Math Is Fun Forum

Mathegocart

Member

Registered: 2012-04-29

Posts: 2,226

The Three Hat Logic Problem

I have been bombarded with a particularly perplexing logic problem which states as follows.
Middle, Rear and Front are standing in a line. A eccentric millionaire challenges them to a game. Choosing arbitrarily from 3 white hats and 2 black hats, the millionaire puts a hat on each of the three men. The men cannot see their own hats. The millionaire then asks each of the three the color of their hats. Rear, who can see two other hats, declines to give an answer(he cannot determine his hat's colour.). Middle, who can see only Front's hat, also declines to give an answer(he cannot determine his hat's colour.). Finally, Front, who cannot see any hats, states the color of his own hat.
How?
If somebody decides to link me to the MIF "Three Hats Puzzle", keep in mind we have THREE white hats and TWO black ones.
Now I disprove that scenarios with Front having a black hat could happen in these conditions(Either Middle or Rear would be able to clearly define the colour of their hat.), and thus Front has a white hat. Wikipedia concurs.
However, I have noticed that I end with a convoluted, case by case solution. Can anyone provide me with a simpler solution(not case by case..)?

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Fruityloop

Member

Registered: 2009-05-18

Posts: 143

Re: The Three Hat Logic Problem

The person in the back sees that the two people in front of him have at least one white hat on.  The middle person sees the person in front has a white hat, because if it were black then he would know that his own hat is white, because if his own hat were black, then the person in back would know that his own hat is white.  So the person in front knows his hat must be white.