62. Three Men in a Line Wearing Black and White Hats Five hats are available: three black and two white. Three of these hats are chosen at random and placed on the heads of three men who then stand in a straight line all facing the same direction. • Man 3 stands at the back of the line and can see the hats of Man 2 and Man 1. • Man 2 stands in the middle and can see the hat of Man 1 only. • Man 1 stands at the front and cannot see anyone else’s hat. All three men know the total supply of hats (3 black, 2 white) and the rules above. Starting with the man at the back, each man is asked, in turn, “What colour is the hat on your own head?” and must answer truthfully: Man 3 looks at the two hats in front of him and says, “I don’t know.” Man 2 then looks at the single hat in front of him and also says, “I don’t know.” Finally, Man 1, who cannot see any hats, is nevertheless able to state with certainty the colour of the hat on his own head. What colour hat is Man 1 wearing? Added 7 September 2012 Show Hint Show Solution Hint: Ask yourself what Man 3 would have concluded if he had seen two white hats. Solution: Man 1 is wearing a black hat. Reasoning: Man 3 sees the two hats in front of him. If they were both white, he would immediately know his own hat must be black (only two white hats exist). Because he says, “I don’t know,” the two hats he sees cannot both be white. At least one of them is black. Man 2 hears this and looks at Man 1’s hat. If he saw a white hat, he could deduce that his own hat must be black (since the pair cannot be two white hats). But he still doesn’t know, which means the hat he sees on Man 1 cannot be white. Therefore Man 1’s hat must be black. Man 1, having followed the prior statements, realises that his hat has to be black and answers accordingly.
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