### Brain Teasers

# The Dwarvish Feast

In a forest somewhere in Scotland lives a group of 100 dwarves. Each night they meet in the middle of the forest for a grand feast. When morning comes, they all go home. Each dwarf is wearing either a red hat or a blue hat. Curiously, there are no mirrors in the forest, so no dwarf knows the color of his own hat. A dwarf would never take off his hat to see its color, and a major dwarf faux pas is to comment on the color of another dwarf's hat. The dwarves know, however, that there is at least one red hatted dwarf and one blue hatted dwarf. One day, the master dwarf announces that the nightly feast will only be intended for blue hatted dwarves, and as soon as a dwarf knows that he is wearing a red hat, he should not come back the next day, and he should never return.

How many days does it take before there are no dwarves left with red hats at the party? (Assume all the dwarves are equally capable of figuring it out, in other words, there are no smart dwarves, and no stupid dwarves...)

How many days does it take before there are no dwarves left with red hats at the party? (Assume all the dwarves are equally capable of figuring it out, in other words, there are no smart dwarves, and no stupid dwarves...)

### Hint

The number of days depends on the number of red hats.### Answer

It will take one more day than the number of dwarves with red hats.How this works is: assume that there is only one red-hatted dwarf. He would go to the party only to see 99 blue hats. He knows that there is at least one red hat. Since all he sees is blue, he knows it must be him, so he knows not to come back the next day. Meanwhile, the blue hatted dwarves can only see one red hat, so they think that if he comes back tomorrow, then he must have seen another red hat. Since each blue hatted dwarf only sees one red hat, he would assume that if the red hatted dwarf came back tomorrow, then he must be the other red hat that the first red hatted dwarf saw. So the second day, all the blue hatted dwarves would come expecting to see if the red hatted dwarf came back. Which he wouldn't, because he knows he has a red hat. So the first day there are no dwarves with red hats is day 2, or one day more than the number of dwarves with red hats.

If there were two dwarves with red hats, each red hatted dwarf would realize on the second day that he was wearing a red hat, because he would see that the other red hat had come back, and that means that he must have seen another red hat. Since the red hats would see 98 blue hats, they would know on the second day, and the third day would be the first day that no red hatted dwarves came to the feast. This same system works for any number of red hats, because each time a blue hatted dwarf is thinking that if the red hats come back tomorrow, then he must have a red hat, the red hatted dwarves already know, and so they don't come the next day.

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