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.
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.
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.
