3-Digit Math Problem

Let us solve the following math problem using QUBO++.

Math Problem: Find all three-digit odd integers whose product of digits is 252.

Let $x$, $y$, and $z$ be the hundreds, tens, and ones digits of the integer, respectively. More specifically:

• $x$ is an integer in $[1, 9]$,
• $y$ is an integer in $[0, 9]$,
• $t$ is an integer in $[0, 4]$,
• $z = 2t + 1$ (so $z$ is odd).

Then the value $v$ of the three-digit integer $xyz$ is

We find all solutions satisfying:

QUBO++ program

The following QUBO++ program finds all solutions:

#include <set>

#include "qbpp.hpp"
#include "qbpp_exhaustive_solver.hpp"

int main() {
  auto x = 1 <= qbpp::var_int("x") <= 9;
  auto y = 0 <= qbpp::var_int("y") <= 9;
  auto t = 0 <= qbpp::var_int("t") <= 4;
  auto z = 2 * t + 1;
  auto v = x * 100 + y * 10 + z;

  auto f = x * y * z == 252;

  f.simplify_as_binary();
  auto solver = qbpp::exhaustive_solver::ExhaustiveSolver(f);
  auto sols = solver.search_optimal_solutions();
  std::set<int> s;
  for (const auto& sol : sols) {
    s.insert(static_cast<int>(sol(v)));
  }
  for (auto v : s) {
    std::cout << v << " ";
  }
  std::cout << std::endl;
}

In this program, x, y, and t are defined as integer variables with the ranges above. Then z, v, and f are defined as expressions. We create an Exhaustive Solver instance for f and store all optimal solutions in sols.

Because x, y, and t are encoded by multiple binary variables, different binary assignments can represent the same integer values. As a result, the same digit triple (x,y,z) may appear multiple times in sols. Therefore, we use a std::set<int> named s to remove duplicates by collecting only the resulting integer values v.

The integers in s are printed as follows:

479 497 667 749 947