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"A human pyramid is a way of stacking people vertically in a triangle. With the exception of the people in the bottom row, each person splits their weight evenly on the two people below them in the pyramid. Their total weight, however, includes both their own body weight and the weight they are supporting above them. For example, in the pyramid above, person A splits her weight across persons B and C$,$ and person H splits his weight$$plus the accumulated weight of the people he$$s supporting$$onto people L and M$.$ In this question, you'll explore just how much weight that is$.$

For simplicity we will assume that everyone in the pyramid weighs exactly $200$ pounds. Person A at the top of the pyramid has no weight on her back. People B and C are each carrying half of person A$\text{'}$s weight. That means that each of them is shouldering $100$ pounds.

Now, let's look at the people in the third row. Let$$s begin by focusing on person E$.$ How much weight is she supporting? Well, she$$s directly supporting half the weight of person B $(100$ pounds$)$ and half the weight of person E $(100$ pounds$),$ so she$$s supporting at least $200$ pounds. On top of this, she$$s feeling some of the weight that people B and C are carrying. Half of the weight that person B is shouldering $(50$ pounds$)$ gets transmitted down onto person E and half the weight that person C is shouldering $(50$ pounds$)$ similarly gets sent down to person E$,$ so person E ends up feeling an extra $100$ pounds. That means she$$s supporting a net total of $300$ pounds.

Not everyone in that third row is feeling the same amount, though. Look at person D for example. The only weight on person D comes from person B$.$ Person D therefore ends up supporting

half of person B$$s body weight $(100$ pounds$),$ plus

half of the weight person B is holding up $(50$ pounds$),$ for a total of $150$ pounds, only half of what E is feeling!

Going deeper in the pyramid, how much weight is person H feeling? Well, person H is supporting

half of person D$$s body weight $(100$ pounds$),$

half of person E$$s body weight $(100$ pounds$),$ plus

half of the weight person D is holding up $(75$ pounds$),$ plus half of the weight person E is holding up $(150)$ pounds.

The net effect is that person H is carrying $425$ pounds$$ouch$!$ A similar calculation shows that person I is also carrying $425$ pounds$$can you see why?

Person G is supporting

half of person D$$s body weight $(100$ pounds$),$ plus

half of the weight person D is holding up $(75$ pounds$)$ or a net total of $175$ pounds.

Finally, let$$s look at person M in the middle of the bottom row. How is she doing? Well, she$$s supporting

half of person H$$s body weight $(100$ pounds$),$

half of person I$$s body weight $(100$ pounds$),$

half of the weight person H is holding up $(212\mathrm{.}5$ pounds$),$ and

half of the weight person I is holding up $($also $212\mathrm{.}5$ pounds$),$ for a net total of $625$ pounds!

In $1410$ and $2420,$ you have become familiar with various forms of recursive problems. The pyramid problem is an example of problems that are difficult to solve with loops and iteration, but elegantly solved with recursion. This problem is broken into $3$ parts. If you did this problem in $1410,$ then you have already done most of this problem. If you haven$$t$,$ then you$$ll need to do the whole thing. For our purposes in $2420,$ the focus is on caching, as you will see below in Part $3.$

Keep in mind that the first and last people in each row calculate their weight differently than people in interior positions. The weight on any person can be computed recursively, with the base case being the person at the top the person at the top of the pyramid $($in row $0),$ who is shouldering $0$ pounds.

Part $1---$ weight$\_$on$\_$cacheless$()$

Write a recursive $($without loops$)$ function, weight$\_$on$\_$cacheless$($r$,$c$),$ which returns the weight on the back of the person in row r and and column c$.$ Rows and columns are $0-$based, so the top position is $(0,0),$ for example, and person H is in position $(3,1).$ The following also hold: