One hundred extremely intelligent male prisoners are imprisoned in solitary cells and on death row. Each cell is soundproofed and completely windowless. There is a separate room with one hundred small boxes numbered and labeled from 1 to 100. Inside each of these boxes is a slip of paper with one of the prisoners' names on it. Each prisoner's name only appears once and is in only one of the one hundred boxes. The warden decides he is going to play a game with all of the prisoners. If they win, they will all be let free, but if they lose the game, they will all be immediately executed. The hundred prisoners are allowed to enter this separate room with 100 boxes in any predetermined order they wish, but each can only enter the room once and the game ends as soon as the hundredth person enters the room. (At any time, only one prisoner is allowed to enter and remain in this room.) Once a prisoner enters the room, he is allowed to open and look inside as many as Xboxes, where X is a positive integer not greater than 100. After a prisoner has opened the boxes (not more than X) and looked inside, he must shut them and leave everything exactly the way it was before he entered. The prisoners are not allowed to communicate with each other in any way. If every prisoner is able to enter the room and open the box that contains his own name, they will all be released from prison immediately! However, if even just one prisoner enters the room, opens X boxes, and does not open the box containing his own name, they will all be executed immediately. Luckily for the prisoners, the warden has decided to allow the first prisoner in the room to open all one hundred boxes if necessary and switch the two names in any two boxes if he would like to. The first prisoner must shut all the boxes he opened and leave everything exactly the way it was before he entered, with the possible exception of the two names he chose to switch. Again, in order to win this game, all one hundred prisoners need to enter the room and open the box with their name in it. The warden allows the prisoners to get together in the courtyard the week before this game begins to discuss and come up with a plan. The prisoners come up with a plan to guarantee that they would win the game. Of course, if X is big enough, for example X= 100, then they will all be released from prison. What is the least possible value of X? Note: (1) No two or more prisoners share the same name. (2) The first prisoner in the room is an exception, meaning that he is able to open all one hundred boxes if necessary and switch the two names in any two boxes if he would like to. Each of the other 99 prisoners is allowed to open and look inside as many as Xboxes, where X is a positive integer not greater than 100. And they can't switch anything or do anything else. (3) No prisoner is allowed to make any kind of mark, sign, or hint. The prisoners are not allowed to communicate with each other in any way. (4) Here's part of their plan: First the prisoners will decide the order in which they will enter the room and number each prisoner from 1 to 100, with Prisoner 1 being the first to enter the room and Prisoner 100 being the last. Then, each prisoner must memorize all other prisoners' positions and names in line, which they are able to do.