Pigeonhole and Extremal Arguments · Challenge problem

The numbers $1, \ldots, 64$ fill an $8 \times 8$ board. Walking from the cell with $1$ to the cell with $64$ through edge-neighbours takes at most $14$ steps, so the $63$ units of increase are spread over at most $14$ steps. What lower bound does this argument give for the largest difference between two edge-neighbours?