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Server Time: 11/21/2008 9:26:39 AM PACIFIC |
 The Mathematicians' Hat Problem, Wren, 8. May 2003 14:40 | ||
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| Three mthematicians get picked for a contest in which they have a chance at splitting a $10,000 prize. The contest works in the following way: on each mathematician's head is placed either a white or a black hat (the colour is chosen randomly with a 50% probability of either being picked). He/she does not know the colour of his/her own hat. The three mathematicians are then brought into a room where each one can see the other two, and hence the colour of the other two hats. On the count of three, each mathematician must simultaneously either guess the colour of his/her own hat, or say nothing at all. The contest is won (and the three split the prize) if AT LEAST one mathematician guesses correctly, and NO mathematician guesses incorrectly. Two important points: (1) The mathematicians are allowed an initial strategy session together before the hats are placed on their heads. (2) When they are brought into the room together with their hats on, they are not allowed to signal one another in any way to indicate hat colour. Question: What strategy should they use to maximize the chance that they will win this contest? What success rate does this strategy guarantee? | ||
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| Re: The Mathematicians' Hat Problem, Snorbolus, 8. May 2003 14:59 | ||
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| Thanks Wren. I am feeling a little guilty about posting the Monty Hall thing on the poker group. Not Quite Poker is much more appropriate. My excuse is that my mind was still in turmoil after thinking about problem for too long. I look forward puzzling over the other problems. Snorbolus | ||
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| Re: The Mathematicians' Hat Problem, Wren, 8. May 2003 15:04 | ||
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| I wouldn't worry about posting it over there...if you initially posted it here, a lot of people might have missed it! But hopefully now we'll have people checking out this forum for logic problems, word puzzles and the like. I hope people post more of these 'cause I love 'em! | ||
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| Re: The Mathematicians' Hat Problem, Nathaniel Brous, 8. May 2003 15:18 | ||
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| If you haven't figured it out yet don't read on. spoiler space ss ss ss ss ss ss ss ss ss The first guy looks at the others and if the other two are different then he stays quiet. When one guy sees the other two have the same color (this has to happen) then he breaks the silence and guesses the opposite color. This should happen 3 out of 4 times. bbb bbw bwb bww www wwb wbw wbb - Nathaniel Brous | ||
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| Re: The Mathematicians' Hat Problem, Snorbolus, 8. May 2003 16:41 | ||
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| Answer below: Home from work now. After much thinking and drawing of tree diagrams I believe that the best strategy would be for each mathematician to only guess if they saw two hats of the same colour, in which case she should guess that she was wearing a hat of the opposite colour. If my calculations are correct this stragegy should give a win probability of 3/4. Working: Case 1: BBB all mathematicians guess W. LOOSE p=1/8 Case 2: BBW 2 mathematicians see BW and don't guess, 1 sees BB and guesses W. WIN p=3/8 Case 3: BWW 2 mathematicians see BW and don't guess, 1 sees WW and guesses B. WIN p=3/8 Case 4: WWW all mathematicians guess B. LOOSE p=1/8 By the way, I checked Nathaniel's answer to make sure that it broadly agreed with mine before I posted ;-) | ||
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| Re: The Mathematicians' Hat Problem, stdioh, 9. May 2003 08:25 | ||
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| Where this problem gets really interesting is when you expand it out to more and more people. You then have a standard problem in coding theory where you need to get each mathematician to calculate his hamming number and then decide whether or not to speak up. | ||
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| Re: The Mathematicians' Hat Problem, Banning, 9. May 2003 16:31 | ||
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| I was waiting for someone to bring up hamming code theory...geek!!! reminds me of my math 222 class. Geeks rule and I'm one of them. | ||
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| Re: The Mathematicians' Hat Problem, stdioh, 12. May 2003 10:01 | ||
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| Yup. The scary part is that I never took coding theory...but I did do a bachelor's of Math with a major in Computer Science (so did Wren, incidentally) and I work in Compilers...so these things get discussed. | ||
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