Nonstandard Dice | Python Fiddle

"""
Consider the following puzzle:
	
    Two n-sided dice are labeled 1 to n. 
    When rolled, their sum is a random number between 2 and 2n. 
    Is there a different way to label these dice with positive integers 
    which results in the same random distribution?
    
This script will solve this puzzle. 
Simply change the value of n as desired, then hit run.
"""

n = 6

# OPTIONAL: Rename p to be any positive integer which divides n to get other solutions.
p = 0

def product(A,B):
    return [sum([A[i]*B[z-i]for i in range(max(z-len(B)+1,0),min(len(A),z+1))])\
             for z in range(len(A)+len(B)-1)]

def quotient(A,B):
    Q = []
    while len(A)>= len(B):
        q = A[0] / B[0]
        A =  [A[i] - q*B[i] for i in range(1, len(B) ) ] + A[len(B):]
        Q += [q]
    while (len(Q)>0 and Q[-1]==0): Q = Q[:-1] #remove trailing zeroes
    return Q

def nonstandard_dice(n,p):
    """Gives labels for two dice, whose sum behaves like two dice labeled 1 to n
        The second input p must be a factor of n.
    """
    q = n/p
    die_1 = product([1]*p,[1]*q)
    die_2 = quotient(product([1]*n,[1]*n),die_1)
    return (Faces(die_1),Faces(die_2))

def Faces(A): return sum([[i+1]*A[::-1][i] for i in range(len(A))],[])

if (not isinstance(p,int)) or (p < 1 or n%p): 
    p = 2
    while p<= n**0.5 and n%p: p += 1
    if n%p: p = 1
        
(d1,d2) = nonstandard_dice(n,p)
print "Die 1 Labels: ",d1
print "Die 2 Labels: ",d2

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