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

\[\begin{aligned} v&=100x+10y+z \end{aligned}\]

We find all solutions satisfying:

\[\begin{aligned} xyz &= 252 \end{aligned}\]

QUBO++ program

The following QUBO++ program finds all solutions:

#include <qbpp/qbpp.hpp>
#include <qbpp/exhaustive_solver.hpp>

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

  auto f = qbpp::cons(x * y * z == 252);

  f.simplify();
  auto solver = qbpp::ExhaustiveSolver(f);
  auto sols = solver.search({{"best_energy_sols", 0}});
  for (const auto& sol : sols) {
    std::cout << sol(v) << " ";
  }
  std::cout << std::endl;
}

In this program, x, y, and t are defined as native integer variables with the ranges above. Then z, v, and f are defined as expressions. f declares the condition $xyz = 252$ as a constraint with qbpp::cons(), so its energy takes the minimum value 0 only when the condition holds. We create an Exhaustive Solver instance for f and store all optimal solutions in sols.

Because x, y, and t are handled as integers, each solution corresponds to exactly one digit triple (x,y,z), and no triple appears more than once. Printing the value of v for each solution gives:

479 497 667 749 947