Show, by means of a counterexample, that the following "greedy" strategy does not always determine an optimal way to cut rods. Define the density of a rod of length i to be p_i/i, that is, its value per inch. The greedy strategy for a rod of length n cuts off a first piece of length i, where 1 ≤  i ≤ n, having maximum density. It then continues by applying the greedy strategy to the remaining piece of length n – i.