a_puzzle.cs

//
// Copyright 2012 Hakan Kjellerstrand
//
// 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.

using System;
using System.Linq;
using Google.OrTools.ConstraintSolver;

public class APuzzle
{
  /**
   *
   * From "God plays dice"
   * "A puzzle"
   * http://gottwurfelt.wordpress.com/2012/02/22/a-puzzle/
   * And the sequel "Answer to a puzzle"
   * http://gottwurfelt.wordpress.com/2012/02/24/an-answer-to-a-puzzle/
   *
   * This problem instance was taken from the latter blog post.
   * (Problem 1)
   *
   * """
   * 8809 = 6
   * 7111 = 0
   * 2172 = 0
   * 6666 = 4
   * 1111 = 0
   * 3213 = 0
   * 7662 = 2
   * 9312 = 1
   * 0000 = 4
   * 2222 = 0
   * 3333 = 0
   * 5555 = 0
   * 8193 = 3
   * 8096 = 5
   * 7777 = 0
   * 9999 = 4
   * 7756 = 1
   * 6855 = 3
   * 9881 = 5
   * 5531 = 0
   *
   * 2581 = ?
   * """
   *
   * Note:
   * This model yields 10 solutions, since x4 is not
   * restricted in the constraints.
   * All solutions has x assigned to the correct result.
   *
   *
   * (Problem 2)
   * The problem stated in "A puzzle"
   * http://gottwurfelt.wordpress.com/2012/02/22/a-puzzle/
   * is
   * """
   * 8809 = 6
   * 7662 = 2
   * 9312 = 1
   * 8193 = 3
   * 8096 = 5
   * 7756 = 1
   * 6855 = 3
   * 9881 = 5
   *
   * 2581 = ?
   * """
   * This problem instance yields two different solutions of x,
   * one is the same (correct) as for the above problem instance,
   * and one is not.
   * This is because here x0,x1,x4 and x9 are underdefined.
   *
   *
   */
  private static void Solve(int p = 1)
  {
    Solver solver = new Solver("APuzzle");

    Console.WriteLine("\nSolving p{0}", p);


    //
    // Data
    //
    int n = 10;

    //
    // Decision variables
    //
    IntVar x0 = solver.MakeIntVar(0, n-1, "x0");
    IntVar x1 = solver.MakeIntVar(0, n-1, "x1");
    IntVar x2 = solver.MakeIntVar(0, n-1, "x2");
    IntVar x3 = solver.MakeIntVar(0, n-1, "x3");
    IntVar x4 = solver.MakeIntVar(0, n-1, "x4");
    IntVar x5 = solver.MakeIntVar(0, n-1, "x5");
    IntVar x6 = solver.MakeIntVar(0, n-1, "x6");
    IntVar x7 = solver.MakeIntVar(0, n-1, "x7");
    IntVar x8 = solver.MakeIntVar(0, n-1, "x8");
    IntVar x9 = solver.MakeIntVar(0, n-1, "x9");

    IntVar[] all = {x0,x1,x2,x3,x4,x5,x6,x7,x8,x9};

    // The unknown, i.e. 2581 = x
    IntVar x = solver.MakeIntVar(0, n-1, "x");


    //
    // Constraints
    //

    // Both problem are here shown in two
    // approaches:
    //   - using equations
    //   - using a a matrix and Sum of each row

    if (p == 1) {

      // Problem 1
      solver.Add(x8+x8+x0+x9 == 6);
      solver.Add(x7+x1+x1+x1 == 0);
      solver.Add(x2+x1+x7+x2 == 0);
      solver.Add(x6+x6+x6+x6 == 4);
      solver.Add(x1+x1+x1+x1 == 0);
      solver.Add(x3+x2+x1+x3 == 0);
      solver.Add(x7+x6+x6+x2 == 2);
      solver.Add(x9+x3+x1+x2 == 1);
      solver.Add(x0+x0+x0+x0 == 4);
      solver.Add(x2+x2+x2+x2 == 0);
      solver.Add(x3+x3+x3+x3 == 0);
      solver.Add(x5+x5+x5+x5 == 0);
      solver.Add(x8+x1+x9+x3 == 3);
      solver.Add(x8+x0+x9+x6 == 5);
      solver.Add(x7+x7+x7+x7 == 0);
      solver.Add(x9+x9+x9+x9 == 4);
      solver.Add(x7+x7+x5+x6 == 1);
      solver.Add(x6+x8+x5+x5 == 3);
      solver.Add(x9+x8+x8+x1 == 5);
      solver.Add(x5+x5+x3+x1 == 0);

      // The unknown
      solver.Add(x2+x5+x8+x1 == x);

    } else if (p == 2) {

      // Another representation of Problem 1
      int[,] problem1 = {
        {8,8,0,9, 6},
        {7,1,1,1, 0},
        {2,1,7,2, 0},
        {6,6,6,6, 4},
        {1,1,1,1, 0},
        {3,2,1,3, 0},
        {7,6,6,2, 2},
        {9,3,1,2, 1},
        {0,0,0,0, 4},
        {2,2,2,2, 0},
        {3,3,3,3, 0},
        {5,5,5,5, 0},
        {8,1,9,3, 3},
        {8,0,9,6, 5},
        {7,7,7,7, 0},
        {9,9,9,9, 4},
        {7,7,5,6, 1},
        {6,8,5,5, 3},
        {9,8,8,1, 5},
        {5,5,3,1, 0}
      };

      for(int i = 0; i < problem1.GetLength(0); i++) {
        solver.Add( (from j in Enumerable.Range(0, 4)
              select all[problem1[i,j]]
              ).ToArray().Sum() == problem1[i,4] );
      }

      solver.Add(all[2]+all[5]+all[8]+all[1] == x);

    } else if (p == 3) {

      // Problem 2
      solver.Add(x8+x8+x0+x9 == 6);
      solver.Add(x7+x6+x6+x2 == 2);
      solver.Add(x9+x3+x1+x2 == 1);
      solver.Add(x8+x1+x9+x3 == 3);
      solver.Add(x8+x0+x9+x6 == 5);
      solver.Add(x7+x7+x5+x6 == 1);
      solver.Add(x6+x8+x5+x5 == 3);
      solver.Add(x9+x8+x8+x1 == 5);

      // The unknown
      solver.Add(x2+x5+x8+x1 == x);

    } else {

      // Another representation of Problem 2
      int[,] problem2 = {
        {8,8,0,9, 6},
        {7,6,6,2, 2},
        {9,3,1,2, 1},
        {8,1,9,3, 3},
        {8,0,9,6, 5},
        {7,7,5,6, 1},
        {6,8,5,5, 3},
        {9,8,8,1, 5}
      };

      for(int i = 0; i < problem2.GetLength(0); i++) {
        solver.Add( (from j in Enumerable.Range(0, 4)
              select all[problem2[i,j]]
              ).ToArray().Sum() == problem2[i,4] );
      }


      solver.Add(all[2]+all[5]+all[8]+all[1] == x);
    }



    //
    // Search
    //
    DecisionBuilder db = solver.MakePhase(all,
        Solver.INT_VAR_DEFAULT,
        Solver.INT_VALUE_DEFAULT);


    solver.NewSearch(db);
    while (solver.NextSolution()) {
      Console.Write("x: {0}  x0..x9: ", x.Value());
      for(int i = 0; i < n; i++) {
        Console.Write(all[i].Value() + " ");
      }
      Console.WriteLine();
    }

    Console.WriteLine("\nSolutions: {0}", solver.Solutions());
    Console.WriteLine("WallTime: {0}ms", solver.WallTime());
    Console.WriteLine("Failures: {0}", solver.Failures());
    Console.WriteLine("Branches: {0} ", solver.Branches());

    solver.EndSearch();

  }

  public static void Main(String[] args)
  {
    for(int p = 1; p <= 4; p++) {
      Solve(p);
    }
  }
}