#!/usr/bin/env python3
import sys
sys.setrecursionlimit(10000)
tab={(1,1):(0,-1),(2,2):(0,-1),(3,3):(0,-1),(1,2):(3,1),(2,1):(3,-1),(2,3):(1,1),(3,2):(1,-1),(3,1):(2,1),(1,3):(2,-1)}
def mul(a,b):
    if a[0]==0: return (b[0],a[1]*b[1])
    if b[0]==0: return (a[0],a[1]*b[1])
    k,s=tab[(a[0],b[0])]
    return (k,a[1]*b[1]*s)
letters=[((1,1),2,1),((1,-1),-2,-1),((2,1),-1,2),((2,-1),1,-2)]
inv=[1,0,3,2]
n=12
from fractions import Fraction
N=S=0
nonN=nonS=0
primN=primS=0
achN=achS=0
Eprim=Fraction(0)
Eall=Fraction(0)
def rec(w,q,x,y):
    global N,S,nonN,nonS,primN,primS,achN,achS,Eprim,Eall
    if len(w)==n:
        if w[0]==inv[w[-1]]: return
        if q!=(0,-1) or y%5: return
        e=(abs(x)+abs(y)+abs(-x-y))//2
        N+=1;S+=e
        periods=[k for k in range(1,n+1) if all(w[i]==w[(i+k)%n] for i in range(n))]
        d=periods[0]
        if d<n:
            nonN+=1;nonS+=e
        else:
            primN+=1;primS+=e
        rev=tuple(inv[w[n-1-i]] for i in range(n))
        ach=any(rev==tuple(w[(i+k)%n] for i in range(n)) for k in range(n))
        if ach:
            achN+=1;achS+=e
        if d==n:
            Eprim+=Fraction(e,12 if ach else 24)
        Eall+=Fraction(e,d if ach else 2*d)
        return
    for g in range(4):
        if g==inv[w[-1]]: continue
        qq,a,b=letters[g]
        rec(w+(g,),mul(q,qq),x+a,y+b)
for e,(q,a,b) in enumerate(letters):
    rec((e,),q,a,b)
print("total",N,S,"non",nonN,nonS,"prim",primN,primS,"ach",achN,achS)
print("Eprim",Eprim,"Eall",Eall)