This question arises from a discussion with my friends on a commonly encountered IQ test questions: "What's the next number in this series 2,6,12,20,...". Here a "number" usually means an integer. I was wondering whether there is a systematical way to solve such problems.Let us call a point on a plane integer point if all the components of it are integers. I want to know the following:


Give a finite set of integer points, Can we always find a corresponding polynomial that passes all these points and maps integers to integers? 



Integral interpolation by polynomial 
Let the points be $(x\_j, y\_j), j=1\ldots n$. If the $x\_j$ are consecutive, the Lagrange interpolating polynomial will take integers to integers: easy proof by induction, using the difference operator $\Delta(p)(x) = p(x+1) - p(x)$. If not, choose arbitrary integers for the $y$ values to fill in the gaps. 


