text: # Copyright 2010 Hakan Kjellerstrand hakank@gmail.com
  #
  # Licensed under the Apache License, Version 2.0 (the "License");
  # you may not use this file except in compliance with the License.
  # You may obtain a copy of the License at
  #
  #     http://www.apache.org/licenses/LICENSE-2.0
  #
  # Unless required by applicable law or agreed to in writing, software
  # distributed under the License is distributed on an "AS IS" BASIS,
  # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  # See the License for the specific language governing permissions and
  # limitations under the License.
  """
  
    Crypto problem in Google CP Solver.
  
    Martin Gardner (February 1967):
    '''
    The integers 1,3,8, and 120 form a set with a remarkable property: the
    product of any two integers is one less than a perfect square. Find
    a fifth number that can be added to the set without destroying
    this property.
    '''
  
    Solution: The number is 0.
  
    There are however other sets of five numbers with this property.
    Here are the one in the range of 0.10000:
    [0, 1, 3, 8, 120]
    [0, 1, 3, 120, 1680]
    [0, 1, 8, 15, 528]
    [0, 1, 8, 120, 4095]
    [0, 1, 15, 24, 1520]
    [0, 1, 24, 35, 3480]
    [0, 1, 35, 48, 6888]
    [0, 2, 4, 12, 420]
    [0, 2, 12, 24, 2380]
    [0, 2, 24, 40, 7812]
    [0, 3, 5, 16, 1008]
    [0, 3, 8, 21, 2080]
    [0, 3, 16, 33, 6440]
    [0, 4, 6, 20, 1980]
    [0, 4, 12, 30, 5852]
    [0, 5, 7, 24, 3432]
    [0, 6, 8, 28, 5460]
    [0, 7, 9, 32, 8160]
  
  
    Compare with the following models:
    * MiniZinc: http://www.hakank.org/minizinc/crypta.mzn
    * Comet   : http://www.hakank.org/comet/crypta.co
    * ECLiPSe : http://www.hakank.org/eclipse/crypta.ecl
    * SICStus : http://hakank.org/sicstus/crypta.pl
  
  
    This model was created by Hakan Kjellerstrand (hakank@gmail.com)
    Also see my other Google CP Solver models:
    http://www.hakank.org/google_or_tools/
  """
  from __future__ import print_function
  from ortools.constraint_solver import pywrapcp
  
  
  def decreasing(solver, x):
    for i in range(len(x) - 1):
      solver.Add(x[i] <= x[i + 1])
  
  
  def main():
  
    # Create the solver.
    solver = pywrapcp.Solver("Curious set of integers")
  
    #
    # data
    #
    n = 5
    max_val = 10000
  
    #
    # variables
    #
    x = [solver.IntVar(0, max_val, "x[%i]" % i) for i in range(n)]
  
    #
    # constraints
    #
    solver.Add(solver.AllDifferent(x))
    decreasing(solver, x)
  
    for i in range(n):
      for j in range(n):
        if i != j:
          p = solver.IntVar(0, max_val, "p[%i,%i]" % (i, j))
          solver.Add(p * p - 1 == (x[i] * x[j]))
  
    # This is the original problem:
    # Which is the fifth number?
    v = [1, 3, 8, 120]
    b = [solver.IsMemberVar(x[i], v) for i in range(n)]
    solver.Add(solver.Sum(b) == 4)
  
    #
    # search and result
    #
    db = solver.Phase(x, solver.CHOOSE_MIN_SIZE_LOWEST_MIN,
                      solver.ASSIGN_MIN_VALUE)
  
    solver.NewSearch(db)
  
    num_solutions = 0
    while solver.NextSolution():
      num_solutions += 1
      print("x:", [int(x[i].Value()) for i in range(n)])
  
    solver.EndSearch()
  
    print()
    print("num_solutions:", num_solutions)
    print("failures:", solver.Failures())
    print("branches:", solver.Branches())
    print("WallTime:", solver.WallTime())
  
  
  if __name__ == "__main__":
    main()
  