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Alice is allowed to choose an arbitrary polynomial p(x) of any degree with nonnegative integer coefficients. Bob can infer the coefficients of p(x) by only two evaluations as follows. He chooses a real number a and Alice communicates p(a) to him. He then chooses a real number b and Alice communicates p(b) to him. What values of a and b helped Bob succeed and how?

This is from https://gurmeet.net/puzzles/perplexing-polynomial/index.html

My approach: I was thinking this is impossible unless, I say P(1)=0 and P(-1)=0 which would indicate 0 polynomial but the solution says something else. Please help me figure out the solution.

Charlie
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  • Hint: If you knew that all the coefficients were integers in the range $0\ldots9$, what would $p(10)$ tell you? Anyway, Bob should select the inputs intellligently. The second input depending on what Alice tells him about the first. – Jyrki Lahtonen Aug 13 '23 at 18:04
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    It seems like you could do this with one guess, any transcendental number like $\pi$. Every polynomial with integer coefficients produces a distinct value for a transcendental input -- so just one evaluation should be enough to identify the polynomial – Alex K Aug 13 '23 at 18:19
  • Thanks!, it does answer my question – Charlie Aug 13 '23 at 20:17

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