ألغاز التشفير

الألغاز التشفيرية هي تمرين رياضي تجتمع فيه أرقام بعض يتم تمثيل الأرقام بأحرف (أو رموز). يمثل كل حرف قيمة فريدة رقم. والهدف من ذلك هو إيجاد الأرقام التي تكون بها معادلة رياضية معينة تم التحقق منه:

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

يؤدي تعيين أحرف واحد إلى أرقام إلى الحصول على المعادلة التالية:

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

هناك إجابات أخرى حول هذه المشكلة. سنوضح كيفية إيجاد جميع الحلول.

نمذجة المشكلة

وكما هو الحال مع أي مشكلة متعلقة بالتحسين، سنبدأ بتحديد المتغيرات القيود. المتغيرات هي الأحرف، التي يمكن أن تتخذ أي رقم واحد

بالنسبة لـ CP + IS + FUN = TRUE، تكون القيود كما يلي:

• المعادلة: CP + IS + FUN = TRUE.
• يجب أن يكون كل حرف من الأحرف العشرة رقمًا مختلفًا.
• لا يمكن أن تكون قيم C وI وF وT صفرًا (لأننا لا نكتب الأصفار البادئة في الأرقام).

يمكنك حل المسائل الحسابية باستخدام أداة حلّ CP-SAT الجديدة، أكثر كفاءة، أو أداة حل CP الأصلية. سنعرض لك أمثلة باستخدام كل من أدوات الحلّ، بدءًا من CP-SAT.

حلول CP-SAT

سنعرض المتغيرات والقيود واستدعاء أداة الحلّ، وأخيرًا البرامج الكاملة.

استيراد المكتبات

يستورد الرمز التالي المكتبة المطلوبة.

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;

تعريف النموذج

يوضح الرمز التالي نموذج المشكلة.

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()}");
    }
}

تعريف المتغيرات

وعند استخدام أداة حل CP-SAT، هناك بعض الطرق المساعدة التي من المفيد التعريف. سنستخدم أحد هذه الأرقام، وهو NewIntVar، للإشارة إلى الأرقام (الصحيحة). نحن نميّز بين الأحرف التي يُحتمل أن تكون صفرًا وتلك التي لا يمكن إرسالها (C وI وF وT).

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 };

تحديد القيود

بعد ذلك، القيود. أولاً، نتأكد من أن جميع الأحرف لها قيم مختلفة، باستخدام طريقة AddAllDifferent المساعدة. بعد ذلك، نستخدم مساعد AddEquality لإنشاء قيود تفرض المساواة في 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);

طابعة الحلول

الرمز البرمجي لطابعة الحلول، والذي يعرض كل حل على أنّه أداة الحلّ يعثر عليه، كما هو موضح أدناه.

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_;
}

جارٍ استدعاء أداة الحلّ

وأخيرًا نحل المشكلة ونعرض الحل. كل السحر يكمن في 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);

عند تشغيل البرنامج، يعرض الإخراج التالي، والذي يكون فيه كل صف حلاً:

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

إكمال البرامج

إليك البرامج الكاملة

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()}");
    }
}

حلّ CP الأصلي

وفي هذه الحالة، سنتعامل مع القاعدة بوصفها متغيرًا، بحيث يمكنك حل المعادلة للقواعد الأعلى. (لا يمكن أن تكون هناك حلول أساسية أقل CP + IS + FUN = TRUE بما أنّ الأحرف العشرة يجب أن تكون مختلفة.)

استيراد المكتبات

يستورد الرمز التالي المكتبة المطلوبة.

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;

جارٍ إنشاء أداة الحلّ

الخطوة الأولى هي إنشاء 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!");

تعريف المتغيرات

الخطوة الأولى هي إنشاء IntVar لكل حرف. نفرق بين الأحرف التي يُحتمل أن تكون صفرًا وتلك التي لا يمكن تنفيذها (C أو I أو F أو وT).

بعد ذلك، ننشئ صفيفًا يحتوي على IntVar جديد لكل حرف. هذا فقط ضروريًا لأنه عندما نحدد قيودنا، سنستخدم AllDifferent، لذا نحتاج إلى بعض المصفوفة التي يجب أن يختلف بها كل عنصر.

وأخيرًا، نتحقق من أن قاعدتنا لا تقل عن عدد الأحرف؛ وإلا، فلا يوجد حل.

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");
}

تحديد القيود

الآن بعد أن حددنا المتغيرات، فإن الخطوة التالية هي تحديد القيود. أولاً، نضيف القيد AllDifferent، ما يجب أن يحتوي كل حرف على رقم مختلف.

بعد ذلك، نضيف القيد CP + IS + FUN = TRUE. تقوم نماذج البرامج بذلك بطرق مختلفة.

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);

جارٍ استدعاء أداة الحلّ

الآن بعد أن أصبح لدينا المتغيرات والقيود، نحن جاهزون لحلها.

الرمز البرمجي لطابعة الحلول، والذي يعرض كل حل على أنّه أداة الحلّ يعثر عليه، كما هو موضح أدناه.

نظرًا لوجود أكثر من حل لمشكلتنا، نكرر باستخدام التكرار الحلقي while solver.NextSolution(). إذا كنا نحاول فقط لإيجاد حل واحد، فسنستخدم هذا المصطلح:\

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}");

إكمال البرامج

إليك البرامج الكاملة

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}");
    }
}