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More ways to get Braingle...

The Dwarvish Feast

Logic puzzles require you to think. You will have to be logical in your reasoning.

 

Puzzle ID:#13609
Fun:*** (2.27)
Difficulty:*** (2.86)
Category:Logic
Submitted By:bobdole***
Corrected By:phrebh

 

 

 



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...)





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Comments

jimbo*au*
Jul 08, 2003

What if each dwarf went up to another and said "See you tomorrow." The dwarves seeing the red hatted dwarf would say, "Oh are you coming then?" The red hatted dwarves would know from the others response that they were not expected. Anyway, I like the puzzle and the logical explanation seems good.
Fasil*
Jul 09, 2003

I guess the dwarves didn't notice the color when the put on the hat good logic puzzle though.
peanutAza*
May 04, 2005

This sounds logical but I'm still not seeing it. The logic works for 1 or two red hats but what if there were say 5 - how does it happen?
paul726Aus*
Feb 16, 2006

I don't think it works even with two. Why would not a blue-hatted dwarf think, when he saw all of the red-hatted dwarves return the second day, that he was also red-hatted? Why would only another red-hatted dwarf think that, given essentially the same info the blue-hats have?
dishuAin*
Apr 18, 2006

and what will happen if there are more than 50 red hat dwarfs or lets say 90 red-hat dwarfs.i dont think it will work then
JimShorts*us*
Apr 24, 2012

If you follow the logic out, amazingly enough it actually works for any number of red hats.

As it's explained in the answer, 1 red hatter, seeing only blue hats and no red hats, would know he had the only red hat, so he wouldn't return the next day. So if I see one red hatter and he DOES return the next day (day two), he must have seen a red hat. Since everyone else I see has blue hats, I must have the other red hat. So I wouldn't return the next day, day three (and neither would he, having reached the same conclusion).

So I know that if there were 2 red hatters, each seeing only the one other red hat, then neither one would come back on day three. So if I see 2 red hatters, and they come back on day three, that means that they saw more than just the red hat on each other. Since everyone else I see has a blue hat, that means I must have the other red hat that they're seeing. So on day four all 3 of us red-hatters, having reached the same conclusion, would not return.

So if I see 3 red hatters and they're still here on day four, then they must not have reached the above conclusion, meaning they must each see more than just the other 2 red hats. Since everyone else I see has a blue hat, that means I must have the other red hat that they're seeing. So on day five all 4 of us red-hatters, having reached the same conclusion, would not return.

So if I see 4 and they're still there on day five, they must see a red hat on me, so I won't come back.

If I see 5 and they're still there on day six, they must see a red hat on me, so I won't come back.

And on and on ...

Presumably, since all the dwarves would understand this formula, they would each count the number of red hats they see (even if it's a large number) and wait that number of days +1. If all the red hatters are gone that day, they'd know they were safe, if not then they'd know they couldn't come back the next day.

So if you see 86 red hatters, you know that if you have a blue hat then each of them are only seeing 85 red hats and they'll leave after day 86 comes and goes, but if day 87 comes AAH MY BRAIN FELL OUT!!



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