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There's an infuriating variant, which I have as yet been unable to solve: An infinite sequence of people have either blue or brown eyes. They must shout out a guess as to their own colour of eyes, simultaneously. Is there a way for them to do it so that only finitely many of them guess incorrectly?
Earlier quoted context omitted.
I think you've got it wrong. The information added is that a blue eyed person has been positively identified. In the three person case, each blue eyed person can see two people and is internally modeling their logic about the two person scenario. Once the logic for a two person scenario falls through, they can infer that there are not two people with blue eyes. The brown eyed people are however modeling an N+1 person…
Hmm. That seems right. With 3 blue-eyed people, a blue-eyed person observing that the 2 blue-eyed people did nothing does add the additional knowledge that there are a total of 3 blue-eyed people. So the additional knowledge added for 3 or more blue-eyed people is a lower bound on the total number of blue-eyed people (which grows over time), but it's not exactly clear to me why the foreigner's statement triggers this…
Or, the other interesting thing is what if there were 500 and 500?
It's a very weird logic puzzle for sure.
There's an infuriating variant, which I have as yet been unable to solve: An infinite sequence of people have either blue or brown eyes. They must shout out a guess as to their own colour of eyes, simultaneously. Is there a way for them to do it so that only finitely many of them guess incorrectly?
They could all shout "red". 0 is finite.
- It is very, very, very frequent in mathematics to describe a number as "finite" specifically to indicate that it is nonzero. This is because while zero is boundedly large, it is not boundedly small (it is "infinitesimal").
- The problem statement asks for finitely many to guess incorrectly, not for finitely many to guess correctly.
Well let's try the experiment a few times and see if our results match up with our theories in double-blind tests.
Me: I saw my parents wrapping a Christmas present, but on Christmas
when I received that present, it was labeled "from Santa".
Him: Santa Claus is real.