In this prelab you will formulate some of the ideas necessary to complete Lab 06. Please **type** your solutions, and hand in a paper copy before the beginning of class on Wednesday. Remember, no late prelabs allowed!

## Reading

Read this Scientific American article about how diversity makes us smarter. If you're so inclined, you should also check out the following two embarrassing examples of what can happen when your development and testing teams aren't diverse:
Exhibit A and
Exhibit B.

**1.** Did you do the reading? (Yes or no will suffice.)

## Recursive Output

For the following two problems, you'll work out by hand the result of method invocations of the following two recursive methods.

### Function strange

def strange(x) :
if x <= 0 :
return 1
else :
return 5 * strange(x-1) - 2

### Function weird

def weird(x) :
if x > 0 :
print(x,"",end='')
if x%2 == 0 :
weird(x-3)
weird(x-2)
else :
weird(x-1)

**2.** What is the output of `print(strange(3))`

**3.** What is the output of `weird(8)`

## Part 1 - Recursion with Numbers

As we have seen (or will have seen, depending on when you're looking at this), the factorial function can be computed not only using a loop, but via recursion as well. Recall the the typical definition of factorial looks something like this: 0! = 1, 1! = 1, 2! = 2*1 = 2 and 3! = 3*2*1 = 6, and in general, n! = n*(n-1)*...*3*2*1.

But we could also define the factorial function recursively as follows. We'll use fact(n) to denote our recursive representation of n!. We'll define fact(n) = 1 when n is 0 and fact(n) = n*fact(n-1) otherwise. If you think about it, fact(n) gives the exact same values as n!.

**4.** Give an analogous definition for n^k (n raised to the power of k). That is, you should define `power(n,k) = ???` when `k = ???` and `power(n,k) = ???` otherwise. The last component you need to fill in should involve a recursive call to power. No loops or `**` operator!

**5.** Give an analogous recursive definition for the sum of the first `n` perfect squares. For example, your definition should satisfy `sps(1) = 1`, `sps(2) = 1 + 4 = 5`, and `sps(3) = 1 + 4 + 9 = 14`. No loops!

## Part 2 - Recursion with Strings

Here is pseudocode for recursively printing a string `s` in reverse

If the length of `s` is 0, do nothing
Otherwise, print the last letter of `s` and reverse `s` with the last letter removed.

**6.** Give pseudocode for a single function that recursively determines whether a string `s` is a palindrome (the same forwards as backwards, like `"racecar"`). You can find the length of `s`, look up characters in `s`, and get slices of `s`, but you should not use any loops or the `reverse()` function that sequences have.

## Part 3 - Fractal Images

Consider the following figure.

Let A_{x} and A_{y} denote the x and y coordinates of the corner labelled A in the large triangle above. Let B_{x}, B_{y}, C_{x} and C_{y} be definied similarly. Assume P, Q and R each lie at the midpoint of their corresponding edge in the triangle ABC.

**7.** Specify the x and y coordinates of P, Q and R * in terms of A, B and C's x and y coordinates*. Do *not* assume that any coordinates are 0 or that there is any relationship between the points (the triangle ABC pictured happens to be fairly symmetric, but you shouldn't assume this will necessarily be the case).

## Honor Code

If you followed the Honor Code in this assignment, write the following sentence attesting to the fact at the top of your homework.

I
affirm that I have adhered to the Honor Code in this assignment.