>I just re-read the original problem. Seems like if they lied, they would get killed because they would have their hat colour wrong.

Yeah, that's what stopped me from continuing to try and figure it out. I thought my first suggestion about pausing or not pausing before they said their hat color was pretty good, but it was shot down... :(

If I was one of the sages we would probably all die as I wouldn't be able to understand what we were supposed to do!