RU-ALLRUSSIAN 1984 Problem 21

A 3 x 3 chess board is colored alternately black and white in the usual way with the center square white. Each white square is given the number 1 or the number - 1. A move consists of simultaneously changing each number to the product of the adjacent numbers. So the four corner squares are each changed to the number previously in the center square and the center square is changed to the product of the four numbers in the corners. Show that after finitely many moves all numbers are 1.