The XKCD thing is dumb and wrong.
The Guru doesn't introduce ANY new information. The only thing she provides is a "starting gun". If EVERYONE is perfectly logical, and they want to leave the island, why wouldn't everyone automatically agree "Well, we all see more than fifty people with blue eyes, why not start at day 50 instead? Waiting here for months is boring".
The thing that makes it even more wrong is that there *is not* a way to logically intuit that the starting gun serves that specific purpose! Once it's explained, it seems obvious, but it does. not. logically. follow. that you would start on day 1 after that. What DOES follow is that everyoue agrees to get off the island as soon as possible. "In 100 days" is exactly as arbitrary as "Wait 50 days then start counting down", but "the fastest possible" is unambiguous.
...I've just spent an hour puzzling over this, trying to figure out how far ahead you can jump. I don't rightly know, but it may be obvious and I may be overthinking it. You want to find a number that everyone can agree with that is guaranteed lower than the size of the smallest group. 90 is clearly safe. 97 is safe: Someone with blue eyes sees 99 people with blue eyes, and says "If I have brown then they'll each think there are either 98 or 99 and if I have blue they'll each think there are either 99 or 100". 99 isn't safe. Is 98?.. How do you identify a special number like that?
I was trying to establish a rule where "if both groups can agree that one group is bigger than the other, then it's the smaller group's duty to leave first, so apply rule set A; otherwise apply ruleset B" to simplify it...but it's proving very difficult to even just figure out when the groups agree!
I am CERTAIN that I can improve on the given solution.
Wow, ninja'd 18 times. Okay. To state succinctly: Starting counting from n=1 is not logically defensible, because 1 is an arbitrary number. The ONLY defensible position is to leave in a minimum number of days, which means starting with n=the highest number possible in the circumstances.