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 exceeding 30%.
Comment: If each prisoner examines a random set of 50 boxes, their probability of survival
is an unenviable 1/2100 ∼ 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!
The prisoners should agree on a method to name the boxes. Like taking their names in alphabetical order and assigning to the boxes in order left to right.
Then when in the room each prisoner starts with opening his namesake box and reads the name inside. The name inside will be the next box opened, etc.
Each prisoner will follow the names until he loops back to his name or runs out his 50 tries. The only way for the prisoners to fail is if there is a loop with more that 50 names.
After some searching I found that we already had a version of this puzzle with Fatal Guess
. But how do we get 30%? While I was searching perplexus I kept conflating this puzzle with Shoelaces
, which asks how many loops do we expect. In that puzzle it was determined that for 100 links we expect 3.28 loops. It turns out that 30% is close to 100%/3.28=30.49%