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!
(In reply to re: Loops
Having all cycles of length 50 or less works because each prisoner is guaranteed to be in their own loop.
If box 1 has prisoner 2's name,
box 2 has prisoner 4
box 4 has prisoner 3 and
box 3 has prisoner 1 then each of these prisoners will find their name in exactly 4 tries.
None of the other prisoners would ever have to check any of the boxes of this cycle.
Posted by Jer
on 2017-04-12 13:27:25