[Prakash Panangaden told me this problem.]
There is a polynomial and you have access to a function that evaluates that polynomial at a given number. You don’t know the degree of the polynomial, nor do you know any of the coefficients of its terms. However, you are told that all coefficients are non-negative integers. How many times do you need to call the evaluation function in order to identify the polynomial (that is, to figure out the values of its coefficients)?