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 can all see everyone else's eyes. That is, person N can see person M's eyes, for all M,N.
[I don't know whether it's possible or not - it feels not, but it has been hinted to me that it is possible.]
- 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.
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?