In a distant, dark forest, lives a population of 400 highly intelligent dwarfs. The dwarfs all look exactly alike, but only differ in the fact that they are wearing either a red or a blue hat. There are 250 dwarfs with a red hat and 150 dwarfs with a blue hat. Striking however, is that the dwarfs don’t know these numbers themselves and that none of them knows what the colour of his hat is (there are for example no mirrors in this forest). But the dwarfs do know that there is at least one dwarf with a red hat.
During a certain period of their year, there is a big party in this village, to which initially all dwarfs will go. However, this party is only intended for dwarfs wearing a blue hat. Dwarfs with a red hat are supposed never to return to the party again, as soon as they know that they are wearing a red hat.
How many days does it take before there are no more dwarfs with a red hat left at the party?
The answer is: after 250 days there are no dwarfs with red hats left at the party.
It is easiest to start from the situation where there is only 1 dwarf with a red hat. In that case, this dwarf would arrive at the party on the first day, and notice that he doesn’t see any red hats. Since he knows that there should be at least 1 dwarf with a red hat, he concludes that he must have a red hat himself. As a consequence, the next day he doesn’t show up at the party anymore.
If there would be two dwarfs with red hats, they both would see 1 dwarf with a red hat on the first day. They know that if this dwarf doesn’t return on the next day, he must have been the only one. If he returns, then there is no other conclusion then that they both must have a red hat. As a consequence, after 2 days they don’t return to the party.
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.