mathematics - 100 Prisoners' Names in Boxes

• The names of 100 prisoners are placed in 100 wooden boxes, one name to a box, and the boxes are lined up on a table in a room.


• One by one, the prisoners are led into the room; each may look in at most 50 boxes, but must leave the room exactly as he found it and is permitted no further communication with the others.


• The prisoners have a chance to plot their strategy in advance, and they are going to need it, because unless every single prisoner finds his own name all will subsequently be executed.

Find a strategy for them which has probability of success (all prisoners' survival) exceeding 30%.

Comment: If each prisoner examines a random set of 50 boxes, their probability of survival is an unenviable $1 /2^{100} \approx 0.0000000000000000000000000000008 $. They could do worse—if they all look in the same 50 boxes, their chances drop to zero. 30% seems ridiculously out of reach—but yes, you heard the problem correctly.

This problem was sourced from the excellent, "Seven Puzzles You Think You Must Not Have Heard Correctly," (Note - link includes solutions!) Compiled by Peter Winkler

This is the hardest brainteaser that I've ever actually been able to solve. (Eventually, anyway - I spent my spare time for a couple of weeks on it, and at the time, even though I found the solution, I lacked the math skills to calculate the precise odds of the optimal solution working.) Can you explain the solution without too much mathematical background (less than what the linked solution assumes)?