3-Digit Math Problem

Let us solve the following math problem using PyQBPP.

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:

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

PyQBPP program

The following PyQBPP program finds all solutions:

import pyqbpp as qbpp

x = qbpp.between(qbpp.var_int("x"), 1, 9)
y = qbpp.between(qbpp.var_int("y"), 0, 9)
t = qbpp.between(qbpp.var_int("t"), 0, 4)
z = 2 * t + 1
v = x * 100 + y * 10 + z

f = x * y * z == 252

f.simplify_as_binary()
solver = qbpp.ExhaustiveSolver(f)
result = solver.search({"best_energy_sols": 0})
results = set()
for sol in result.sols():
    results.add(sol(v))
for val in sorted(results):
    print(val, end=" ")
print()

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 set to remove duplicates by collecting only the resulting integer values v.

The integers are printed as follows:

479 497 667 749 947