A Puzzle: The Unfair Millionaire
This is a logic puzzle I came up with over twenty years ago, when I was in high school. I will post the solution in a separate post in a week or so, if nobody solves it before then.
Once upon a time, a bored and whimsical millionaire invited five of the world's most prominent logicians to his house for a game. The winners were promised a prize of five million dollars. (The losers, needless to say, were promised nothing.) "But you must use logic," said the millionaire. "This is no game of chance!"
The millionaire placed a hat on each of their heads such that nobody could see the color of their own hat. They could only see the colors of the hats on the other logicians' heads. They were told that the hats were randomly selected from a batch consisting of five white, three red and one black hat. The logicians were then numbered one through five and told that they would be asked, in order, if they knew the color of their own hat. They would have to prove it. Guessing was not allowed. Chatting or passing messages was not allowed, either. They were allowed to answer the question, but not talk in any other way, since their comments would be heard by all.
The first logician was asked, "Do you know the color of your hat?"
The logician was noticably annoyed and replied, "Your game is unfair! I cannot know the color of my hat, and none of the other logicians can know the color of theirs, except for the last one. In fact, the last one can know even if she is blind!"
"Even if she is blind?!" The millionaire responded in disbelief. "If you can prove what you say is true, I will award you the five million dollars!"
The first logician obliged (and, legend has it, shared the reward equally with the other four logicians).
What was the proof?
Once upon a time, a bored and whimsical millionaire invited five of the world's most prominent logicians to his house for a game. The winners were promised a prize of five million dollars. (The losers, needless to say, were promised nothing.) "But you must use logic," said the millionaire. "This is no game of chance!"
The millionaire placed a hat on each of their heads such that nobody could see the color of their own hat. They could only see the colors of the hats on the other logicians' heads. They were told that the hats were randomly selected from a batch consisting of five white, three red and one black hat. The logicians were then numbered one through five and told that they would be asked, in order, if they knew the color of their own hat. They would have to prove it. Guessing was not allowed. Chatting or passing messages was not allowed, either. They were allowed to answer the question, but not talk in any other way, since their comments would be heard by all.
The first logician was asked, "Do you know the color of your hat?"
The logician was noticably annoyed and replied, "Your game is unfair! I cannot know the color of my hat, and none of the other logicians can know the color of theirs, except for the last one. In fact, the last one can know even if she is blind!"
"Even if she is blind?!" The millionaire responded in disbelief. "If you can prove what you say is true, I will award you the five million dollars!"
The first logician obliged (and, legend has it, shared the reward equally with the other four logicians).
What was the proof?
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