Difficult Sudoku Puzzles Created by Replica Exchange Monte Carlo Method

TL;DR

This paper introduces a novel algorithm using replica exchange Monte Carlo to generate extremely difficult Sudoku puzzles by minimizing a specially designed Hamiltonian, achieving some of the hardest puzzles known.

Contribution

It presents a new method combining Ising spin-glass models and replica exchange Monte Carlo to systematically create difficult Sudoku puzzles, including the hardest known to date.

Findings

Successfully created the world's hardest Sudoku puzzle by their method.

The method efficiently searches for low-energy (difficult) puzzles.

The difficulty measure used may not accurately reflect human solvability.

Abstract

An algorithm to create difficult Sudoku puzzles is proposed. An Ising spin-glass like Hamiltonian describing difficulty of puzzles is defined, and difficult puzzles are created by minimizing the energy of the Hamiltonian. We adopt the replica exchange Monte Carlo method with simultaneous temperature adjustments to search lower energy states efficiently, and we succeed in creating a puzzle which is the world hardest ever created in our definition, to our best knowledge. (Added on Mar. 11, the created puzzle can be solved easily by hand. Our definition of the difficulty is inappropriate.)