#!/usr/local/bin/python3
def rk(M,n):
 M=[r[:] for r in M];r=0
 for c in range(n):
  p=next((i for i in range(r,n) if M[i][c]),-1)
  if p<0:continue
  M[r],M[p]=M[p],M[r]
  for i in range(r+1,n):
   if M[i][c]:
    for j in range(c,n):M[i][j]^=M[r][j]
  r+=1
 return r
def dt(M,n):return rk(M,n)==n
def iv(M,n):
 A=[M[i][:]+[int(i==j) for j in range(n)] for i in range(n)];r=0
 for c in range(n):
  p=next((i for i in range(r,n) if A[i][c]),-1)
  if p<0:return None
  A[r],A[p]=A[p],A[r]
  for i in range(n):
   if i!=r and A[i][c]:
    for j in range(c,2*n):A[i][j]^=A[r][j]
  r+=1
 return [q[n:] for q in A]
def ker(M,n):
 A=[r[:] for r in M];piv=[];r=0
 for c in range(n):
  p=next((i for i in range(r,n) if A[i][c]),-1)
  if p<0:continue
  A[r],A[p]=A[p],A[r]
  for i in range(n):
   if i!=r and A[i][c]:
    for j in range(c,n):A[i][j]^=A[r][j]
  piv.append(c);r+=1
 free=[c for c in range(n) if c not in piv]
 bs=[]
 for f in free:
  v=[0]*n;v[f]=1
  for i,c in enumerate(piv):v[c]=A[i][f]
  bs.append(v)
 return bs
n=6
E=[(i,j) for i in range(n) for j in range(i+1,n)];cnt=0;pat={};Ws={}
for m in range(1<<len(E)):
 A=[[0]*n for _ in range(n)]
 for k,(i,j) in enumerate(E):
  if m>>k&1:A[i][j]=A[j][i]=1
 if dt(A,n):
  B=[r[:] for r in A]
  for i in range(n):B[i][i]^=1
  if dt(B,n):
   C=iv(B,n)
   if all(C[i][i]==(i<2) for i in range(n)):
    cnt+=1
    D=[r[:] for r in C]
    D[0][0]^=1;D[1][1]^=1
    bs=ker(D,n)
    M2=[[bs[0][0],bs[1][0]],[bs[0][1],bs[1][1]]]
    # invert 2x2
    d=M2[0][0]*M2[1][1]^M2[0][1]*M2[1][0]
    r1=[(M2[1][1]*bs[0][k]^M2[0][1]*bs[1][k])*d^1 for k in range(n)]
    r2=[(M2[0][0]*bs[1][k]^M2[1][0]*bs[0][k])*d^1 for k in range(n)]
    L=[(r1[j],r2[j]) for j in range(2,n)]
    for p in L:pat[p]=pat.get(p,0)+1
    W=[[D[i][j] for j in range(2,n)] for i in range(2,n)]
    Ws[str(W)]=Ws.get(str(W),0)+1
print('cnt',cnt)
print('pat',pat)
print('Wnum',len(Ws))
print('Wmult',sorted(Ws.values())[:5],sorted(Ws.values())[-5:])