I can NOT do this puzzle: "Consider an army with 10 generals...."

Steven G

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Consider an army with 10 generals. One wants a security system such that any three of them can determine the code to launch nuclear missiles, but no two of them can. It is possible to devise such a system by using a quadratic polynomial, such as a x^2 + bx + c; to launch the missiles, one must input (a,b,c). One cannot just tell each general one of a, b, or c (as then it is possible that some subset of three generals won’t know a, b and c); however, if a general knows two of (a,b,c), then a set of two generals can launch the missiles! What information should be given to the generals so that any three can find (a,b,c) but no two can? What about the general situation with N generals and any M can launch (but no set of M-1) can?
 
Consider an army with 10 generals. One wants a security system such that any three of them can determine the code to launch nuclear missiles, but no two of them can. It is possible to devise such a system by using a quadratic polynomial, such as a x^2 + bx + c; to launch the missiles, one must input (a,b,c). One cannot just tell each general one of a, b, or c (as then it is possible that some subset of three generals won’t know a, b and c); however, if a general knows two of (a,b,c), then a set of two generals can launch the missiles! What information should be given to the generals so that any three can find (a,b,c) but no two can? What about the general situation with N generals and any M can launch (but no set of M-1) can?

Hint: what does it take three of to determine a quadratic? (We're assuming the generals either are good mathematicians, or have a computer program to do the work for them.)
 
Hint: what does it take three of to determine a quadratic? (We're assuming the generals either are good mathematicians, or have a computer program to do the work for them.)
Three points (the two x-intercepts and the vertex). But i do not think this is what you are looking for because they are not the same--as in what does it take three of. I thought of three symmetric points, but two will work.

Another hint?

Edit: The three coefficients. But how does this help?
 
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Three points (the two x-intercepts and the vertex). But i do not think this is what you are looking for because they are not the same--as in what does it take three of. I thought of three symmetric points, but two will work.

Another hint?

Edit: The three coefficients. But how does this help?

Why not any three points?
 
Why not any three points?
Yes, sure any three points. The problem I see is that if you tell the generals the values p(r), p(s) and p(t) I think that you have the same problem as if you just told them a, b and c (you can't tell the generals either one, two or all three of p(r), p(s) and p(t).
 
Yes, sure any three points. The problem I see is that if you tell the generals the values p(r), p(s) and p(t) I think that you have the same problem as if you just told them a, b and c (you can't tell the generals either one, two or all three of p(r), p(s) and p(t).

You're missing the "point".

You need to give each of the 10 generals something different, such that any three of them can come together and use them to find a, b, and c. What you give each of them does not have to be a single number.
 
You're missing the "point".

You need to give each of the 10 generals something different, such that any three of them can come together and use them to find a, b, and c. What you give each of them does not have to be a single number.
Got it. Is wasn't all that hard after all. Thank you.
 
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