The article says "the maximum number of moves that will ever be required for a cube of side n is proportional to n^2/logn". It also says "standard Rubik's cube can be solved in 20 moves or less" and then it says Demaine's "first task is to work out how to turn that into an exact number for given sizes of cube." but if the number of moves for n=3 is 20 and the number of moves is proportional to n^2/logn then the number of moves for cubes of any length =kn^2/logn where k=20/(3^2/LOG(3)) = 1.1 . So maximum number of moves = 1.1 n^2/logn (approximately) and Demaine's first task is completed unless he needs exact numbers for each n.
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The article says "the maximum number of moves that will ever be required for a cube of side n is proportional to n^2/logn". It also says "standard Rubik's cube can be solved in 20 moves or less" and then it says Demaine's "first task is to work out how to turn that into an exact number for given sizes of cube." but if the number of moves for n=3 is 20 and the number of moves is proportional to n^2/logn then the number of moves for cubes of any length =kn^2/logn where k=20/(3^2/LOG(3)) = 1.1 . So maximum number of moves = 1.1 n^2/logn (approximately) and Demaine's first task is completed unless he needs exact numbers for each n.