Solutions to the 11³, 13³ and 17³ Queens Puzzle

#!/usr/bin/env python

from itertools import permutations, product

class WrongSize(Exception):
  pass

# Finds out how many dimensions the solution trace has.
def Dimensions(trace, side):
  power, exponent = 1, 1
  while power < len(trace):
    power *= side
    exponent += 1
  if power != len(trace):
    raise WrongSize()
  return exponent

# Generates functions that transform a cell's coordinates to its image.
# For a number of dimensions D, there are {factorial(D) * 2**D} symmetries.
def Symmetries(dimensions, side):
  return (lambda cell: (cell[p[i]] if s[i] else side - cell[p[i]] - 1
                        for i in range(len(cell)))
          for p, s in product(permutations(range(dimensions)),
                              product(*((True, False),) * dimensions)))

# Transforms an expanded solution (full coordinates of each cell) to a solution
# trace containing the significant (lowest order) coordinates only.
def SolutionToTrace(solution):
  return (cell[-1] for cell in sorted(solution))

# Transforms a solution trace containing the significant (lowest order)
# coordinates only to an expanded solution (full coordinates of each cell).
def TraceToSolution(trace, side):
  dimensions = Dimensions(trace, side)
  return (p + (s,)
          for p, s in zip(product(*(range(side),) * (dimensions - 1)), trace))

# Generates all symmetries for the given trace, based on its dimensionality.
# For example, there will be 8 symmetries in 2D, 48 in 3D, 384 in 4D etc.
def ExpandSymmetries(traces, side):
  for t, trace in enumerate(traces):
    for s, sym in enumerate(Symmetries(Dimensions(trace, side), side)):
      yield SolutionToTrace(tuple(sym(tuple(c)))
                            for c in TraceToSolution(trace, side)), t, s

# Based on a set of canonical solutions (solution seeds), generates all existing
# solutions using symmetries and identifies duplicates (while describing which
# canonical solutions and symmetries each duplicate came from).
def SummarizeSolutions(traces, side):
  traces_to_recipes = {}
  for trace, t, s in ExpandSymmetries(traces, side):
    traces_to_recipes.setdefault(tuple(trace), []).append((t, s))
  n_unique, n_total = 0, 0
  for n, trace in enumerate(sorted(traces_to_recipes)):
    recipes = traces_to_recipes[trace]
    n_total += len(recipes)
    n_unique += 1
    recipe = ' or '.join(f'canonical {t} and symetry {s}' for t, s in recipes)
    print(f'Solution {n} from {recipe}:\n{trace}\n')
  print(f'{len(traces)} canonical solutions\n'
        f'{n_unique} unique solutions\n'
        f'{n_total} canonical solutions × symmetries\n'
        f'{n_total - n_unique} overlaps')

CANONICALS_11 = [
  [0, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 4, 6, 8, 10, 1, 3, 5, 7, 9, 0, 2, 8, 10, 1, 3, 5, 7, 9, 0, 2, 4, 6, 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10, 5, 7, 9, 0, 2, 4, 6, 8, 10, 1, 3, 9, 0, 2, 4, 6, 8, 10, 1, 3, 5, 7, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 0, 6, 8, 10, 1, 3, 5, 7, 9, 0, 2, 4, 10, 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10, 1, 7, 9, 0, 2, 4, 6, 8, 10, 1, 3, 5],
  [0, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 5, 7, 9, 0, 2, 4, 6, 8, 10, 1, 3, 10, 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 4, 6, 8, 10, 1, 3, 5, 7, 9, 0, 2, 9, 0, 2, 4, 6, 8, 10, 1, 3, 5, 7, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10, 1, 8, 10, 1, 3, 5, 7, 9, 0, 2, 4, 6, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 0, 7, 9, 0, 2, 4, 6, 8, 10, 1, 3, 5, 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10, 6, 8, 10, 1, 3, 5, 7, 9, 0, 2, 4],
  [0, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 6, 8, 10, 1, 3, 5, 7, 9, 0, 2, 4, 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10, 7, 9, 0, 2, 4, 6, 8, 10, 1, 3, 5, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 0, 8, 10, 1, 3, 5, 7, 9, 0, 2, 4, 6, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10, 1, 9, 0, 2, 4, 6, 8, 10, 1, 3, 5, 7, 4, 6, 8, 10, 1, 3, 5, 7, 9, 0, 2, 10, 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 5, 7, 9, 0, 2, 4, 6, 8, 10, 1, 3],
  [0, 5, 10, 4, 9, 3, 8, 2, 7, 1, 6, 8, 2, 7, 1, 6, 0, 5, 10, 4, 9, 3, 5, 10, 4, 9, 3, 8, 2, 7, 1, 6, 0, 2, 7, 1, 6, 0, 5, 10, 4, 9, 3, 8, 10, 4, 9, 3, 8, 2, 7, 1, 6, 0, 5, 7, 1, 6, 0, 5, 10, 4, 9, 3, 8, 2, 4, 9, 3, 8, 2, 7, 1, 6, 0, 5, 10, 1, 6, 0, 5, 10, 4, 9, 3, 8, 2, 7, 9, 3, 8, 2, 7, 1, 6, 0, 5, 10, 4, 6, 0, 5, 10, 4, 9, 3, 8, 2, 7, 1, 3, 8, 2, 7, 1, 6, 0, 5, 10, 4, 9],
  [1, 4, 7, 10, 2, 5, 8, 0, 3, 6, 9, 6, 9, 1, 4, 7, 10, 2, 5, 8, 0, 3, 0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8, 5, 8, 0, 3, 6, 9, 1, 4, 7, 10, 2, 10, 2, 5, 8, 0, 3, 6, 9, 1, 4, 7, 4, 7, 10, 2, 5, 8, 0, 3, 6, 9, 1, 9, 1, 4, 7, 10, 2, 5, 8, 0, 3, 6, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8, 0, 8, 0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 2, 5, 8, 0, 3, 6, 9, 1, 4, 7, 10, 7, 10, 2, 5, 8, 0, 3, 6, 9, 1, 4],
  [1, 6, 0, 5, 10, 4, 9, 3, 8, 2, 7, 10, 4, 9, 3, 8, 2, 7, 1, 6, 0, 5, 8, 2, 7, 1, 6, 0, 5, 10, 4, 9, 3, 6, 0, 5, 10, 4, 9, 3, 8, 2, 7, 1, 4, 9, 3, 8, 2, 7, 1, 6, 0, 5, 10, 2, 7, 1, 6, 0, 5, 10, 4, 9, 3, 8, 0, 5, 10, 4, 9, 3, 8, 2, 7, 1, 6, 9, 3, 8, 2, 7, 1, 6, 0, 5, 10, 4, 7, 1, 6, 0, 5, 10, 4, 9, 3, 8, 2, 5, 10, 4, 9, 3, 8, 2, 7, 1, 6, 0, 3, 8, 2, 7, 1, 6, 0, 5, 10, 4, 9]]

CANONICALS_13 = [
  [0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7],
  [0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6],
  [0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5],
  [0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4],
  [0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 0, 10, 12, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3],
  [0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4],
  [0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2],
  [0, 4, 8, 12, 3, 7, 11, 2, 6, 10, 1, 5, 9, 6, 10, 1, 5, 9, 0, 4, 8, 12, 3, 7, 11, 2, 12, 3, 7, 11, 2, 6, 10, 1, 5, 9, 0, 4, 8, 5, 9, 0, 4, 8, 12, 3, 7, 11, 2, 6, 10, 1, 11, 2, 6, 10, 1, 5, 9, 0, 4, 8, 12, 3, 7, 4, 8, 12, 3, 7, 11, 2, 6, 10, 1, 5, 9, 0, 10, 1, 5, 9, 0, 4, 8, 12, 3, 7, 11, 2, 6, 3, 7, 11, 2, 6, 10, 1, 5, 9, 0, 4, 8, 12, 9, 0, 4, 8, 12, 3, 7, 11, 2, 6, 10, 1, 5, 2, 6, 10, 1, 5, 9, 0, 4, 8, 12, 3, 7, 11, 8, 12, 3, 7, 11, 2, 6, 10, 1, 5, 9, 0, 4, 1, 5, 9, 0, 4, 8, 12, 3, 7, 11, 2, 6, 10, 7, 11, 2, 6, 10, 1, 5, 9, 0, 4, 8, 12, 3],
  [1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6],
  [1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5],
  [1, 5, 9, 0, 4, 8, 12, 3, 7, 11, 2, 6, 10, 8, 12, 3, 7, 11, 2, 6, 10, 1, 5, 9, 0, 4, 2, 6, 10, 1, 5, 9, 0, 4, 8, 12, 3, 7, 11, 9, 0, 4, 8, 12, 3, 7, 11, 2, 6, 10, 1, 5, 3, 7, 11, 2, 6, 10, 1, 5, 9, 0, 4, 8, 12, 10, 1, 5, 9, 0, 4, 8, 12, 3, 7, 11, 2, 6, 4, 8, 12, 3, 7, 11, 2, 6, 10, 1, 5, 9, 0, 11, 2, 6, 10, 1, 5, 9, 0, 4, 8, 12, 3, 7, 5, 9, 0, 4, 8, 12, 3, 7, 11, 2, 6, 10, 1, 12, 3, 7, 11, 2, 6, 10, 1, 5, 9, 0, 4, 8, 6, 10, 1, 5, 9, 0, 4, 8, 12, 3, 7, 11, 2, 0, 4, 8, 12, 3, 7, 11, 2, 6, 10, 1, 5, 9, 7, 11, 2, 6, 10, 1, 5, 9, 0, 4, 8, 12, 3],
  [1, 6, 11, 3, 8, 0, 5, 10, 2, 7, 12, 4, 9, 12, 4, 9, 1, 6, 11, 3, 8, 0, 5, 10, 2, 7, 10, 2, 7, 12, 4, 9, 1, 6, 11, 3, 8, 0, 5, 8, 0, 5, 10, 2, 7, 12, 4, 9, 1, 6, 11, 3, 6, 11, 3, 8, 0, 5, 10, 2, 7, 12, 4, 9, 1, 4, 9, 1, 6, 11, 3, 8, 0, 5, 10, 2, 7, 12, 2, 7, 12, 4, 9, 1, 6, 11, 3, 8, 0, 5, 10, 0, 5, 10, 2, 7, 12, 4, 9, 1, 6, 11, 3, 8, 11, 3, 8, 0, 5, 10, 2, 7, 12, 4, 9, 1, 6, 9, 1, 6, 11, 3, 8, 0, 5, 10, 2, 7, 12, 4, 7, 12, 4, 9, 1, 6, 11, 3, 8, 0, 5, 10, 2, 5, 10, 2, 7, 12, 4, 9, 1, 6, 11, 3, 8, 0, 3, 8, 0, 5, 10, 2, 7, 12, 4, 9, 1, 6, 11],
  [1, 7, 0, 6, 12, 5, 11, 4, 10, 3, 9, 2, 8, 11, 4, 10, 3, 9, 2, 8, 1, 7, 0, 6, 12, 5, 8, 1, 7, 0, 6, 12, 5, 11, 4, 10, 3, 9, 2, 5, 11, 4, 10, 3, 9, 2, 8, 1, 7, 0, 6, 12, 2, 8, 1, 7, 0, 6, 12, 5, 11, 4, 10, 3, 9, 12, 5, 11, 4, 10, 3, 9, 2, 8, 1, 7, 0, 6, 9, 2, 8, 1, 7, 0, 6, 12, 5, 11, 4, 10, 3, 6, 12, 5, 11, 4, 10, 3, 9, 2, 8, 1, 7, 0, 3, 9, 2, 8, 1, 7, 0, 6, 12, 5, 11, 4, 10, 0, 6, 12, 5, 11, 4, 10, 3, 9, 2, 8, 1, 7, 10, 3, 9, 2, 8, 1, 7, 0, 6, 12, 5, 11, 4, 7, 0, 6, 12, 5, 11, 4, 10, 3, 9, 2, 8, 1, 4, 10, 3, 9, 2, 8, 1, 7, 0, 6, 12, 5, 11],
  [1, 8, 2, 9, 3, 10, 4, 11, 5, 12, 6, 0, 7, 11, 5, 12, 6, 0, 7, 1, 8, 2, 9, 3, 10, 4, 8, 2, 9, 3, 10, 4, 11, 5, 12, 6, 0, 7, 1, 5, 12, 6, 0, 7, 1, 8, 2, 9, 3, 10, 4, 11, 2, 9, 3, 10, 4, 11, 5, 12, 6, 0, 7, 1, 8, 12, 6, 0, 7, 1, 8, 2, 9, 3, 10, 4, 11, 5, 9, 3, 10, 4, 11, 5, 12, 6, 0, 7, 1, 8, 2, 6, 0, 7, 1, 8, 2, 9, 3, 10, 4, 11, 5, 12, 3, 10, 4, 11, 5, 12, 6, 0, 7, 1, 8, 2, 9, 0, 7, 1, 8, 2, 9, 3, 10, 4, 11, 5, 12, 6, 10, 4, 11, 5, 12, 6, 0, 7, 1, 8, 2, 9, 3, 7, 1, 8, 2, 9, 3, 10, 4, 11, 5, 12, 6, 0, 4, 11, 5, 12, 6, 0, 7, 1, 8, 2, 9, 3, 10]]

CANONICALS_17 = [
[0, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 16, 1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14, 16, 1, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 15, 0, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 0, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14, 16, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11],
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[1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15],
[1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14],
[1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 16, 7, 15, 6, 14, 5, 13],
[1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13],
[1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12],
[1, 11, 4, 14, 7, 0, 10, 3, 13, 6, 16, 9, 2, 12, 5, 15, 8, 15, 8, 1, 11, 4, 14, 7, 0, 10, 3, 13, 6, 16, 9, 2, 12, 5, 12, 5, 15, 8, 1, 11, 4, 14, 7, 0, 10, 3, 13, 6, 16, 9, 2, 9, 2, 12, 5, 15, 8, 1, 11, 4, 14, 7, 0, 10, 3, 13, 6, 16, 6, 16, 9, 2, 12, 5, 15, 8, 1, 11, 4, 14, 7, 0, 10, 3, 13, 3, 13, 6, 16, 9, 2, 12, 5, 15, 8, 1, 11, 4, 14, 7, 0, 10, 0, 10, 3, 13, 6, 16, 9, 2, 12, 5, 15, 8, 1, 11, 4, 14, 7, 14, 7, 0, 10, 3, 13, 6, 16, 9, 2, 12, 5, 15, 8, 1, 11, 4, 11, 4, 14, 7, 0, 10, 3, 13, 6, 16, 9, 2, 12, 5, 15, 8, 1, 8, 1, 11, 4, 14, 7, 0, 10, 3, 13, 6, 16, 9, 2, 12, 5, 15, 5, 15, 8, 1, 11, 4, 14, 7, 0, 10, 3, 13, 6, 16, 9, 2, 12, 2, 12, 5, 15, 8, 1, 11, 4, 14, 7, 0, 10, 3, 13, 6, 16, 9, 16, 9, 2, 12, 5, 15, 8, 1, 11, 4, 14, 7, 0, 10, 3, 13, 6, 13, 6, 16, 9, 2, 12, 5, 15, 8, 1, 11, 4, 14, 7, 0, 10, 3, 10, 3, 13, 6, 16, 9, 2, 12, 5, 15, 8, 1, 11, 4, 14, 7, 0, 7, 0, 10, 3, 13, 6, 16, 9, 2, 12, 5, 15, 8, 1, 11, 4, 14, 4, 14, 7, 0, 10, 3, 13, 6, 16, 9, 2, 12, 5, 15, 8, 1, 11],
[2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 14, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 0, 9, 1, 6, 15, 7, 16, 8, 0, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14]]

print(f'\n{'=' * 30}11{'=' * 30}\n')
SummarizeSolutions(CANONICALS_11, 11)
print(f'\n{'=' * 30}13{'=' * 30}\n')
SummarizeSolutions(CANONICALS_13, 13)
print(f'\n{'=' * 30}17{'=' * 30}\n')
SummarizeSolutions(CANONICALS_17, 17)