Clock problem

2025-11-11 · Source: Yet Another Math Programming Consultant · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Software Development & Engineering, Data Science & Analytics · Depth: Advanced, long

Summary

This analysis presents a mathematical programming model to determine the time displayed on a clock with unmarked hands of equal length. The problem involves three smoothly moving hands, with two positioned exactly on hour markers and one between a minute and hour marker. The model defines hand positions using relative positions and an offset, incorporating modular arithmetic to handle the 12-hour cycle. It assigns roles (hour, minute, second) to hands using a fragment of an assignment model and linearizes the product of binary and continuous variables to maintain linearity. The solution, derived from a feasibility problem, indicates the time is 4:50, with the hour hand at 4.833 and the minute hand at 10.000 (decimal hours). The analysis also contrasts this approach with a ChatGPT-generated Pyomo model, highlighting differences in unit representation, linearization, constraint tightness, and variable typing.

Key takeaway

For Data Scientists or Operations Research professionals developing optimization models, this analysis demonstrates robust techniques for handling cyclical systems and nonlinearities. You should prioritize explicit interval definitions for uncertain inputs and employ exact linearization methods for products of binary and continuous variables. This approach ensures model accuracy and robustness, contrasting with less precise LLM-generated solutions that may require significant debugging and refinement.

Key insights

A clock puzzle can be precisely solved using mathematical programming, addressing modular arithmetic and variable assignment.

Principles

• Modular arithmetic is essential for cyclical systems.
• Linearization can transform nonconvex quadratic problems into linear ones.
• Tight bounds are crucial for robust mathematical programming models.

Method

Model hand positions using relative offsets and modular arithmetic, assign roles via binary variables, and linearize products of binary and continuous variables to solve as a Mixed-Integer Program.

In practice

• Apply linearization techniques for products of binary and continuous variables.
• Use explicit intervals for uncertain input data in models.
• Define variable domains carefully to prevent ambiguity (e.g., [0, 11.999]).

Topics

• Mathematical Programming
• Mixed-Integer Programming
• Clock Puzzle
• Linearization Techniques
• GAMS

Best for: Research Scientist, Data Scientist, Software Engineer

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