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A few researchers are trying to crack a code which involves discovering the values of three integers. They know they are between 1 and 100 (inclusive), and that they may be the same.

They each have a different piece of information;

Alice knows the geometric mean

Bob knows the arithmetic mean

Chris knows the arithmetic mean of the squares

They get together to share information and crack the code, but they are being very secretive in an attempt to conceal their results from anyone else. You listen in to their conversation;

Chris: “I don't know all of the numbers”

Alice: “I don't know any of the numbers”

Chris: “You didn't need to tell me that, I knew that already”

Alice: “Well now I know all the numbers!”

Chris: “I still don't know all the numbers”

Bob: “I don't know all of the numbers either”

Chris: “Argh, I still don't know all the numbers”

Bob: “Ah - now I know all of the numbers”

“A ha!” you cry. For they have accidentally revealed the numbers to you. What are they?

You should assume these researchers always tell the truth and make perfect deductions and calculations. However, they might not always be as precise as they could be; “I don't know all the numbers” does not necessarily mean that they know any of the numbers.

Note: You might want to use a computer to solve this.

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    Did you make this up? Do you know if this has a solution?2012-02-01
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    This is a riddle that was proposed to me in a serious context, and is supposed to have an answer. I did not make it up, but have similar suspicions about the existence of a solution. Then, I would like to prove that it has no solution.2012-02-01
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    @Aryabhata I know this has a solution. This is found in texts available in India for people preparing for Math Olympiads in the form of Census-Taker and ages and blah. Too lazy to write down a solution, though!2012-02-01
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    Aha, thanks Kannappan. Perhaps when you have a little more energy, you could enlighten us? Or, perhaps even just a clue? Thanks for your input, cheers.2012-02-01
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    In trying to find out where this problem came from, I ran across some Matlab code intended to solve it: http://www.mathworks.jp/matlabcentral/fileexchange/34575-two-cool-problems-solved-by-matlab-codes/content/MyCode_Prob2.m2012-02-01
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    Thanks, BR! That's really good. Although, this guy makes a mistake: % Part 1 : C says I don't know all of the numbers. % 2. It means that he knows at least one number which he deduced that % from the mean of square.2012-02-01
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    while, the rule states: “I don't know all the numbers” does not necessarily mean that they know any of the numbers. (this is of course mathematically true apriori)2012-02-01
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    @BR: the solution to this problem, as well as the other one posted by that guy on matlabcentral are bunk!2012-02-01
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    confused, I agree, though by reading his solution, I see how to program a correct one! (Well, I can't guarantee it is correct, since I'm not going to actually program it.)2012-02-02
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    I agree, BR. It is a matter of brut force. This will be the way to prove it has an answer or perhaps has no answer... thanks, cheers2012-02-02
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    Just a comment: Do not remove the question by editing, and replacing it with "Thanks I got it!". Since other people will come to this thread, leaving it that way makes it unreadable for everyone. (In the event that you want to remove the question, it is possible to delete it by clicking the delete button.)2012-02-04
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    And if you *have* found the solution yourself, then instead of deleting the question, the best thing for all concerned is if you post the solution as an answer to your own question and accept it!2012-02-04
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    @BR As the original author of this puzzle I can tell you that I devised this as a deliberate variant on a much older puzzle: http://en.wikipedia.org/wiki/Impossible_Puzzle. Ironically, the reason for its construction was because the 2 number puzzle was too easily solved by looking on the internet!2013-05-15

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