sicherman_dice.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.Collections;
using System.Collections.Generic;
using System.Linq;
using Google.OrTools.ConstraintSolver;

public class SichermanDice
{
  /**
   *
   * Sicherman Dice.
   *
   * From http://en.wikipedia.org/wiki/Sicherman_dice
   * ""
   * Sicherman dice are the only pair of 6-sided dice which are not normal dice,
   * bear only positive integers, and have the same probability distribution for
   * the sum as normal dice.
   *
   * The faces on the dice are numbered 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8.
   * ""
   *
   * I read about this problem in a book/column by Martin Gardner long
   * time ago, and got inspired to model it now by the WolframBlog post
   * "Sicherman Dice": http://blog.wolfram.com/2010/07/13/sicherman-dice/
   *
   * This model gets the two different ways, first the standard way and
   * then the Sicherman dice:
   *
   *  x1 = [1, 2, 3, 4, 5, 6]
   *  x2 = [1, 2, 3, 4, 5, 6]
   *  ----------
   *  x1 = [1, 2, 2, 3, 3, 4]
   *  x2 = [1, 3, 4, 5, 6, 8]
   *
   *
   * Extra: If we also allow 0 (zero) as a valid value then the
   * following two solutions are also valid:
   *
   * x1 = [0, 1, 1, 2, 2, 3]
   * x2 = [2, 4, 5, 6, 7, 9]
   * ----------
   * x1 = [0, 1, 2, 3, 4, 5]
   * x2 = [2, 3, 4, 5, 6, 7]
   *
   * These two extra cases are mentioned here:
   * http://mathworld.wolfram.com/SichermanDice.html
   *
   *
   * Also see http://www.hakank.org/or-tools/sicherman_dice.py
   *
   */
  private static void Solve()
  {
    Solver solver = new Solver("SichermanDice");

    //
    // Data
    //
    int n = 6;
    int m = 10;
    int lowest_value = 0;

    // standard distribution
    int[] standard_dist = {1,2,3,4,5,6,5,4,3,2,1};


    IEnumerable<int> RANGE = Enumerable.Range(0, n);
    IEnumerable<int> RANGE1 = Enumerable.Range(0, n-1);


    //
    // Decision variables
    //

    IntVar[] x1 = solver.MakeIntVarArray(n, lowest_value, m, "x1");
    IntVar[] x2 = solver.MakeIntVarArray(n, lowest_value, m, "x2");

    //
    // Constraints
    //
    for(int k = 0; k < standard_dist.Length; k++) {
      solver.Add((from i in RANGE
                  from j in RANGE
                  select x1[i] + x2[j] == k + 2
                  ).ToArray().Sum() == standard_dist[k]);
    }

    // symmetry breaking
    foreach(int i in RANGE1) {
      solver.Add(x1[i] <= x1[i+1]);
      solver.Add(x2[i] <= x2[i+1]);
      solver.Add(x1[i] <= x2[i]);
    }


    //
    // Search
    //
    DecisionBuilder db = solver.MakePhase(x1.Concat(x2).ToArray(),
                                          Solver.INT_VAR_DEFAULT,
                                          Solver.INT_VALUE_DEFAULT);

    solver.NewSearch(db);

    while (solver.NextSolution()) {
      Console.Write("x1: ");
      foreach(int i in RANGE) {
        Console.Write(x1[i].Value() + " ");
      }
      Console.Write("\nx2: ");
      foreach(int i in RANGE) {
        Console.Write(x2[i].Value() + " ");
      }
      Console.WriteLine("\n");
    }

    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)
  {
    Solve();
  }
}