Kryptarithmetische Rätsel

kryptarithmetisches Puzzle ist eine mathematische Übung, bei der die Ziffern einiger Zahlen werden durch Buchstaben (oder Symbole) dargestellt. Jeder Buchstabe steht für eine eindeutige Ziffern. Das Ziel ist es, die Ziffern so zu finden, dass eine gegebene mathematische Gleichung Bestätigt:

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

Eine Zuweisung von Buchstaben zu Ziffern ergibt die folgende Gleichung:

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

Es gibt andere Antworten auf dieses Problem. Wir zeigen Ihnen, wie Sie alle Lösungen finden können.

Problem modellieren

Wie bei jedem Optimierungsproblem beginnen wir mit der Identifizierung von Variablen und Einschränkungen. Die Variablen sind die Buchstaben, die eine beliebige einstellige Zahl annehmen können. Wert.

Für CP + IS + FUN = TRUE gelten die folgenden Einschränkungen:

• Die Gleichung: CP + IS + FUN = TRUE.
• Jeder der zehn Buchstaben muss eine andere Ziffer sein.
• C, I, F und T dürfen nicht Null sein (da wir keine führenden Nullen in Zahlen).

Kryptarithmetische Probleme lassen sich mit dem neuen CP-SAT-Löser lösen. effizienter oder der ursprüngliche CP-Löser ist. Wir zeigen Ihnen Beispiele mit beiden Matherechnern, beginnend mit CP-SAT.

CP-SAT-Lösung

Wir zeigen die Variablen, Einschränkungen, den Solver-Aufruf und schließlich alle Programme ansehen.

Bibliotheken importieren

Mit dem folgenden Code wird die erforderliche Bibliothek importiert.

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;

Modell deklarieren

Mit dem folgenden Code wird das Modell für das Problem deklariert.

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()}");
    }
}

Variablen definieren

Bei der Verwendung des CP-SAT-Lösers sind bestimmte Hilfsmethoden nützlich, definieren. Wir verwenden eine davon, NewIntVar, um die Ziffern (Ganzzahl) zu deklarieren. Wir unterscheiden zwischen Buchstaben, die potenziell Null sein können, und Buchstaben, nicht möglich (C, I, F und 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 };

Einschränkungen definieren

Als Nächstes: Einschränkungen. Zunächst stellen wir sicher, dass alle Buchstaben unterschiedliche Werte haben, mithilfe der Hilfsmethode AddAllDifferent. Dann verwenden wir den Helper AddEquality zum Erstellen von Einschränkungen, die die CP + IS + FUN = TRUE-Gleichheit erzwingen.

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);

Lösungsdrucker

Der Code für den Lösungsdrucker, der jede Lösung als Löser anzeigt finden, wie unten dargestellt.

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_;
}

Löser aufrufen

Schließlich lösen wir das Problem und zeigen die Lösung an. Die ganze Magie steckt in 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);

Wenn Sie das Programm ausführen, wird die folgende Ausgabe angezeigt, in der jede Zeile eine Lösung darstellt:

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

Programme abschließen

Hier sind die vollständigen Programme.

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()}");
    }
}

Ursprüngliche CP-Lösung

In diesem Fall behandeln wir die Basis als Variable, damit Sie die Gleichung lösen können. für höhere Basen. (Es gibt keine Lower-Base-Lösungen für CP + IS + FUN = TRUE, da die zehn Buchstaben unterschiedlich sein müssen.)

Bibliotheken importieren

Mit dem folgenden Code wird die erforderliche Bibliothek importiert.

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;

Rechner erstellen

Der erste Schritt besteht darin, Solver zu erstellen.

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!");

Variablen definieren

Der erste Schritt besteht darin, eine IntVar für jeden Buchstaben zu erstellen. Wir unterscheiden zwischen Die Buchstaben, die potenziell Null sein können, und die, die dies nicht können (C, I, F, und T).

Als Nächstes erstellen wir ein Array, das einen neuen IntVar für jeden Buchstaben enthält. Dies ist nur notwendig, denn wenn wir unsere Einschränkungen definieren, AllDifferent. Wir benötigen also ein Array, bei dem sich jedes Element unterscheiden muss.

Zum Schluss überprüfen wir, ob unsere Basis mindestens so groß ist wie die Anzahl der Buchstaben. sonst gibt es keine Lösung.

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");
}

Einschränkungen definieren

Nachdem wir nun die Variablen definiert haben, besteht der nächste Schritt im Definieren der Einschränkungen. Zuerst fügen wir die Einschränkung AllDifferent hinzu und erzwingen, dass jeder Buchstabe mit einer anderen Ziffer.

Als Nächstes fügen wir die Einschränkung CP + IS + FUN = TRUE hinzu. Die Beispielprogramme tun dies, auf unterschiedliche Weise nutzen.

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);

Löser aufrufen

Da wir nun die Variablen und Einschränkungen haben, können wir sie lösen.

Der Code für den Lösungsdrucker, der jede Lösung als Löser anzeigt gefunden hat, sehen Sie unten.

Da es mehr als eine Lösung für unser Problem gibt, iterieren wir die Lösungen mit einer while solver.NextSolution()-Schleife. Wenn wir nur versuchen würden, suchen wir mit dieser Schreibweise:\

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}");

Programme abschließen

Hier sind die vollständigen Programme.

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}");
    }
}