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Filling an infinite square grid

Source: Japan Mathematical Olympiad Finals 2018 Q4

February 13, 2018
combinatoricsgrid

Problem Statement

Let nn be an odd positive integer, and consider an infinite square grid. Prove that it is impossible to fill in one of 1,21,2 or 33 in every cell, which simultaneously satisfies the following conditions: (1) Any two cells which share a common side does not have the same number filled in them. (2) For any 1×31\times 3 or 3×13\times 1 subgrid, the numbers filled does not contain 1,2,31,2,3 in that order be it reading from top to bottom, bottom to top, or left to right, or right to left. (3) The sum of numbers of any n×nn\times n subgrid is the same.
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