A group of terrorists have taken 100 men as hostages and numbered them as hostage 1,2,3...,100. The terrorists tell them that each one of them is asked to enter a room (one at a time) where there are 100 boxes each containing a paper with one number in it (each number is placed to a random box and each box contains only one paper).

Once a hostage is asked to enter a room he chooses 50 boxes to be opened and if he finds his number on any of the papers he will be escorted to a waiting room. If however the hostage fails to find his number all of the hostages are brutally killed. If all of the hostages succeed in finding their own number they all will be spared.

After telling this the terrorists give some time for the men to consider for an optimal strategy to survive. Once they have agreed on their strategy all communication is forbidden.

If all of the hostages choose 50 boxes at random their probability of survival is kind of weak (.5^100). What kind of strategy should they use to survive at least with 30% probability?

Case 1. If the hostages are allowed to leave any sign, then it is simple: hostage-1 will check the boxes 1-50, finds his number with 50% probability. If he finds paper-2 in these boxes, then he moves forward box-1, otherwise he moves forward box-51. Hostage-2 will know whether to check boxes 1-50 or 51-100. He will also look for paper-3 and so on. Survival rate is 50%.

Case 2. If the hostages are not allowed to leave any sign: there is no way to get 30% survival. The first hostage will have 50% (50/100) chance of success by checking boxes 1-50 (or any 50 boxes). Hostage-2 will know that paper-1 is in boxes 1-50, so he has a better chance of checking boxes 51-100. His chance is 50/99. The total chance of surviving after hostage-2 is now less than 30%: (50/100)*(50/99).

Posted by Art M on 2007-01-30 05:51:51