In this problem you have to devise a strategy to beat the computer in a "guess the polynomial" game.
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I want you to play a little game with the computer. The computer is going to think of a polynomial with non-negative, integer coefficients. Let's say \(p(n)\) is the secret polynomial the computer is thinking of.
I want you to find out what \(p(n)\) is, and the only thing you can do is to ask for hints in the form of values of \(p(n)\) for \(n \geq 0\) integer. For example, you can ask what \(p(0)\) or \(p(49)\) is, but you can't ask for the value of \(p(-1)\) or \(p(0.5)\). You have to come up with the best strategy possible, that allows you to determine \(p(n)\) with the least number of hints possible.
You can test your strategy with the computer below. The computer will only think of polynomials with degree at most \(3\) and the coefficients will be at most \(3\) as well, but that is just to make testing your strategy easier. The strategy should work for higher degrees and larger coefficients.
With the restrictions for the computer test, we have
\[ p(n) = c_0 + c_1n + c_2n^2 + c_3n^3, 0 \leq c_i \leq 3\]
If you need any clarification whatsoever, feel free to ask in the comment section below.
You can read the solution here to compare with your own solution. You can also use that link to post your own solution in the comments! Please do not post spoilers in the comments here.
This problem was brought to my attention by MathGurl.
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