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I believe the statement as made is correct, although not necessarily as clear as it could be. For example, your suggested "strategy" is wrong - the first priso

I believe the statement as made is correct, although not necessarily as clear as it could be. For example, your suggested "strategy" is wrong - the first prisoner as you've stated only has a 50% chance. Further, after each prisoner the boxes are closed so that each prisoner finds the room is exactly the same state, with no communication between them.

Someone 16y ago
Apparently, my formulation isn't as clear as it could be, either :-) My 'solution' has all of the prisoners point out 'their' box after the second one has opened the last 50 boxes. As to the 'no communication between': if there is no communication, it seems each of them cannot do better than 51/100 (50 opened boxes, and if their name isn't there, a 2% gamble) However that isn't true. Suppose the first 51 each open the first 50 boxes and decide to choose the 51st box if they do not find their name, Then, all others, working under the assumption that the first 51 found their name (if they didn't, the game is lost anyways), can just open the last 49 and be certain to find their name. That would bring the probability to about 2^-50. I doubt that is the optimal strategy, though. There seems to be much more room between 1/100! and 0.01^100.
- RiderOfGiraffes 16y ago
  I still, really don't understand what you're suggesting. Let's go carefully. The first person opens 50 boxes. Let's suppose they find their own name (otherwise we're dead anyway). They leave the room, everything is put back exactly as it was, they're not allowed to communicate with the others. It seems to me you're suggesting the next person opens the second batch of 50. OK. But then they leave the room, everything is put back exactly as it was, and they're not allowed to communicate with the others. What does the third person do? They're confronted by 100 closed boxes with no idea of anything, except that the first person's name is in the first 50 boxes, and the second person's name is in the second 50 boxes. So, what now?
  - dedward 16y ago
    How do they even know that? Does the game continue until the end - or does it end as soon as one person fails to find their name?
    - RiderOfGiraffes 16y ago
      They don't know that, but you may as well assume they do, because otherwise anything they do is pointless and has no effect.
      - dedward 16y ago
        True - but you only know that at the end of the puzzle. The point of the exercise is to maximize their strategy of winning. There is no "winning" strategy - but there is one that is significantly better than what immediately comes to mind.
        RiderOfGiraffes 16y ago
        But that's not the point. The point is that without loss of generality each person can assume that the people before have all found their own names. If not then it makes no difference what they do, they're all doomed. Unless I've misunderstood you.
  - Someone 16y ago
    The word 'close' does not appear in the problem formulation at all. Apparently, we must deduce that all boxes get closed between visits from "they will not know anything about the previous prisoners' experiences ahead of time". I am OK with that, but I still do not think it is a clear way to describe the problem. So, ignoring that IMO fairly implicit message, the 'answer' I proposed is: prisoner #1 opens 50 boxes, prisoner 2 opens the other 50, and then each prisoner trivially searches for his name. Nitpicking the text further: what is that ahead of time doing there? What time? What can they learn afterwards?
    - 16y ago

Someone 16y ago

Apparently, my formulation isn't as clear as it could be, either :-) My 'solution' has all of the prisoners point out 'their' box after the second one has opened the last 50 boxes. As to the 'no communication between': if there is no communication, it seems each of them cannot do better than 51/100 (50 opened boxes, and if their name isn't there, a 2% gamble) However that isn't true. Suppose the first 51 each open the first 50 boxes and decide to choose the 51st box if they do not find their name, Then, all others, working under the assumption that the first 51 found their name (if they didn't, the game is lost anyways), can just open the last 49 and be certain to find their name. That would bring the probability to about 2^-50. I doubt that is the optimal strategy, though. There seems to be much more room between 1/100! and 0.01^100.

RiderOfGiraffes 16y ago
I still, really don't understand what you're suggesting. Let's go carefully. The first person opens 50 boxes. Let's suppose they find their own name (otherwise we're dead anyway). They leave the room, everything is put back exactly as it was, they're not allowed to communicate with the others. It seems to me you're suggesting the next person opens the second batch of 50. OK. But then they leave the room, everything is put back exactly as it was, and they're not allowed to communicate with the others. What does the third person do? They're confronted by 100 closed boxes with no idea of anything, except that the first person's name is in the first 50 boxes, and the second person's name is in the second 50 boxes. So, what now?
- dedward 16y ago
  How do they even know that? Does the game continue until the end - or does it end as soon as one person fails to find their name?
  - RiderOfGiraffes 16y ago
    They don't know that, but you may as well assume they do, because otherwise anything they do is pointless and has no effect.
    - dedward 16y ago
      True - but you only know that at the end of the puzzle. The point of the exercise is to maximize their strategy of winning. There is no "winning" strategy - but there is one that is significantly better than what immediately comes to mind.
      - RiderOfGiraffes 16y ago
        But that's not the point. The point is that without loss of generality each person can assume that the people before have all found their own names. If not then it makes no difference what they do, they're all doomed. Unless I've misunderstood you.
- Someone 16y ago
  The word 'close' does not appear in the problem formulation at all. Apparently, we must deduce that all boxes get closed between visits from "they will not know anything about the previous prisoners' experiences ahead of time". I am OK with that, but I still do not think it is a clear way to describe the problem. So, ignoring that IMO fairly implicit message, the 'answer' I proposed is: prisoner #1 opens 50 boxes, prisoner 2 opens the other 50, and then each prisoner trivially searches for his name. Nitpicking the text further: what is that ahead of time doing there? What time? What can they learn afterwards?
  - 16y ago