The riddle goes like this:
$\qquad$ There are $100$ prisoners standing in line, each with a number on their back. The numbers are all different, and range from $1$ to $101$ (i.e. one number is missing). The person in the back of the line can see all 99 prisoners ahead of him's numbers. The prisoner in the front of the line sees no numbers. Starting with the prisoner in the back, each prisoner must shout the number on their own back, but each number can be said aloud only once.
$\qquad$The first prisoner clearly only has a $50$% chance of guessing his number correctly. However, once he has guessed his number correctly, the remaining 99 prisoners have a $100$% chance of correctly guessing the number on their backs.
What is their strategy to achieve this?
I have been given a hint that the solution "does not just involve modulo sums." There is more to the solution than just a modular trick.