The 'dot-town suicides' is a more general version of a puzzle I know, "The Island with Blue-Eyed People". The solution is an induction, which is unusual in these kinds of problems.
https://en.wikipedia.org/wiki/Common_knowledge_(logic)
Spivak has a version of it in his Calculus book, phrased as 17 (heh!) professors who must resign if a flaw is found in their published work (hehehe), and all have a flaw in their papers, known to each other except each author.
For instance, if the number of blues is 25, and the (merciful) stranger says that the number of blues is not prime every day, no one ever has enough information.
And in fact, even that doesn't seem to be enough. The non-trivial part must be that at least one person knows more than they did before the statement, otherwise a merciful stranger could say 'The number of blues is less than <big number>' each day, where the big number is more than the population.
Unfortunately that stringent of a definition for non-trivial sort of ruins the problem since in the formulation is the idea that at least one person in the village gets (at least) one number closer to knowing the exact number of blues everyday, and as soon as they know the exact number of blues they are eliminated.
The new information in that scenario is that the other guy with blue eyes knows that someone has blue eyes.
I think that's what they meant by non-trivial. If the stranger tells them something they already know, that doesn't count.
You can assume that an individual person can count the blue dots they see, but they can't count their own dot. If they count 9 dots, then the true number must be either 9 or 10, so they already know the number can't be prime.
The fact that this happens even though it doesn't sound like new information is what makes it an interesting puzzle!
Also:
> For instance, if the number of blues is 25, and the (merciful) stranger says that the number of blues is not prime every day, no one ever has enough information.
This is not correct. Saying "the number of blues isn't prime" just once would be sufficient to start the process, as would "there is at least one blue", etc. It doesn't matter that it's not new information to any one villager.
Sorry, can you explain this part? If there were 10 blues and 10 reds, and the stranger said "there is at least one blue", why would that result in a suicide in any of the dot-town people?
If you'd like a hint, imagine that there are only two villagers - you're one of them, and the other one has a blue dot. A visitor tells you both "there is at least one blue". You already knew that, but did he? And what happens the following day, when you see him still alive (or don't)?
Some of the questions incorporate a "new idea" or allow you to change elements. Triangle boxes, prisoners who can make requests, tired tennis players.... I introduced a yellow dot or am I supposed to change the variables to two?
Edit: had a paren instead of opening quote.
In a classroom with 30 people right now and I couldn't tell you how many of each gender there are unless I actively try. If that meant certain death, why would I count?
The distinction comes naturally to some people, it's a real struggle for others.