Remember the rod cutting problem, the following is an instance of it:

$\backslash$table$[[$length $i,1,2,3,4,5,6,7,8,9,10],[$price $p_i,1,4,8,9,10,17,18,22,22,30]]$

a$.(15$ pts$)$ Define the density of a rod of length $i$ to be $\frac{p}{i},$ that is$,$ value per inch. A greedy strategy for cutting a rod of length $n$ cuts first the piece with the highest density.

b$.(15$ pts$)$ Show by a counterexample $($at least for one $n)$ that this greedy strategy does not always determine an optimal way to cut rods.