Originally Posted by dreams
Thank you, pdurrant! That was so great of you to write that out and make it so understandable (well, at least with the last part you wrote, I finally understood.
I read several other people's explanations of (equivalent) puzzles before I wrote mine. Many of them we much more mathematical, and raised the subject of induction formally, and probably don't help people who haven't done formal mathematics including induction proofs.
So I decided that it would be better to write it out in words. But for many people without mathematical training, simply referring back to a previous proof smacks of sleight of hand — the "with one bound he was free" school of problem solving. (Even though, of course, it's entirely valid.)
Which is why I wrote out the fourth case in full. It is still a little hard to read, but I think it shows how a chain of imagined responses to a situation can be changed by the governor's speech, even though the knowledge of the fact he mentions is not changed for any of the actual prisoners.
In effect, the governor by his speech tells an imaginary prisoner that there are both blue and green, and this anchors the chain of reasoning. Without the speech, we can't think that the imaginary prisoner who sees only one colour will knows that there are two colours and apply for release, and so the chain breaks.
I'm very pleased to hear that it did make sense to you.