|Spoiler Inside: Solution to Puzzle||SelectShow>|
The assumption is that the chessboard is oriented so that the rows and columns are oriented north-south and east-west in some way.
Instead, orient the chessboard so that the diagonals run north-south and east-west. Colour the squares black and white in a chessboard pattern. In each of the black squares, put a positive integer, say 4. In each of the white squares, put a different positive integer, say 5. The conditions of the problem are met, but there are two distinct values.