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> Well, first of all, nothing rules out the first prisoner getting free--look more carefully at how the sequences work. Except that in the infinite colors case

> Well, first of all, nothing rules out the first prisoner getting free--look more carefully at how the sequences work. Except that in the infinite colors case it's infinitely improbable. > Thinking more: it seems like you've confused the claims: "there is a finite number n such that any sequence of prisoners can guarantee that they will only have n prisoners remaining" (false) with "for each sequence of prisoners, there is a finite number n such that they will only have n prisoners remaining" ( true). The latter is the claim that I understood to be effectively false. Can you actually present an example sequence of prisoners for which there is a finite standard integer n? If so, what is the value of n in the example?

hyperpape 12y ago
The first of the two claims I stated implies the second, but not vice versa. I don't really understand what you mean by choosing an example sequence. Here's one: all prisoners have black hats. They all guess black. They all go home. Nothing in the problem rules that out, though nothing guarantees that the prisoners' strategy will get them this result. I believe there are actually an uncountable infinity of strategies that the prisoners could choose, with an infinite number of distinct choices that the prisoners could make for any given sequence that they get arranged in. What the strategy guarantees is that starting with the first prisoner, some (zero or more) go home, and some (zero or more) remain captives, and then after some (finite, standard) number of prisoners, all the rest go home.
- rwallace 12y ago
  Right, that's a valid example case with n=0, though such cases are of measure zero in the same way the rational numbers are of measure zero in the real numbers. Here's a counterexample: give all the prisoners white hats. Now the same strategy produces no prisoners going home. In fact, if the population of hat distributions is restricted to {all black, all white}, this still suffices to show that there is no strategy that guarantees all prisoners will go home after some finite standard number.
  - ColinWright 12y ago
    > ... if the population of hat distributions is > restricted to {all black, all white}, this still > suffices to show that there is no strategy that > guarantees all prisoners will go home ... The strategy given in the submitted article is such that if all prisoners are given the same color hat then they will all go home, so I'm not sure what you're saying here.
    - rwallace 12y ago
      I'm saying the strategy in the article does not actually do this at all, and it's easy to prove this by noting that a strategy that has them all going home with black hats has none of them going home with white hats.
      - hyperpape 12y ago
        You're confused about what the strategies are. A strategy is not "all say black" or so on. The prisoners will make different choices depending on the infinite sequence of hats they see in front of them.
      - ColinWright 12y ago
        I've worked through the strategy, and what you say seems simply to be false. As I work through the strategy I find that if they all have black hats the strategy has them all saying "Black", and if they all have white hats then the strategy has them all saying "White." I don't understand why you claim otherwise.

hyperpape 12y ago

The first of the two claims I stated implies the second, but not vice versa. I don't really understand what you mean by choosing an example sequence. Here's one: all prisoners have black hats. They all guess black. They all go home. Nothing in the problem rules that out, though nothing guarantees that the prisoners' strategy will get them this result. I believe there are actually an uncountable infinity of strategies that the prisoners could choose, with an infinite number of distinct choices that the prisoners could make for any given sequence that they get arranged in. What the strategy guarantees is that starting with the first prisoner, some (zero or more) go home, and some (zero or more) remain captives, and then after some (finite, standard) number of prisoners, all the rest go home.

rwallace 12y ago
Right, that's a valid example case with n=0, though such cases are of measure zero in the same way the rational numbers are of measure zero in the real numbers. Here's a counterexample: give all the prisoners white hats. Now the same strategy produces no prisoners going home. In fact, if the population of hat distributions is restricted to {all black, all white}, this still suffices to show that there is no strategy that guarantees all prisoners will go home after some finite standard number.
- ColinWright 12y ago
  > ... if the population of hat distributions is > restricted to {all black, all white}, this still > suffices to show that there is no strategy that > guarantees all prisoners will go home ... The strategy given in the submitted article is such that if all prisoners are given the same color hat then they will all go home, so I'm not sure what you're saying here.
  - rwallace 12y ago
    I'm saying the strategy in the article does not actually do this at all, and it's easy to prove this by noting that a strategy that has them all going home with black hats has none of them going home with white hats.
    - hyperpape 12y ago
      You're confused about what the strategies are. A strategy is not "all say black" or so on. The prisoners will make different choices depending on the infinite sequence of hats they see in front of them.
    - ColinWright 12y ago
      I've worked through the strategy, and what you say seems simply to be false. As I work through the strategy I find that if they all have black hats the strategy has them all saying "Black", and if they all have white hats then the strategy has them all saying "White." I don't understand why you claim otherwise.