#!/usr/bin/env python3
from fractions import Fraction
from itertools import product
def comps(n,r):
    if r==1:
        yield (n,)
    else:
        for a in range(1,n-r+2):
            for z in comps(n-a,r-1):
                yield (a,)+z
def calc(I,J,even=False):
    tot=Fraction(0)
    for r in range(1,min(I,J)+1):
        for A in comps(I,r):
            for B in comps(J,r):
                K=sum(B[l]*sum(A[:l+1]) for l in range(r))% 2
                for sa in product((-1,1),repeat=r):
                    x=sum(sa[l]*A[l] for l in range(r))
                    for sb in product((-1,1),repeat=r):
                        y=sum(sb[l]*B[l] for l in range(r))
                        if (x+2*y)% 5: continue
                        if even:
                            ok=True
                        else:
                            delta=(1 if I% 4==2 else 0)+(1 if J% 4==2 else 0)
                            ok=((x+y)//2+K+delta)% 2==1
                        if ok:
                            e=(abs(2*x-y)+abs(x+2*y)+abs(-3*x-y))//2
                            tot+=Fraction(e,2*r)
    return tot
print(calc(2,10),calc(10,2),calc(4,8),calc(8,4),calc(6,6))
print(calc(6,6,True))
print(sum((calc(I,12-I,True) for I in [2,4,6,8,10]),Fraction(0)))