• Unlike hw7a, this portion of the hw is COLLABORATIVE.  You may work in groups of up to 3 other students who are in this course this semester (and nobody else except current CA's), so long as you do not simply copy their work but truly work together.  In fact, you are encouraged (though not required) to work in groups, provided you all contribute and nobody simply copies someone else's work.
   
• If you work collaboratively, you still must submit your own solutions.
   
• This portion is worth 25 pts (5 questions at 5 pts each).
   
• Submit a physical copy (handwritten, not typed).  Show your supporting calculations as appropriate.  Write neatly.  Staple multiple pages.  Include your name, andrew id, and section clearly at the top of the first page, and the names and andrew id's of your collaborators.
   
• Since this requires a physical submission, if you cannot attend office hours on Sunday night, you may drop off your submission at my office (GHC 5001) on Monday morning up until 9am (but no later!), and your submission will count as on time.

1.  Reasoning About Code  [25 pts; 5 pts each]

For each of the following functions, find values of the parameters so that the functions will return True.  You should assume that x, y, and z are ints, and that s is a string.  Hint:  reason your way through this.  How you find your answer is just as important, if not more so, than what the answer is!  While you may confirm your answer by running it in a Python interpreter, do not use the Python interpreter at all when solving the problem.  You will be given similar problems such as these on written quizzes without access to Python interpreters!  Also, consider only the possible inputs.  For example, if you are given that x+y==16, since x>y>0, we only need consider (15,1), (14,2), ..., (9,7).  That's only 7 (x,y) points in total, one of which must be correct!  Also, CIRCLE YOUR ANSWERS!

def f1(a):
    n = len(a)
    assert(n == 3)
    b = [([0]*n) for row in xrange(n)]
    for i in xrange(n):
        b[i][i] = i
        b[i][n-1-i] = i+1
    return (b == a)       


def f2(a):
    assert ((len(a) == 2) and (len(a[0]) == 3))
    for c in xrange(len(a[0])):
        s = 0
        for r in xrange(len(a)):
            if (r > 0): assert(a[r][c] != a[r-1][c])
            s += a[r][c]
        assert(s == 5)
    return True  


def f3(a):
    assert(len(a) * len(a[0]) == 6)
    x = 0
    for r in xrange(3):
        for c in xrange(2):
            if ((r+c) % 2 == 0):
                x = 10*x + (a[r][c] % 10)
            else:
                assert(a[r][c] == 0)
    return (x == 42)


def f4(a):
    (rr,cc,r,c) = (len(a), len(a[0]), 0, 1)
    assert((rr*cc == 15) and (rr>cc>1))
    for i in xrange(rr*cc):
        assert(a[r][c] == i%7)
        if (c == 0): (r,c) = ((r-1)%rr,cc-1)
        else: (r,c) = (r, c-1)
    return True


def f5(c):
    # Note that c is a non-rectangular ("ragged") 2d list!
    n = len(c)
    a = [[1]]
    for i in xrange(1,n):
        a.append([1]*(1+len(a[i-1])))
        for j in xrange(1,i):
            a[i][j] = a[i-1][j-1]+a[i-1][j]
    assert(a == c)
    b = [1]
    for i in xrange(1,n):
        s = 0
        for j in xrange(i):
            if (j < len(a[i-j])):
                s += a[i-j][j]
        b.append(s)
    return b[-1] == 8

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