Mr S and Mr P: From http://programmingpraxis.com/2009/10/23/mr-s-and-mr-p/ and told to me by Mike Stringer. My answer in pure python.

gistfile1.txt

"""From http://programmingpraxis.com/2009/10/23/mr-s-and-mr-p/ and
told to me by Mike Stringer. John McCarthy, who discovered Lisp,
attributes this puzzle to Hans Freudenthal:

We pick two numbers a and b, so that 100>a,b>1. We tell
Mr. P. the product ab and Mr. S. the sum a+b. Then Mr. S. and
Mr. P. engage in the following dialog:

Mr. P.: I don't know the numbers.
Mr. S.: I knew you didn't know. I don't know either.
Mr. P.: Now I know the numbers.
Mr. S.: Now I know them too.

Find the numbers a and b.

Here is my answer -- in pure python. 

Gabriel Gaster, 2015
"""

def products(n, N=100):
    return set(tuple(sorted((i,n/i))) for i in xrange(2,n/2+1) 
               if (n%i == 0) and 1<i<N and 1<n/i<N)

def sums(n, N=100):
    return set(tuple(sorted((i,n-i))) for i in xrange(n) 
               if 1<i<N and 1<n-i<N)

def know(possibilities):
    return len(possibilities)==1

def a_and_b(N=100):
    first = ((a,b) for a in xrange(2,N) for b in xrange(a,N) 
             if statement1(a*b, N=N))
    second = set((a,b) for a,b in first if statement2(a+b, N))
    third = set((a,b) for a,b in second 
                if statement3(a*b, second, N=N))
    fourth = [(a,b) for a,b in third 
              if statement4(a+b, third, N=N)]
    return fourth

def statement1(p, N=100):
    return not know(products(p, N=N))

def statement2(s, N=100):
    return not know(sums(s, N)) and all(statement1(a*b, N=N) 
                                        for a,b in 
                                        sums(s, N=N))

def statement3(p, second, N=100):
    return know(products(p, N=N) & second)

def statement4(s, third, N=100):
    return know(sums(s, N=N) & third)

if __name__=="__main__":
    N = 100
    the_numbers = a_and_b(N=N)
    msg = "do" if know(the_numbers) else "do not"
    print "we %s know the numbers <%i" % (msg, N)
    print the_numbers

    if know(the_numbers):
        a,b = the_numbers.pop()
        print "Nums:",a,b
        print "Mr S: %i" % (a+b)
        print "Mr P: %i" % (a*b)