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The chaos within Sudoku.

Mária Ercsey-Ravasz1, Zoltán Toroczkai

  • 1Faculty of Physics, Babeş-Bolyai University , Str. Kogalniceanu Nr. 1, RO-400084 Cluj-Napoca, Romania. ercsey.ravasz@phys.ubbcluj.ro

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PubMed
Summary
This summary is machine-generated.

Sudoku puzzle difficulty is mathematically linked to chaotic dynamics in a novel dynamical system. An escape rate invariant quantifies puzzle hardness, correlating with human ratings and defining a new scale.

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Area of Science:

  • Complexity Science
  • Computational Mathematics
  • Statistical Mechanics

Background:

  • Sudoku puzzles represent complex constraint satisfaction problems with applications in diverse scientific fields.
  • Understanding the inherent difficulty of such problems is crucial for computational and theoretical advancements.

Purpose of the Study:

  • To establish a mathematical framework connecting Sudoku puzzle difficulty to dynamical systems theory.
  • To develop a quantitative, objective measure of Sudoku puzzle hardness.

Main Methods:

  • Exact mapping of Sudoku puzzles to a deterministic, continuous-time dynamical system.
  • Analysis of transient chaotic behavior within the dynamical system.
  • Calculation of the escape rate (κ) as an invariant of transient chaos.

Main Results:

  • Sudoku puzzle difficulty directly correlates with the transient chaotic behavior of the mapped dynamical system.
  • The escape rate (κ) serves as a reliable scalar measure of puzzle hardness.
  • A new logarithmic scale (η = -log₁₀κ) for puzzle hardness is proposed, aligning with human difficulty perception.

Conclusions:

  • Transient chaos in dynamical systems provides a powerful tool for quantifying the difficulty of constraint satisfaction problems like Sudoku.
  • The proposed 'Richter'-type scale offers an objective method for classifying Sudoku puzzle hardness, from easy to ultra-hard.