic homework - to compute the value of $2^{50}$. He had, in the days prior, painstakingly calculated every last digit of that value - and was, to be quite honest, very proud of his result.
Unfortunately, this was also before pencils. It was also before little John would fully develop his fine motor skills. Consequently, as he wrote the last digit, he tipped over his inkwell, and spilled the blackest India ink all over his work! He was able to decipher most of the number, but he still couldn't make out one digit $\blacksquare$ towards the middle which had been completely obscured.
$$112589990\blacksquare 842624.$$
He cried all day. All night too. Here was his magnum opus (he wasn't very ambitious) ruined! His parents weren't much help - they didn't want to redo the computation because, honestly, it was pretty tedious, and they were too busy doing whatever it was people did before computers.
So, they took him, and his ruined paper to the local doctor (of mathematics). The parents explained the situation and the doctor (of mathematics), seeing the pitiful boy, took a look at the paper, and, after making a few calculations (far too few to have calculated the value of $2^{50}$ anew), wrote the correct digit above the ink-splotch.
Examining the paper, the parents discovered that the doctor (of mathematics) had used every other digit on the child's homework as part of the computation and had never written a number greater than $16$ (or less than $0$).
How did the doctor (of mathematics) find the missing digit?