OK, POW#1 one knew because he could tell the other two couldn't work out what theirs were from what they could see. POW#3 didn't know what colour his was... so therefore #1+#3 either BOTH had Black, or one had Black, and one had White, because if they both had white he would have to have Black. POW#2 realises that POW#3 doesn't know, so knows that his could either be black or white because he knows that POW#1's beanie is black, so his could be black or white, so he doesn't know. POW#1 therefore realises that as neither of the other two know, his must be black.