The Prisoners' Dilemma

The warden of Alligator Isle Prison, Sad Sammy, liked playing with the prisoners in his charge. He couldn't usually toy with them terminally, of course, or he would have no prisoners left; but one faction of 35 prisoners had been giving him no end of grief. He didn't have an excuse to kill them off without any kind of recourse, so he posed the following challenge:

"Tomorrow morning I will put thirty-five slips of paper in a hat; each slip will have a number from 1-35. At least one of the numbers will appear twice (e.g. there might be two 33s, and I might not use 32 at all). You will then be led, one at a time, to a room with a blackboard. The blackboard has a single number on it; initially this number will be 1, but you can alter the number. You can perform other computations on the blackboard, but when you leave there should again be a single number in the middle of it. It may be any real number, but not a more complicated expression. Because I rather want to feed you scurvy knaves to the alligators, I am adding the condition that the number must be of fewer than 35 digits. Simply assigning each man a digit will not suffice for this puzzle.

"While in the room you are not allowed to alter anything else: if you try to do so, the guard will drag you out and throw you to the alligators.

"Using this number, you must determine whether another prisoner with your number has already been through the room. You don't know who received which number, so the blackboard is your only source of information. I know who has all the numbers, and if the second of a pair comes out without saying anything (he doesn't realize he was the second), you will all have failed the task. If someone comes out and says he is the second, and is mistaken, you will all have failed the task. If, however, someone correctly states that he is the second of the first pair to visit the room, you will all have succeeded.

"If you fail, the alligators in the lagoon haven't been fed in awhile.
If you succeed, you will go free.

"Because I am such a fair, benevolent warden, I will not force you to take this challenge. I will give you an opportunity to discuss, and tomorrow morning anyone who does not want to take the challenge may stand down, and I will modify the problem appropriately by removing one non-duplicate number. I will toss you in solitary for a month for marring my fun, but that is all."

What scheme can the prisoners come up with that insures that they can successfully answer the warden's challenge?

* "inspired by" indicates that this is not the puzzle submitted, but would not exist without the submission. There is generally a family resemblance in either the problem or the solution, but not both.

Solution

Note: this is actually a revision of the true puzzle 1; the original problem definition was rather looser, allowing a lovely but thoroughly complicated solution (ruled out in this version); but also allowing quite a simple, straightforward solution (also ruled out in this version) which I completely overlooked. If you really want to see the original definition, go here.