Derangements

[S3 Maths and Up: Combinatorics]

Kept you waiting, huh?

The 100 prisoners' problem is a fascinating problem to tackle.

It's hard to wrap the head around that idea, but it's simply using a property of the patterns available to find the answer. As long as the number chain isn't too long, typically finding the right answer is theoretically possible.

The puzzle's trick hinges on the fact that the prisoners get half the chance to find their number among all boxes. Searching 30 out of 100 boxes would result in a very different formula for calculating compared to searching 50 out of 100 boxes. It only feels like fairness is employed when we give them 'half' the chances to search for the appropriate box.

Here's a question for you to attempt: Suppose the warden wants to be fair, and increase these 30.7% chance to 50%. How many boxes would wardens need to allow prisoners to search for the solution's percentage to reach 50%?