AUSTRIAN-POLISH 1998 Problem 8

In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called admissible, where $n > 2$ is a given natural number. The value of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon $P$ is congruent to $P$ .)