Problem of the Week 1032

A Polynomial Mystery

Alice: I'm thinking of a polynomial with nonnegative integer coefficients. Can you tell me what it is?

Bob: Well, I need more info than that. Can I ask you for some values of the polynomial?

Alice: That seems reasonable. How many would you like?

Bob: Well, N + 1 would be nice, where N is the degree of your polynomial, for then I could just solve N + 1 equations in N + 1 unknowns.

Alice: That would take too long, both for me and for you. Moreover, you don't know the degree and I don't want to reveal it. But I am willing to tell you two values of the polynomial for integer arguments that you choose.

Can Bob determine Alice's polynomial P(x) asking Alice only twice for integer values of P(x)?

Source: CMJ p. 100 Mar 2005, where it is attributed to I. B. Keene at the University of Michigan.

© Copyright 2005 Stan Wagon. Reproduced with permission.