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 by Charlie)
Each prisoner starts with his namesake box. That box must be in the same loop as the goal box with his name inside it, specifically the goal box will be the last box in the loop. If we ignore the 50 tries limit then every prisoner will find his name in a number of tries equal to the size of the loop he is in.
To illustrate a scenario with 10 prisoners: Lets say the boxes in order have names B G H A F C J E I D inside them and the prisoners name the boxes A B C D E F G H I J. Then there are the following loops: {A-B-G-J-D}, {C-H-E-F}, {I}
Lets follow Prisoner E. He starts with the box named E and finds name F. His next box is F, where he finds C. His third box is C, where he finds H. His fourth box is H, where he finds his name C.
C, F, and H will all have similar results finding their name in that loop in the fourth try, the size of his loop. A,B,D,G,J will all find their names on the fifth try. I will be lucky since his name is in the box named for him. In this specific 10 prisoner scenario they will each succeed in finding their own names within 5 tries.
re(2): Loops
