Crack the passcode

2025-12-09 · Source: Yet Another Math Programming Consultant · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Advanced, medium

Summary

This analysis demonstrates how to solve a 3-digit passcode puzzle using a Mixed Integer Programming (MIP) model, where each digit is an integer from 0 to 9. The approach uses binary variables, $\color{darkred}x_{i,k}$, to represent whether digit $i$ has value $k$. Constraints are formulated for five hints provided: "One number is correct and well placed" (e.g., for $|6|8|2|$), "One number is correct but wrongly placed" (e.g., for $|6|1|4|$), "Two numbers are correct but wrongly placed" (for $|2|0|6|$), "Nothing is correct" (for $|7|3|8|$), and "One number is correct but wrongly placed" (for $|7|8|0|$). The MIP model, implemented in GAMS, yields a unique solution of $|0|4|2|$, with the presolver eliminating all rows and columns, indicating the problem was solved without needing the branch & bound algorithm. The analysis also debunks a claim that hints 4 and 5 are unnecessary, showing that removing them leads to incorrect or multiple solutions.

Key takeaway

For Data Scientists or AI Researchers tackling logic puzzles or constraint satisfaction problems, consider formulating them as Mixed Integer Programming (MIP) models. This approach ensures precision in problem interpretation and leverages powerful solvers to find unique solutions, often more robustly than manual deduction. Your ability to translate natural language rules into formal mathematical constraints is key to unlocking automated solutions for complex logical challenges.

Key insights

MIP modeling offers a precise, systematic approach to solve logic puzzles by translating natural language into mathematical constraints.

Principles

• Binary variables simplify complex logical constraints.
• Mathematical models force precise problem interpretation.
• Presolvers can solve many MIP problems directly.

Method

Translate each puzzle hint into a distinct mathematical constraint using binary variables $\color{darkred}x_{i,k}$, then combine these into a MIP model with a dummy objective for feasibility, and solve.

In practice

• Use binary variables for digit-based puzzles.
• Formulate constraints hint-by-hint.
• Verify solution uniqueness with a "no-good" constraint.

Topics

• Mixed Integer Programming
• Mathematical Modeling
• Constraint Programming
• GAMS
• Puzzle Solving

Best for: AI Researcher, Data Scientist, AI Student

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Editorial summary, takeaway, and curation by AIssential. Original article published by Yet Another Math Programming Consultant.