Assume the contrary, so there is a numbering of the squares in which the difference between any two adjacent entries is at most

. Suppose we move a king from one square to another. (Kings in chess move one square horizontally, vertically, or diagonally.) The number on the square he moves to increases by at most

from the previous square, so in

moves, the entries increase by at most

. Chessplayers (of the

variety, at least) know that the king can move from any square on the board to any other square in at most

moves.

So what does this say about a king moving from square

to square

?