Brain Teaser #49
Colorful Dwarfs
In a distant, dark forest, lives a population of 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. Striking however, is that none of the dwarfs knows what the color of his or her hat is. This is because it is forbidden to speak about the color of the hats, and there no mirrors in the forest. Nevertheless, the dwarfs do know that there is at least one dwarf with a red hat.
Once every century, 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 not to return to the party the next day, as soon as they know that they are wearing a red hat.
The question: How many days did it take before there were no more dwarfs with a red hat left at the last party, if you know that there were then 250 dwarfs with a red hat and 150 dwarfs with a blue hat?
Solution to: Colorful Dwarfs
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.
