The answer is: after 250 days, there were no dwarfs with red hats left at the party.
It is easiest to start from the situation where there is only one dwarf with a red hat.
In that case, this dwarf would arrive at the party on the first day, and notice that he does not see any red hats.
Since he knows that there should be at least one dwarf with a red hat, he concludes that he must have a red hat himself.
Consequently, he does not show up at the party anymore the next day.
If there would be two dwarfs with red hats, they both would see one dwarf with a red hat on the first day.
They know that if this dwarf does not return on the next day, he must have been the only one.
If he returns, then there is no other conclusion possible than that they both must have a red hat.
Consequently, they do not return to the party after 2 days.
The situation where there are 250 red hats is identical,
only now it takes 250 days before all dwarfs with red hats can conclude that they must have a red hat.
back to the puzzle