Teka-Teki Kriptaritmatika

teka-teki kriptografis adalah latihan matematika yang melibatkan angka beberapa angka yang diwakili oleh huruf (atau simbol). Setiap huruf mewakili digit unik. Tujuannya adalah untuk menemukan angka sehingga persamaan matematika tertentu diverifikasi:

      CP
+     IS
+    FUN
--------
=   TRUE

Satu penetapan huruf ke angka menghasilkan persamaan berikut:

      23
+     74
+    968
--------
=   1065

Ada jawaban lain untuk masalah ini. Kami akan menunjukkan cara menemukan semua solusi.

Membuat model masalah

Seperti halnya masalah pengoptimalan lainnya, kita akan mulai dengan mengidentifikasi variabel dan batasan. Variabelnya adalah huruf, yang dapat berisi nilai satu digit mana pun.

Untuk CP + IS + FUN = TRUE, batasannya adalah sebagai berikut:

• Persamaan: CP + IS + FUN = TRUE.
• Masing-masing dari sepuluh huruf harus berupa digit yang berbeda.
• C, I, F, dan T tidak boleh nol (karena kita tidak menuliskan angka nol di awal dalam angka).

Anda dapat menyelesaikan masalah kriptografis dengan pemecah CP-SAT baru, yang lebih efisien, atau pemecah CP asli. Kami akan menampilkan contoh menggunakan kedua pemecah masalah, dimulai dengan CP-SAT.

Solusi CP-SAT

Kita akan menampilkan variabel, batasan, pemanggilan pemecah, dan terakhir program lengkap.

Mengimpor library

Kode berikut mengimpor library yang diperlukan.

Python

from ortools.sat.python import cp_model

C++

#include <stdlib.h>

#include <cstdint>

#include "ortools/base/logging.h"
#include "ortools/sat/cp_model.h"
#include "ortools/sat/cp_model.pb.h"
#include "ortools/sat/cp_model_solver.h"
#include "ortools/sat/model.h"
#include "ortools/sat/sat_parameters.pb.h"
#include "ortools/util/sorted_interval_list.h"

Java

import com.google.ortools.Loader;
import com.google.ortools.sat.CpModel;
import com.google.ortools.sat.CpSolver;
import com.google.ortools.sat.CpSolverSolutionCallback;
import com.google.ortools.sat.IntVar;
import com.google.ortools.sat.LinearExpr;

C#

using System;
using Google.OrTools.Sat;

Mendeklarasikan model

Kode berikut mendeklarasikan model untuk masalah.

Python

model = cp_model.CpModel()

C++

CpModelBuilder cp_model;

Java

CpModel model = new CpModel();

C#

        CpModel model = new CpModel();

        int kBase = 10;

        IntVar c = model.NewIntVar(1, kBase - 1, "C");
        IntVar p = model.NewIntVar(0, kBase - 1, "P");
        IntVar i = model.NewIntVar(1, kBase - 1, "I");
        IntVar s = model.NewIntVar(0, kBase - 1, "S");
        IntVar f = model.NewIntVar(1, kBase - 1, "F");
        IntVar u = model.NewIntVar(0, kBase - 1, "U");
        IntVar n = model.NewIntVar(0, kBase - 1, "N");
        IntVar t = model.NewIntVar(1, kBase - 1, "T");
        IntVar r = model.NewIntVar(0, kBase - 1, "R");
        IntVar e = model.NewIntVar(0, kBase - 1, "E");

        // We need to group variables in a list to use the constraint AllDifferent.
        IntVar[] letters = new IntVar[] { c, p, i, s, f, u, n, t, r, e };

        // Define constraints.
        model.AddAllDifferent(letters);

        // CP + IS + FUN = TRUE
        model.Add(c * kBase + p + i * kBase + s + f * kBase * kBase + u * kBase + n ==
                  t * kBase * kBase * kBase + r * kBase * kBase + u * kBase + e);

        // Creates a solver and solves the model.
        CpSolver solver = new CpSolver();
        VarArraySolutionPrinter cb = new VarArraySolutionPrinter(letters);
        // Search for all solutions.
        solver.StringParameters = "enumerate_all_solutions:true";
        // And solve.
        solver.Solve(model, cb);

        Console.WriteLine("Statistics");
        Console.WriteLine($"  conflicts : {solver.NumConflicts()}");
        Console.WriteLine($"  branches  : {solver.NumBranches()}");
        Console.WriteLine($"  wall time : {solver.WallTime()} s");
        Console.WriteLine($"  number of solutions found: {cb.SolutionCount()}");
    }
}

Menentukan variabel

Saat menggunakan pemecah CP-SAT, ada metode bantuan tertentu yang perlu ditentukan. Kita akan menggunakan salah satunya, NewIntVar, untuk mendeklarasikan digit (bilangan bulat). Kita membedakan huruf yang berpotensi nol dan huruf yang tidak bisa (C, I, F, dan T).

Python

base = 10

c = model.new_int_var(1, base - 1, "C")
p = model.new_int_var(0, base - 1, "P")
i = model.new_int_var(1, base - 1, "I")
s = model.new_int_var(0, base - 1, "S")
f = model.new_int_var(1, base - 1, "F")
u = model.new_int_var(0, base - 1, "U")
n = model.new_int_var(0, base - 1, "N")
t = model.new_int_var(1, base - 1, "T")
r = model.new_int_var(0, base - 1, "R")
e = model.new_int_var(0, base - 1, "E")

# We need to group variables in a list to use the constraint AllDifferent.
letters = [c, p, i, s, f, u, n, t, r, e]

# Verify that we have enough digits.
assert base >= len(letters)

C++

const int64_t kBase = 10;

// Define decision variables.
Domain digit(0, kBase - 1);
Domain non_zero_digit(1, kBase - 1);

IntVar c = cp_model.NewIntVar(non_zero_digit).WithName("C");
IntVar p = cp_model.NewIntVar(digit).WithName("P");
IntVar i = cp_model.NewIntVar(non_zero_digit).WithName("I");
IntVar s = cp_model.NewIntVar(digit).WithName("S");
IntVar f = cp_model.NewIntVar(non_zero_digit).WithName("F");
IntVar u = cp_model.NewIntVar(digit).WithName("U");
IntVar n = cp_model.NewIntVar(digit).WithName("N");
IntVar t = cp_model.NewIntVar(non_zero_digit).WithName("T");
IntVar r = cp_model.NewIntVar(digit).WithName("R");
IntVar e = cp_model.NewIntVar(digit).WithName("E");

Java

int base = 10;
IntVar c = model.newIntVar(1, base - 1, "C");
IntVar p = model.newIntVar(0, base - 1, "P");
IntVar i = model.newIntVar(1, base - 1, "I");
IntVar s = model.newIntVar(0, base - 1, "S");
IntVar f = model.newIntVar(1, base - 1, "F");
IntVar u = model.newIntVar(0, base - 1, "U");
IntVar n = model.newIntVar(0, base - 1, "N");
IntVar t = model.newIntVar(1, base - 1, "T");
IntVar r = model.newIntVar(0, base - 1, "R");
IntVar e = model.newIntVar(0, base - 1, "E");

// We need to group variables in a list to use the constraint AllDifferent.
IntVar[] letters = new IntVar[] {c, p, i, s, f, u, n, t, r, e};

C#

        int kBase = 10;

        IntVar c = model.NewIntVar(1, kBase - 1, "C");
        IntVar p = model.NewIntVar(0, kBase - 1, "P");
        IntVar i = model.NewIntVar(1, kBase - 1, "I");
        IntVar s = model.NewIntVar(0, kBase - 1, "S");
        IntVar f = model.NewIntVar(1, kBase - 1, "F");
        IntVar u = model.NewIntVar(0, kBase - 1, "U");
        IntVar n = model.NewIntVar(0, kBase - 1, "N");
        IntVar t = model.NewIntVar(1, kBase - 1, "T");
        IntVar r = model.NewIntVar(0, kBase - 1, "R");
        IntVar e = model.NewIntVar(0, kBase - 1, "E");

        // We need to group variables in a list to use the constraint AllDifferent.
        IntVar[] letters = new IntVar[] { c, p, i, s, f, u, n, t, r, e };

Menentukan batasan

Selanjutnya, batasan. Pertama, kami memastikan bahwa semua huruf memiliki nilai yang berbeda, menggunakan metode helper AddAllDifferent. Kemudian, kita menggunakan metode helper AddEquality untuk membuat batasan yang menerapkan kesetaraan CP + IS + FUN = TRUE.

Python

model.add_all_different(letters)

# CP + IS + FUN = TRUE
model.add(
    c * base + p + i * base + s + f * base * base + u * base + n
    == t * base * base * base + r * base * base + u * base + e
)

C++

// Define constraints.
cp_model.AddAllDifferent({c, p, i, s, f, u, n, t, r, e});

// CP + IS + FUN = TRUE
cp_model.AddEquality(
    c * kBase + p + i * kBase + s + f * kBase * kBase + u * kBase + n,
    kBase * kBase * kBase * t + kBase * kBase * r + kBase * u + e);

Java

model.addAllDifferent(letters);

// CP + IS + FUN = TRUE
model.addEquality(LinearExpr.weightedSum(new IntVar[] {c, p, i, s, f, u, n, t, r, u, e},
                      new long[] {base, 1, base, 1, base * base, base, 1, -base * base * base,
                          -base * base, -base, -1}),
    0);

C#

// Define constraints.
model.AddAllDifferent(letters);

// CP + IS + FUN = TRUE
model.Add(c * kBase + p + i * kBase + s + f * kBase * kBase + u * kBase + n ==
          t * kBase * kBase * kBase + r * kBase * kBase + u * kBase + e);

Printer solusi

Kode untuk printer solusi, yang menampilkan setiap solusi sebagai pemecah masalah, ditunjukkan di bawah ini.

Python

class VarArraySolutionPrinter(cp_model.CpSolverSolutionCallback):
    """Print intermediate solutions."""

    def __init__(self, variables: list[cp_model.IntVar]):
        cp_model.CpSolverSolutionCallback.__init__(self)
        self.__variables = variables
        self.__solution_count = 0

    def on_solution_callback(self) -> None:
        self.__solution_count += 1
        for v in self.__variables:
            print(f"{v}={self.value(v)}", end=" ")
        print()

    @property
    def solution_count(self) -> int:
        return self.__solution_count 

C++

Model model;
int num_solutions = 0;
model.Add(NewFeasibleSolutionObserver([&](const CpSolverResponse& response) {
  LOG(INFO) << "Solution " << num_solutions;
  LOG(INFO) << "C=" << SolutionIntegerValue(response, c) << " "
            << "P=" << SolutionIntegerValue(response, p) << " "
            << "I=" << SolutionIntegerValue(response, i) << " "
            << "S=" << SolutionIntegerValue(response, s) << " "
            << "F=" << SolutionIntegerValue(response, f) << " "
            << "U=" << SolutionIntegerValue(response, u) << " "
            << "N=" << SolutionIntegerValue(response, n) << " "
            << "T=" << SolutionIntegerValue(response, t) << " "
            << "R=" << SolutionIntegerValue(response, r) << " "
            << "E=" << SolutionIntegerValue(response, e);
  num_solutions++;
}));

Java

static class VarArraySolutionPrinter extends CpSolverSolutionCallback {
  public VarArraySolutionPrinter(IntVar[] variables) {
    variableArray = variables;
  }

  @Override
  public void onSolutionCallback() {
    for (IntVar v : variableArray) {
      System.out.printf("  %s = %d", v.getName(), value(v));
    }
    System.out.println();
    solutionCount++;
  }

  public int getSolutionCount() {
    return solutionCount;
  }

  private int solutionCount;
  private final IntVar[] variableArray;
}

C#

public class VarArraySolutionPrinter : CpSolverSolutionCallback
{
    public VarArraySolutionPrinter(IntVar[] variables)
    {
        variables_ = variables;
    }

    public override void OnSolutionCallback()
    {
        {
            foreach (IntVar v in variables_)
            {
                Console.Write(String.Format("  {0}={1}", v.ToString(), Value(v)));
            }
            Console.WriteLine();
            solution_count_++;
        }
    }

    public int SolutionCount()
    {
        return solution_count_;
    }

    private int solution_count_;
    private IntVar[] variables_;
}

Memanggil pemecah masalah

Terakhir, kita selesaikan masalah dan tampilkan solusinya. Semua keajaibannya ada dalam metode operations_research::sat::SolveCpModel().

Python

solver = cp_model.CpSolver()
solution_printer = VarArraySolutionPrinter(letters)
# Enumerate all solutions.
solver.parameters.enumerate_all_solutions = True
# Solve.
status = solver.solve(model, solution_printer)

C++

// Tell the solver to enumerate all solutions.
SatParameters parameters;
parameters.set_enumerate_all_solutions(true);
model.Add(NewSatParameters(parameters));

const CpSolverResponse response = SolveCpModel(cp_model.Build(), &model);
LOG(INFO) << "Number of solutions found: " << num_solutions;

Java

CpSolver solver = new CpSolver();
VarArraySolutionPrinter cb = new VarArraySolutionPrinter(letters);
// Tell the solver to enumerate all solutions.
solver.getParameters().setEnumerateAllSolutions(true);
// And solve.
solver.solve(model, cb);

C#

// Creates a solver and solves the model.
CpSolver solver = new CpSolver();
VarArraySolutionPrinter cb = new VarArraySolutionPrinter(letters);
// Search for all solutions.
solver.StringParameters = "enumerate_all_solutions:true";
// And solve.
solver.Solve(model, cb);

Saat Anda menjalankannya, program akan menampilkan output berikut, yang setiap barisnya berisi solusi:

C=2 P=3 I=7 S=4 F=9 U=6 N=8 T=1 R=0 E=5
C=2 P=4 I=7 S=3 F=9 U=6 N=8 T=1 R=0 E=5
C=2 P=5 I=7 S=3 F=9 U=4 N=8 T=1 R=0 E=6
C=2 P=8 I=7 S=3 F=9 U=4 N=5 T=1 R=0 E=6
C=2 P=8 I=7 S=3 F=9 U=6 N=4 T=1 R=0 E=5
C=3 P=7 I=6 S=2 F=9 U=8 N=5 T=1 R=0 E=4
C=6 P=7 I=3 S=2 F=9 U=8 N=5 T=1 R=0 E=4
C=6 P=5 I=3 S=2 F=9 U=8 N=7 T=1 R=0 E=4
C=3 P=5 I=6 S=2 F=9 U=8 N=7 T=1 R=0 E=4
C=3 P=8 I=6 S=4 F=9 U=2 N=5 T=1 R=0 E=7
C=3 P=7 I=6 S=5 F=9 U=8 N=2 T=1 R=0 E=4
C=3 P=8 I=6 S=5 F=9 U=2 N=4 T=1 R=0 E=7
C=3 P=5 I=6 S=4 F=9 U=2 N=8 T=1 R=0 E=7
C=3 P=4 I=6 S=5 F=9 U=2 N=8 T=1 R=0 E=7
C=3 P=2 I=6 S=5 F=9 U=8 N=7 T=1 R=0 E=4
C=3 P=4 I=6 S=8 F=9 U=2 N=5 T=1 R=0 E=7
C=3 P=2 I=6 S=7 F=9 U=8 N=5 T=1 R=0 E=4
C=3 P=5 I=6 S=8 F=9 U=2 N=4 T=1 R=0 E=7
C=3 P=5 I=6 S=7 F=9 U=8 N=2 T=1 R=0 E=4
C=2 P=5 I=7 S=6 F=9 U=8 N=3 T=1 R=0 E=4
C=2 P=5 I=7 S=8 F=9 U=4 N=3 T=1 R=0 E=6
C=2 P=6 I=7 S=5 F=9 U=8 N=3 T=1 R=0 E=4
C=2 P=4 I=7 S=8 F=9 U=6 N=3 T=1 R=0 E=5
C=2 P=3 I=7 S=8 F=9 U=6 N=4 T=1 R=0 E=5
C=2 P=8 I=7 S=5 F=9 U=4 N=3 T=1 R=0 E=6
C=2 P=8 I=7 S=4 F=9 U=6 N=3 T=1 R=0 E=5
C=2 P=6 I=7 S=3 F=9 U=8 N=5 T=1 R=0 E=4
C=2 P=5 I=7 S=3 F=9 U=8 N=6 T=1 R=0 E=4
C=2 P=3 I=7 S=5 F=9 U=4 N=8 T=1 R=0 E=6
C=2 P=3 I=7 S=5 F=9 U=8 N=6 T=1 R=0 E=4
C=2 P=3 I=7 S=6 F=9 U=8 N=5 T=1 R=0 E=4
C=2 P=3 I=7 S=8 F=9 U=4 N=5 T=1 R=0 E=6
C=4 P=3 I=5 S=8 F=9 U=2 N=6 T=1 R=0 E=7
C=5 P=3 I=4 S=8 F=9 U=2 N=6 T=1 R=0 E=7
C=6 P=2 I=3 S=7 F=9 U=8 N=5 T=1 R=0 E=4
C=7 P=3 I=2 S=6 F=9 U=8 N=5 T=1 R=0 E=4
C=7 P=3 I=2 S=8 F=9 U=4 N=5 T=1 R=0 E=6
C=6 P=4 I=3 S=8 F=9 U=2 N=5 T=1 R=0 E=7
C=5 P=3 I=4 S=6 F=9 U=2 N=8 T=1 R=0 E=7
C=4 P=3 I=5 S=6 F=9 U=2 N=8 T=1 R=0 E=7
C=5 P=6 I=4 S=3 F=9 U=2 N=8 T=1 R=0 E=7
C=7 P=4 I=2 S=3 F=9 U=6 N=8 T=1 R=0 E=5
C=7 P=3 I=2 S=4 F=9 U=6 N=8 T=1 R=0 E=5
C=6 P=2 I=3 S=5 F=9 U=8 N=7 T=1 R=0 E=4
C=7 P=3 I=2 S=5 F=9 U=4 N=8 T=1 R=0 E=6
C=6 P=4 I=3 S=5 F=9 U=2 N=8 T=1 R=0 E=7
C=6 P=5 I=3 S=4 F=9 U=2 N=8 T=1 R=0 E=7
C=7 P=5 I=2 S=3 F=9 U=4 N=8 T=1 R=0 E=6
C=4 P=6 I=5 S=3 F=9 U=2 N=8 T=1 R=0 E=7
C=6 P=5 I=3 S=8 F=9 U=2 N=4 T=1 R=0 E=7
C=6 P=5 I=3 S=7 F=9 U=8 N=2 T=1 R=0 E=4
C=7 P=5 I=2 S=8 F=9 U=4 N=3 T=1 R=0 E=6
C=7 P=5 I=2 S=6 F=9 U=8 N=3 T=1 R=0 E=4
C=5 P=8 I=4 S=6 F=9 U=2 N=3 T=1 R=0 E=7
C=4 P=8 I=5 S=6 F=9 U=2 N=3 T=1 R=0 E=7
C=4 P=8 I=5 S=3 F=9 U=2 N=6 T=1 R=0 E=7
C=5 P=8 I=4 S=3 F=9 U=2 N=6 T=1 R=0 E=7
C=7 P=8 I=2 S=3 F=9 U=4 N=5 T=1 R=0 E=6
C=7 P=8 I=2 S=3 F=9 U=6 N=4 T=1 R=0 E=5
C=7 P=8 I=2 S=4 F=9 U=6 N=3 T=1 R=0 E=5
C=7 P=8 I=2 S=5 F=9 U=4 N=3 T=1 R=0 E=6
C=6 P=8 I=3 S=5 F=9 U=2 N=4 T=1 R=0 E=7
C=6 P=8 I=3 S=4 F=9 U=2 N=5 T=1 R=0 E=7
C=6 P=7 I=3 S=5 F=9 U=8 N=2 T=1 R=0 E=4
C=7 P=6 I=2 S=5 F=9 U=8 N=3 T=1 R=0 E=4
C=7 P=3 I=2 S=5 F=9 U=8 N=6 T=1 R=0 E=4
C=7 P=4 I=2 S=8 F=9 U=6 N=3 T=1 R=0 E=5
C=7 P=3 I=2 S=8 F=9 U=6 N=4 T=1 R=0 E=5
C=5 P=6 I=4 S=8 F=9 U=2 N=3 T=1 R=0 E=7
C=4 P=6 I=5 S=8 F=9 U=2 N=3 T=1 R=0 E=7
C=7 P=6 I=2 S=3 F=9 U=8 N=5 T=1 R=0 E=4
C=7 P=5 I=2 S=3 F=9 U=8 N=6 T=1 R=0 E=4

Statistics
  - status          : OPTIMAL
  - conflicts       : 110
  - branches        : 435
  - wall time       : 0.014934 ms
  - solutions found : 72

Selesaikan program

Berikut program lengkapnya.

Python

"""Cryptarithmetic puzzle.

First attempt to solve equation CP + IS + FUN = TRUE
where each letter represents a unique digit.

This problem has 72 different solutions in base 10.
"""
from ortools.sat.python import cp_model


class VarArraySolutionPrinter(cp_model.CpSolverSolutionCallback):
    """Print intermediate solutions."""

    def __init__(self, variables: list[cp_model.IntVar]):
        cp_model.CpSolverSolutionCallback.__init__(self)
        self.__variables = variables
        self.__solution_count = 0

    def on_solution_callback(self) -> None:
        self.__solution_count += 1
        for v in self.__variables:
            print(f"{v}={self.value(v)}", end=" ")
        print()

    @property
    def solution_count(self) -> int:
        return self.__solution_count


def main() -> None:
    """solve the CP+IS+FUN==TRUE cryptarithm."""
    # Constraint programming engine
    model = cp_model.CpModel()

    base = 10

    c = model.new_int_var(1, base - 1, "C")
    p = model.new_int_var(0, base - 1, "P")
    i = model.new_int_var(1, base - 1, "I")
    s = model.new_int_var(0, base - 1, "S")
    f = model.new_int_var(1, base - 1, "F")
    u = model.new_int_var(0, base - 1, "U")
    n = model.new_int_var(0, base - 1, "N")
    t = model.new_int_var(1, base - 1, "T")
    r = model.new_int_var(0, base - 1, "R")
    e = model.new_int_var(0, base - 1, "E")

    # We need to group variables in a list to use the constraint AllDifferent.
    letters = [c, p, i, s, f, u, n, t, r, e]

    # Verify that we have enough digits.
    assert base >= len(letters)

    # Define constraints.
    model.add_all_different(letters)

    # CP + IS + FUN = TRUE
    model.add(
        c * base + p + i * base + s + f * base * base + u * base + n
        == t * base * base * base + r * base * base + u * base + e
    )

    # Creates a solver and solves the model.
    solver = cp_model.CpSolver()
    solution_printer = VarArraySolutionPrinter(letters)
    # Enumerate all solutions.
    solver.parameters.enumerate_all_solutions = True
    # Solve.
    status = solver.solve(model, solution_printer)

    # Statistics.
    print("\nStatistics")
    print(f"  status   : {solver.status_name(status)}")
    print(f"  conflicts: {solver.num_conflicts}")
    print(f"  branches : {solver.num_branches}")
    print(f"  wall time: {solver.wall_time} s")
    print(f"  sol found: {solution_printer.solution_count}")


if __name__ == "__main__":
    main()

C++

// Cryptarithmetic puzzle
//
// First attempt to solve equation CP + IS + FUN = TRUE
// where each letter represents a unique digit.
//
// This problem has 72 different solutions in base 10.
#include <stdlib.h>

#include <cstdint>

#include "ortools/base/logging.h"
#include "ortools/sat/cp_model.h"
#include "ortools/sat/cp_model.pb.h"
#include "ortools/sat/cp_model_solver.h"
#include "ortools/sat/model.h"
#include "ortools/sat/sat_parameters.pb.h"
#include "ortools/util/sorted_interval_list.h"

namespace operations_research {
namespace sat {

void CPIsFunSat() {
  // Instantiate the solver.
  CpModelBuilder cp_model;

  const int64_t kBase = 10;

  // Define decision variables.
  Domain digit(0, kBase - 1);
  Domain non_zero_digit(1, kBase - 1);

  IntVar c = cp_model.NewIntVar(non_zero_digit).WithName("C");
  IntVar p = cp_model.NewIntVar(digit).WithName("P");
  IntVar i = cp_model.NewIntVar(non_zero_digit).WithName("I");
  IntVar s = cp_model.NewIntVar(digit).WithName("S");
  IntVar f = cp_model.NewIntVar(non_zero_digit).WithName("F");
  IntVar u = cp_model.NewIntVar(digit).WithName("U");
  IntVar n = cp_model.NewIntVar(digit).WithName("N");
  IntVar t = cp_model.NewIntVar(non_zero_digit).WithName("T");
  IntVar r = cp_model.NewIntVar(digit).WithName("R");
  IntVar e = cp_model.NewIntVar(digit).WithName("E");

  // Define constraints.
  cp_model.AddAllDifferent({c, p, i, s, f, u, n, t, r, e});

  // CP + IS + FUN = TRUE
  cp_model.AddEquality(
      c * kBase + p + i * kBase + s + f * kBase * kBase + u * kBase + n,
      kBase * kBase * kBase * t + kBase * kBase * r + kBase * u + e);

  Model model;
  int num_solutions = 0;
  model.Add(NewFeasibleSolutionObserver([&](const CpSolverResponse& response) {
    LOG(INFO) << "Solution " << num_solutions;
    LOG(INFO) << "C=" << SolutionIntegerValue(response, c) << " "
              << "P=" << SolutionIntegerValue(response, p) << " "
              << "I=" << SolutionIntegerValue(response, i) << " "
              << "S=" << SolutionIntegerValue(response, s) << " "
              << "F=" << SolutionIntegerValue(response, f) << " "
              << "U=" << SolutionIntegerValue(response, u) << " "
              << "N=" << SolutionIntegerValue(response, n) << " "
              << "T=" << SolutionIntegerValue(response, t) << " "
              << "R=" << SolutionIntegerValue(response, r) << " "
              << "E=" << SolutionIntegerValue(response, e);
    num_solutions++;
  }));

  // Tell the solver to enumerate all solutions.
  SatParameters parameters;
  parameters.set_enumerate_all_solutions(true);
  model.Add(NewSatParameters(parameters));

  const CpSolverResponse response = SolveCpModel(cp_model.Build(), &model);
  LOG(INFO) << "Number of solutions found: " << num_solutions;

  // Statistics.
  LOG(INFO) << "Statistics";
  LOG(INFO) << CpSolverResponseStats(response);
}

}  // namespace sat
}  // namespace operations_research

int main(int argc, char** argv) {
  operations_research::sat::CPIsFunSat();
  return EXIT_SUCCESS;
}

Java

package com.google.ortools.sat.samples;
import com.google.ortools.Loader;
import com.google.ortools.sat.CpModel;
import com.google.ortools.sat.CpSolver;
import com.google.ortools.sat.CpSolverSolutionCallback;
import com.google.ortools.sat.IntVar;
import com.google.ortools.sat.LinearExpr;

/** Cryptarithmetic puzzle. */
public final class CpIsFunSat {
  static class VarArraySolutionPrinter extends CpSolverSolutionCallback {
    public VarArraySolutionPrinter(IntVar[] variables) {
      variableArray = variables;
    }

    @Override
    public void onSolutionCallback() {
      for (IntVar v : variableArray) {
        System.out.printf("  %s = %d", v.getName(), value(v));
      }
      System.out.println();
      solutionCount++;
    }

    public int getSolutionCount() {
      return solutionCount;
    }

    private int solutionCount;
    private final IntVar[] variableArray;
  }

  public static void main(String[] args) throws Exception {
    Loader.loadNativeLibraries();
    // Create the model.
    CpModel model = new CpModel();

    int base = 10;
    IntVar c = model.newIntVar(1, base - 1, "C");
    IntVar p = model.newIntVar(0, base - 1, "P");
    IntVar i = model.newIntVar(1, base - 1, "I");
    IntVar s = model.newIntVar(0, base - 1, "S");
    IntVar f = model.newIntVar(1, base - 1, "F");
    IntVar u = model.newIntVar(0, base - 1, "U");
    IntVar n = model.newIntVar(0, base - 1, "N");
    IntVar t = model.newIntVar(1, base - 1, "T");
    IntVar r = model.newIntVar(0, base - 1, "R");
    IntVar e = model.newIntVar(0, base - 1, "E");

    // We need to group variables in a list to use the constraint AllDifferent.
    IntVar[] letters = new IntVar[] {c, p, i, s, f, u, n, t, r, e};

    // Define constraints.
    model.addAllDifferent(letters);

    // CP + IS + FUN = TRUE
    model.addEquality(LinearExpr.weightedSum(new IntVar[] {c, p, i, s, f, u, n, t, r, u, e},
                          new long[] {base, 1, base, 1, base * base, base, 1, -base * base * base,
                              -base * base, -base, -1}),
        0);

    // Create a solver and solve the model.
    CpSolver solver = new CpSolver();
    VarArraySolutionPrinter cb = new VarArraySolutionPrinter(letters);
    // Tell the solver to enumerate all solutions.
    solver.getParameters().setEnumerateAllSolutions(true);
    // And solve.
    solver.solve(model, cb);

    // Statistics.
    System.out.println("Statistics");
    System.out.println("  - conflicts : " + solver.numConflicts());
    System.out.println("  - branches  : " + solver.numBranches());
    System.out.println("  - wall time : " + solver.wallTime() + " s");
    System.out.println("  - solutions : " + cb.getSolutionCount());
  }

  private CpIsFunSat() {}
}

C#

// Cryptarithmetic puzzle
//
// First attempt to solve equation CP + IS + FUN = TRUE
// where each letter represents a unique digit.
//
// This problem has 72 different solutions in base 10.
using System;
using Google.OrTools.Sat;

public class CpIsFunSat
{
    public class VarArraySolutionPrinter : CpSolverSolutionCallback
    {
        public VarArraySolutionPrinter(IntVar[] variables)
        {
            variables_ = variables;
        }

        public override void OnSolutionCallback()
        {
            {
                foreach (IntVar v in variables_)
                {
                    Console.Write(String.Format("  {0}={1}", v.ToString(), Value(v)));
                }
                Console.WriteLine();
                solution_count_++;
            }
        }

        public int SolutionCount()
        {
            return solution_count_;
        }

        private int solution_count_;
        private IntVar[] variables_;
    }

    // Solve the CP+IS+FUN==TRUE cryptarithm.
    static void Main()
    {
        // Constraint programming engine
        CpModel model = new CpModel();

        int kBase = 10;

        IntVar c = model.NewIntVar(1, kBase - 1, "C");
        IntVar p = model.NewIntVar(0, kBase - 1, "P");
        IntVar i = model.NewIntVar(1, kBase - 1, "I");
        IntVar s = model.NewIntVar(0, kBase - 1, "S");
        IntVar f = model.NewIntVar(1, kBase - 1, "F");
        IntVar u = model.NewIntVar(0, kBase - 1, "U");
        IntVar n = model.NewIntVar(0, kBase - 1, "N");
        IntVar t = model.NewIntVar(1, kBase - 1, "T");
        IntVar r = model.NewIntVar(0, kBase - 1, "R");
        IntVar e = model.NewIntVar(0, kBase - 1, "E");

        // We need to group variables in a list to use the constraint AllDifferent.
        IntVar[] letters = new IntVar[] { c, p, i, s, f, u, n, t, r, e };

        // Define constraints.
        model.AddAllDifferent(letters);

        // CP + IS + FUN = TRUE
        model.Add(c * kBase + p + i * kBase + s + f * kBase * kBase + u * kBase + n ==
                  t * kBase * kBase * kBase + r * kBase * kBase + u * kBase + e);

        // Creates a solver and solves the model.
        CpSolver solver = new CpSolver();
        VarArraySolutionPrinter cb = new VarArraySolutionPrinter(letters);
        // Search for all solutions.
        solver.StringParameters = "enumerate_all_solutions:true";
        // And solve.
        solver.Solve(model, cb);

        Console.WriteLine("Statistics");
        Console.WriteLine($"  conflicts : {solver.NumConflicts()}");
        Console.WriteLine($"  branches  : {solver.NumBranches()}");
        Console.WriteLine($"  wall time : {solver.WallTime()} s");
        Console.WriteLine($"  number of solutions found: {cb.SolutionCount()}");
    }
}

Solusi CP Asli

Dalam hal ini kita akan memperlakukan basis sebagai variabel sehingga Anda dapat menyelesaikan persamaan untuk basis yang lebih tinggi. (Tidak ada solusi dasar yang lebih rendah untuk CP + IS + FUN = TRUE karena semua hurufnya harus berbeda.)

Mengimpor library

Kode berikut mengimpor library yang diperlukan.

Python

from ortools.constraint_solver import pywrapcp

C++

#include <cstdint>
#include <vector>

#include "absl/flags/flag.h"
#include "absl/log/flags.h"
#include "ortools/base/init_google.h"
#include "ortools/base/logging.h"
#include "ortools/constraint_solver/constraint_solver.h"

Java

C#

using System;
using Google.OrTools.ConstraintSolver;

Membuat pemecah masalah

Langkah pertama adalah membuat Solver.

Python

solver = pywrapcp.Solver("CP is fun!")

C++

Solver solver("CP is fun!");

Java

Solver solver = new Solver("CP is fun!");

C#

Solver solver = new Solver("CP is fun!");

Menentukan variabel

Langkah pertama adalah membuat IntVar untuk setiap huruf. Kita membedakan huruf-huruf yang berpotensi nol dan yang tidak (C, I, F, dan T).

Selanjutnya, kita membuat array yang berisi IntVar baru untuk setiap huruf. Ini hanya diperlukan karena saat menentukan batasan, kita akan menggunakan AllDifferent, jadi kita memerlukan beberapa array yang setiap elemennya perlu dibedakan.

Terakhir, kita memverifikasi bahwa basis kita setidaknya sebesar jumlah huruf; jika tidak, tidak ada solusi.

Python

base = 10

# Decision variables.
digits = list(range(0, base))
digits_without_zero = list(range(1, base))
c = solver.IntVar(digits_without_zero, "C")
p = solver.IntVar(digits, "P")
i = solver.IntVar(digits_without_zero, "I")
s = solver.IntVar(digits, "S")
f = solver.IntVar(digits_without_zero, "F")
u = solver.IntVar(digits, "U")
n = solver.IntVar(digits, "N")
t = solver.IntVar(digits_without_zero, "T")
r = solver.IntVar(digits, "R")
e = solver.IntVar(digits, "E")

# We need to group variables in a list to use the constraint AllDifferent.
letters = [c, p, i, s, f, u, n, t, r, e]

# Verify that we have enough digits.
assert base >= len(letters)

C++

const int64_t kBase = 10;

// Define decision variables.
IntVar* const c = solver.MakeIntVar(1, kBase - 1, "C");
IntVar* const p = solver.MakeIntVar(0, kBase - 1, "P");
IntVar* const i = solver.MakeIntVar(1, kBase - 1, "I");
IntVar* const s = solver.MakeIntVar(0, kBase - 1, "S");
IntVar* const f = solver.MakeIntVar(1, kBase - 1, "F");
IntVar* const u = solver.MakeIntVar(0, kBase - 1, "U");
IntVar* const n = solver.MakeIntVar(0, kBase - 1, "N");
IntVar* const t = solver.MakeIntVar(1, kBase - 1, "T");
IntVar* const r = solver.MakeIntVar(0, kBase - 1, "R");
IntVar* const e = solver.MakeIntVar(0, kBase - 1, "E");

// We need to group variables in a vector to be able to use
// the global constraint AllDifferent
std::vector<IntVar*> letters{c, p, i, s, f, u, n, t, r, e};

// Check if we have enough digits
CHECK_GE(kBase, letters.size());

Java

final int base = 10;

// Decision variables.
final IntVar c = solver.makeIntVar(1, base - 1, "C");
final IntVar p = solver.makeIntVar(0, base - 1, "P");
final IntVar i = solver.makeIntVar(1, base - 1, "I");
final IntVar s = solver.makeIntVar(0, base - 1, "S");
final IntVar f = solver.makeIntVar(1, base - 1, "F");
final IntVar u = solver.makeIntVar(0, base - 1, "U");
final IntVar n = solver.makeIntVar(0, base - 1, "N");
final IntVar t = solver.makeIntVar(1, base - 1, "T");
final IntVar r = solver.makeIntVar(0, base - 1, "R");
final IntVar e = solver.makeIntVar(0, base - 1, "E");

// Group variables in a vector so that we can use AllDifferent.
final IntVar[] letters = new IntVar[] {c, p, i, s, f, u, n, t, r, e};

// Verify that we have enough digits.
if (base < letters.length) {
  throw new Exception("base < letters.Length");
}

C#

const int kBase = 10;

// Decision variables.
IntVar c = solver.MakeIntVar(1, kBase - 1, "C");
IntVar p = solver.MakeIntVar(0, kBase - 1, "P");
IntVar i = solver.MakeIntVar(1, kBase - 1, "I");
IntVar s = solver.MakeIntVar(0, kBase - 1, "S");
IntVar f = solver.MakeIntVar(1, kBase - 1, "F");
IntVar u = solver.MakeIntVar(0, kBase - 1, "U");
IntVar n = solver.MakeIntVar(0, kBase - 1, "N");
IntVar t = solver.MakeIntVar(1, kBase - 1, "T");
IntVar r = solver.MakeIntVar(0, kBase - 1, "R");
IntVar e = solver.MakeIntVar(0, kBase - 1, "E");

// Group variables in a vector so that we can use AllDifferent.
IntVar[] letters = new IntVar[] { c, p, i, s, f, u, n, t, r, e };

// Verify that we have enough digits.
if (kBase < letters.Length)
{
    throw new Exception("kBase < letters.Length");
}

Menentukan batasan

Setelah kita menentukan variabel, langkah berikutnya adalah menentukan batasan. Pertama, kita tambahkan batasan AllDifferent, yang memaksa setiap huruf memiliki digit yang berbeda.

Selanjutnya, tambahkan batasan CP + IS + FUN = TRUE. Program-program sampel melakukannya dengan cara yang berbeda.

Python

solver.Add(solver.AllDifferent(letters))

# CP + IS + FUN = TRUE
solver.Add(
    p + s + n + base * (c + i + u) + base * base * f
    == e + base * u + base * base * r + base * base * base * t
)

C++

// Define constraints.
solver.AddConstraint(solver.MakeAllDifferent(letters));

// CP + IS + FUN = TRUE
IntVar* const term1 = MakeBaseLine2(&solver, c, p, kBase);
IntVar* const term2 = MakeBaseLine2(&solver, i, s, kBase);
IntVar* const term3 = MakeBaseLine3(&solver, f, u, n, kBase);
IntVar* const sum_terms =
    solver.MakeSum(solver.MakeSum(term1, term2), term3)->Var();

IntVar* const sum = MakeBaseLine4(&solver, t, r, u, e, kBase);

solver.AddConstraint(solver.MakeEquality(sum_terms, sum));

Java

solver.addConstraint(solver.makeAllDifferent(letters));

// CP + IS + FUN = TRUE
final IntVar sum1 =
    solver
        .makeSum(new IntVar[] {p, s, n,
            solver.makeProd(solver.makeSum(new IntVar[] {c, i, u}).var(), base).var(),
            solver.makeProd(f, base * base).var()})
        .var();
final IntVar sum2 = solver
                        .makeSum(new IntVar[] {e, solver.makeProd(u, base).var(),
                            solver.makeProd(r, base * base).var(),
                            solver.makeProd(t, base * base * base).var()})
                        .var();
solver.addConstraint(solver.makeEquality(sum1, sum2));

C#

solver.Add(letters.AllDifferent());

// CP + IS + FUN = TRUE
solver.Add(p + s + n + kBase * (c + i + u) + kBase * kBase * f ==
           e + kBase * u + kBase * kBase * r + kBase * kBase * kBase * t);

Memanggil pemecah masalah

Sekarang setelah kita memiliki variabel dan batasan, kita siap untuk mengatasinya.

Kode untuk printer solusi, yang menampilkan setiap solusi sebagai pemecah masalah, ditunjukkan di bawah ini.

Karena ada lebih dari satu solusi untuk masalah ini, kita melakukan iterasi solusi dengan loop while solver.NextSolution(). Jika kita hanya mencoba untuk menemukan solusi tunggal, kita akan menggunakan idiom ini:

if (solver.NextSolution()) {
    // Print solution.
} else {
    // Print that no solution could be found.
}

Python

solution_count = 0
db = solver.Phase(letters, solver.INT_VAR_DEFAULT, solver.INT_VALUE_DEFAULT)
solver.NewSearch(db)
while solver.NextSolution():
    print(letters)
    # Is CP + IS + FUN = TRUE?
    assert (
        base * c.Value()
        + p.Value()
        + base * i.Value()
        + s.Value()
        + base * base * f.Value()
        + base * u.Value()
        + n.Value()
        == base * base * base * t.Value()
        + base * base * r.Value()
        + base * u.Value()
        + e.Value()
    )
    solution_count += 1
solver.EndSearch()
print(f"Number of solutions found: {solution_count}")

C++

int num_solutions = 0;
// Create decision builder to search for solutions.
DecisionBuilder* const db = solver.MakePhase(
    letters, Solver::CHOOSE_FIRST_UNBOUND, Solver::ASSIGN_MIN_VALUE);
solver.NewSearch(db);
while (solver.NextSolution()) {
  LOG(INFO) << "C=" << c->Value() << " " << "P=" << p->Value() << " "
            << "I=" << i->Value() << " " << "S=" << s->Value() << " "
            << "F=" << f->Value() << " " << "U=" << u->Value() << " "
            << "N=" << n->Value() << " " << "T=" << t->Value() << " "
            << "R=" << r->Value() << " " << "E=" << e->Value();

  // Is CP + IS + FUN = TRUE?
  CHECK_EQ(p->Value() + s->Value() + n->Value() +
               kBase * (c->Value() + i->Value() + u->Value()) +
               kBase * kBase * f->Value(),
           e->Value() + kBase * u->Value() + kBase * kBase * r->Value() +
               kBase * kBase * kBase * t->Value());
  num_solutions++;
}
solver.EndSearch();
LOG(INFO) << "Number of solutions found: " << num_solutions;

Java

int countSolution = 0;
// Create the decision builder to search for solutions.
final DecisionBuilder db =
    solver.makePhase(letters, Solver.CHOOSE_FIRST_UNBOUND, Solver.ASSIGN_MIN_VALUE);
solver.newSearch(db);
while (solver.nextSolution()) {
  System.out.println("C=" + c.value() + " P=" + p.value());
  System.out.println(" I=" + i.value() + " S=" + s.value());
  System.out.println(" F=" + f.value() + " U=" + u.value());
  System.out.println(" N=" + n.value() + " T=" + t.value());
  System.out.println(" R=" + r.value() + " E=" + e.value());

  // Is CP + IS + FUN = TRUE?
  if (p.value() + s.value() + n.value() + base * (c.value() + i.value() + u.value())
          + base * base * f.value()
      != e.value() + base * u.value() + base * base * r.value()
          + base * base * base * t.value()) {
    throw new Exception("CP + IS + FUN != TRUE");
  }
  countSolution++;
}
solver.endSearch();
System.out.println("Number of solutions found: " + countSolution);

C#

int SolutionCount = 0;
// Create the decision builder to search for solutions.
DecisionBuilder db = solver.MakePhase(letters, Solver.CHOOSE_FIRST_UNBOUND, Solver.ASSIGN_MIN_VALUE);
solver.NewSearch(db);
while (solver.NextSolution())
{
    Console.Write("C=" + c.Value() + " P=" + p.Value());
    Console.Write(" I=" + i.Value() + " S=" + s.Value());
    Console.Write(" F=" + f.Value() + " U=" + u.Value());
    Console.Write(" N=" + n.Value() + " T=" + t.Value());
    Console.Write(" R=" + r.Value() + " E=" + e.Value());
    Console.WriteLine();

    // Is CP + IS + FUN = TRUE?
    if (p.Value() + s.Value() + n.Value() + kBase * (c.Value() + i.Value() + u.Value()) +
            kBase * kBase * f.Value() !=
        e.Value() + kBase * u.Value() + kBase * kBase * r.Value() + kBase * kBase * kBase * t.Value())
    {
        throw new Exception("CP + IS + FUN != TRUE");
    }
    SolutionCount++;
}
solver.EndSearch();
Console.WriteLine($"Number of solutions found: {SolutionCount}");

Selesaikan program

Berikut program lengkapnya.

Python

"""Cryptarithmetic puzzle.

First attempt to solve equation CP + IS + FUN = TRUE
where each letter represents a unique digit.

This problem has 72 different solutions in base 10.
"""
from ortools.constraint_solver import pywrapcp


def main():
    # Constraint programming engine
    solver = pywrapcp.Solver("CP is fun!")

    base = 10

    # Decision variables.
    digits = list(range(0, base))
    digits_without_zero = list(range(1, base))
    c = solver.IntVar(digits_without_zero, "C")
    p = solver.IntVar(digits, "P")
    i = solver.IntVar(digits_without_zero, "I")
    s = solver.IntVar(digits, "S")
    f = solver.IntVar(digits_without_zero, "F")
    u = solver.IntVar(digits, "U")
    n = solver.IntVar(digits, "N")
    t = solver.IntVar(digits_without_zero, "T")
    r = solver.IntVar(digits, "R")
    e = solver.IntVar(digits, "E")

    # We need to group variables in a list to use the constraint AllDifferent.
    letters = [c, p, i, s, f, u, n, t, r, e]

    # Verify that we have enough digits.
    assert base >= len(letters)

    # Define constraints.
    solver.Add(solver.AllDifferent(letters))

    # CP + IS + FUN = TRUE
    solver.Add(
        p + s + n + base * (c + i + u) + base * base * f
        == e + base * u + base * base * r + base * base * base * t
    )

    solution_count = 0
    db = solver.Phase(letters, solver.INT_VAR_DEFAULT, solver.INT_VALUE_DEFAULT)
    solver.NewSearch(db)
    while solver.NextSolution():
        print(letters)
        # Is CP + IS + FUN = TRUE?
        assert (
            base * c.Value()
            + p.Value()
            + base * i.Value()
            + s.Value()
            + base * base * f.Value()
            + base * u.Value()
            + n.Value()
            == base * base * base * t.Value()
            + base * base * r.Value()
            + base * u.Value()
            + e.Value()
        )
        solution_count += 1
    solver.EndSearch()
    print(f"Number of solutions found: {solution_count}")


if __name__ == "__main__":
    main()

C++

// Cryptarithmetic puzzle
//
// First attempt to solve equation CP + IS + FUN = TRUE
// where each letter represents a unique digit.
//
// This problem has 72 different solutions in base 10.
#include <cstdint>
#include <vector>

#include "absl/flags/flag.h"
#include "absl/log/flags.h"
#include "ortools/base/init_google.h"
#include "ortools/base/logging.h"
#include "ortools/constraint_solver/constraint_solver.h"

namespace operations_research {

// Helper functions.
IntVar* MakeBaseLine2(Solver* s, IntVar* const v1, IntVar* const v2,
                      const int64_t base) {
  return s->MakeSum(s->MakeProd(v1, base), v2)->Var();
}

IntVar* MakeBaseLine3(Solver* s, IntVar* const v1, IntVar* const v2,
                      IntVar* const v3, const int64_t base) {
  std::vector<IntVar*> tmp_vars;
  std::vector<int64_t> coefficients;
  tmp_vars.push_back(v1);
  coefficients.push_back(base * base);
  tmp_vars.push_back(v2);
  coefficients.push_back(base);
  tmp_vars.push_back(v3);
  coefficients.push_back(1);

  return s->MakeScalProd(tmp_vars, coefficients)->Var();
}

IntVar* MakeBaseLine4(Solver* s, IntVar* const v1, IntVar* const v2,
                      IntVar* const v3, IntVar* const v4, const int64_t base) {
  std::vector<IntVar*> tmp_vars;
  std::vector<int64_t> coefficients;
  tmp_vars.push_back(v1);
  coefficients.push_back(base * base * base);
  tmp_vars.push_back(v2);
  coefficients.push_back(base * base);
  tmp_vars.push_back(v3);
  coefficients.push_back(base);
  tmp_vars.push_back(v4);
  coefficients.push_back(1);

  return s->MakeScalProd(tmp_vars, coefficients)->Var();
}

void CPIsFunCp() {
  // Instantiate the solver.
  Solver solver("CP is fun!");

  const int64_t kBase = 10;

  // Define decision variables.
  IntVar* const c = solver.MakeIntVar(1, kBase - 1, "C");
  IntVar* const p = solver.MakeIntVar(0, kBase - 1, "P");
  IntVar* const i = solver.MakeIntVar(1, kBase - 1, "I");
  IntVar* const s = solver.MakeIntVar(0, kBase - 1, "S");
  IntVar* const f = solver.MakeIntVar(1, kBase - 1, "F");
  IntVar* const u = solver.MakeIntVar(0, kBase - 1, "U");
  IntVar* const n = solver.MakeIntVar(0, kBase - 1, "N");
  IntVar* const t = solver.MakeIntVar(1, kBase - 1, "T");
  IntVar* const r = solver.MakeIntVar(0, kBase - 1, "R");
  IntVar* const e = solver.MakeIntVar(0, kBase - 1, "E");

  // We need to group variables in a vector to be able to use
  // the global constraint AllDifferent
  std::vector<IntVar*> letters{c, p, i, s, f, u, n, t, r, e};

  // Check if we have enough digits
  CHECK_GE(kBase, letters.size());

  // Define constraints.
  solver.AddConstraint(solver.MakeAllDifferent(letters));

  // CP + IS + FUN = TRUE
  IntVar* const term1 = MakeBaseLine2(&solver, c, p, kBase);
  IntVar* const term2 = MakeBaseLine2(&solver, i, s, kBase);
  IntVar* const term3 = MakeBaseLine3(&solver, f, u, n, kBase);
  IntVar* const sum_terms =
      solver.MakeSum(solver.MakeSum(term1, term2), term3)->Var();

  IntVar* const sum = MakeBaseLine4(&solver, t, r, u, e, kBase);

  solver.AddConstraint(solver.MakeEquality(sum_terms, sum));

  int num_solutions = 0;
  // Create decision builder to search for solutions.
  DecisionBuilder* const db = solver.MakePhase(
      letters, Solver::CHOOSE_FIRST_UNBOUND, Solver::ASSIGN_MIN_VALUE);
  solver.NewSearch(db);
  while (solver.NextSolution()) {
    LOG(INFO) << "C=" << c->Value() << " " << "P=" << p->Value() << " "
              << "I=" << i->Value() << " " << "S=" << s->Value() << " "
              << "F=" << f->Value() << " " << "U=" << u->Value() << " "
              << "N=" << n->Value() << " " << "T=" << t->Value() << " "
              << "R=" << r->Value() << " " << "E=" << e->Value();

    // Is CP + IS + FUN = TRUE?
    CHECK_EQ(p->Value() + s->Value() + n->Value() +
                 kBase * (c->Value() + i->Value() + u->Value()) +
                 kBase * kBase * f->Value(),
             e->Value() + kBase * u->Value() + kBase * kBase * r->Value() +
                 kBase * kBase * kBase * t->Value());
    num_solutions++;
  }
  solver.EndSearch();
  LOG(INFO) << "Number of solutions found: " << num_solutions;
}

}  // namespace operations_research

int main(int argc, char** argv) {
  InitGoogle(argv[0], &argc, &argv, true);
  absl::SetFlag(&FLAGS_stderrthreshold, 0);
  operations_research::CPIsFunCp();
  return EXIT_SUCCESS;
}

Java

// Cryptarithmetic puzzle
//
// First attempt to solve equation CP + IS + FUN = TRUE
// where each letter represents a unique digit.
//
// This problem has 72 different solutions in base 10.
package com.google.ortools.constraintsolver.samples;
import com.google.ortools.Loader;
import com.google.ortools.constraintsolver.DecisionBuilder;
import com.google.ortools.constraintsolver.IntVar;
import com.google.ortools.constraintsolver.Solver;

/** Cryptarithmetic puzzle. */
public final class CpIsFunCp {
  public static void main(String[] args) throws Exception {
    Loader.loadNativeLibraries();
    // Instantiate the solver.
    Solver solver = new Solver("CP is fun!");

    final int base = 10;

    // Decision variables.
    final IntVar c = solver.makeIntVar(1, base - 1, "C");
    final IntVar p = solver.makeIntVar(0, base - 1, "P");
    final IntVar i = solver.makeIntVar(1, base - 1, "I");
    final IntVar s = solver.makeIntVar(0, base - 1, "S");
    final IntVar f = solver.makeIntVar(1, base - 1, "F");
    final IntVar u = solver.makeIntVar(0, base - 1, "U");
    final IntVar n = solver.makeIntVar(0, base - 1, "N");
    final IntVar t = solver.makeIntVar(1, base - 1, "T");
    final IntVar r = solver.makeIntVar(0, base - 1, "R");
    final IntVar e = solver.makeIntVar(0, base - 1, "E");

    // Group variables in a vector so that we can use AllDifferent.
    final IntVar[] letters = new IntVar[] {c, p, i, s, f, u, n, t, r, e};

    // Verify that we have enough digits.
    if (base < letters.length) {
      throw new Exception("base < letters.Length");
    }

    // Define constraints.
    solver.addConstraint(solver.makeAllDifferent(letters));

    // CP + IS + FUN = TRUE
    final IntVar sum1 =
        solver
            .makeSum(new IntVar[] {p, s, n,
                solver.makeProd(solver.makeSum(new IntVar[] {c, i, u}).var(), base).var(),
                solver.makeProd(f, base * base).var()})
            .var();
    final IntVar sum2 = solver
                            .makeSum(new IntVar[] {e, solver.makeProd(u, base).var(),
                                solver.makeProd(r, base * base).var(),
                                solver.makeProd(t, base * base * base).var()})
                            .var();
    solver.addConstraint(solver.makeEquality(sum1, sum2));

    int countSolution = 0;
    // Create the decision builder to search for solutions.
    final DecisionBuilder db =
        solver.makePhase(letters, Solver.CHOOSE_FIRST_UNBOUND, Solver.ASSIGN_MIN_VALUE);
    solver.newSearch(db);
    while (solver.nextSolution()) {
      System.out.println("C=" + c.value() + " P=" + p.value());
      System.out.println(" I=" + i.value() + " S=" + s.value());
      System.out.println(" F=" + f.value() + " U=" + u.value());
      System.out.println(" N=" + n.value() + " T=" + t.value());
      System.out.println(" R=" + r.value() + " E=" + e.value());

      // Is CP + IS + FUN = TRUE?
      if (p.value() + s.value() + n.value() + base * (c.value() + i.value() + u.value())
              + base * base * f.value()
          != e.value() + base * u.value() + base * base * r.value()
              + base * base * base * t.value()) {
        throw new Exception("CP + IS + FUN != TRUE");
      }
      countSolution++;
    }
    solver.endSearch();
    System.out.println("Number of solutions found: " + countSolution);
  }

  private CpIsFunCp() {}
}

C#

// Cryptarithmetic puzzle
//
// First attempt to solve equation CP + IS + FUN = TRUE
// where each letter represents a unique digit.
//
// This problem has 72 different solutions in base 10.
using System;
using Google.OrTools.ConstraintSolver;

public class CpIsFunCp
{
    public static void Main(String[] args)
    {
        // Instantiate the solver.
        Solver solver = new Solver("CP is fun!");

        const int kBase = 10;

        // Decision variables.
        IntVar c = solver.MakeIntVar(1, kBase - 1, "C");
        IntVar p = solver.MakeIntVar(0, kBase - 1, "P");
        IntVar i = solver.MakeIntVar(1, kBase - 1, "I");
        IntVar s = solver.MakeIntVar(0, kBase - 1, "S");
        IntVar f = solver.MakeIntVar(1, kBase - 1, "F");
        IntVar u = solver.MakeIntVar(0, kBase - 1, "U");
        IntVar n = solver.MakeIntVar(0, kBase - 1, "N");
        IntVar t = solver.MakeIntVar(1, kBase - 1, "T");
        IntVar r = solver.MakeIntVar(0, kBase - 1, "R");
        IntVar e = solver.MakeIntVar(0, kBase - 1, "E");

        // Group variables in a vector so that we can use AllDifferent.
        IntVar[] letters = new IntVar[] { c, p, i, s, f, u, n, t, r, e };

        // Verify that we have enough digits.
        if (kBase < letters.Length)
        {
            throw new Exception("kBase < letters.Length");
        }

        // Define constraints.
        solver.Add(letters.AllDifferent());

        // CP + IS + FUN = TRUE
        solver.Add(p + s + n + kBase * (c + i + u) + kBase * kBase * f ==
                   e + kBase * u + kBase * kBase * r + kBase * kBase * kBase * t);

        int SolutionCount = 0;
        // Create the decision builder to search for solutions.
        DecisionBuilder db = solver.MakePhase(letters, Solver.CHOOSE_FIRST_UNBOUND, Solver.ASSIGN_MIN_VALUE);
        solver.NewSearch(db);
        while (solver.NextSolution())
        {
            Console.Write("C=" + c.Value() + " P=" + p.Value());
            Console.Write(" I=" + i.Value() + " S=" + s.Value());
            Console.Write(" F=" + f.Value() + " U=" + u.Value());
            Console.Write(" N=" + n.Value() + " T=" + t.Value());
            Console.Write(" R=" + r.Value() + " E=" + e.Value());
            Console.WriteLine();

            // Is CP + IS + FUN = TRUE?
            if (p.Value() + s.Value() + n.Value() + kBase * (c.Value() + i.Value() + u.Value()) +
                    kBase * kBase * f.Value() !=
                e.Value() + kBase * u.Value() + kBase * kBase * r.Value() + kBase * kBase * kBase * t.Value())
            {
                throw new Exception("CP + IS + FUN != TRUE");
            }
            SolutionCount++;
        }
        solver.EndSearch();
        Console.WriteLine($"Number of solutions found: {SolutionCount}");
    }
}