Dulce
secondary
student

There are 1000 dwarves in Dwarfland. Now as you know, dwarves are born with either red or white hats on.

The dwarves in Dwarfland love to party. Every night, they all flock to the same club to dance the night away.

One day, a sign is posted at the club: "NO RED-HATTED DWARVES ALLOWED". A certain number of days pass, after which all of the white-hatted dwarves are in attendance,and none of the reds.

How did the red hats know not to come?

Assumptions:

1. The number of red and white-hatted dwarves need not be equal. There are at least 1 of each color.
   
   
2. There are no reflections or mirrors in Dwarfland (i.e. the dwarves can't see the color of their own hat), and no dwarf can reveal through direct or indirect means the color of another dwarf's hat.
   
   
3. There is NO COMMUNICATION among dwarves
   
   
4. No white-hatted dwarf ever misses a night of partying

Additionnal assumptions:

5. No red-hatted ever ever misses a night of partying either, unless he knows he's not allowed.
   
   
6. Dwarves can read, count, distinguish colors and think logically.
   
   
7. Dwarves are aware of the above assumptions.

Conclusions:


1. If there were only one red-hatted dwarf in Dwarfland:
   
   Then seeing the sign on the first day, he would have looked around and upon seeing only white-shatted dwarfs he would have realized that the interdiction applied only to him and gone home to sulk.
   
   
2. If there were only two red-hatted dwarves in Dwarfland:
   
   Then seeing the sign on the first day, each would have looked around and upon seeing only one red-shatted dwarf, each would have hoped that the interdiction applied only to that poor fellow. But when the next day the same fellow came back, each would have realized that this fellow saw another red hat in the crowd. At that point, each would have realized that theinterdiction applied to him also and gone home to sulk.
   
   
3. If there were only three red-hatted dwarves in Dwarfland:
   
   Then seeing the sign on the first day, each would have looked around and upon seeing only two red-shatted dwarves, each would have hoped that the interdiction applied only to these poor fellows. But when two days later the same fellows came back, each would have realized that these fellows saw another red hat in the crowd. At that point, each would have realized that theinterdiction applied to him also and gone home to sulk.
   
   
4. If there were only four red-hatted dwarves in Dwarfland:
   
   Then seeing the sign on the first day, each would have looked around and upon seeing only three red-shatted dwarves, each would have hoped that the interdiction applied only to these poor fellows...
Claude

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