Part $1-$ recursive run of a sample rod cutting problem with memoization

In class we discussed a sample run of the unrefined, recursive rod cutting algorithm, but also hinting at how to do the same problem using memoization. In this exercise, I want you to do memoization for this instance of the problem right away:

Lengths$1234$Prices$1324$

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Run the algorithm on paper. Make sure to show every detail of that run. Use the notes of the demonstration I gave in class $($go to class notes, Classes$/$Lecture $16-$ Rod Cutting$)$ to see what I expect you to put on paper.

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Based on that run, answer the following questions:

What is the value of the optimal solution? Just name the number.

What is the solution, i$.$e$.$ how does the rod of $4$ inches need to be cut? Show that you did recursive backtracking to get the result.

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Part $2-$ bottom up run of the same, above problem

Now come up with the solution of the same problem above, but now use a bottom up approach. Convince yourself that you need to get the same results. For that reason, I only ask you to run the algorithm on paper, there are no questions as the answers would be the same.