Sudoku Solving and Finding Magic Squares by Probability Models and Markov Chains

TL;DR

This paper introduces probabilistic models and Markov chains for solving Sudoku puzzles and constructing magic squares, demonstrating their effectiveness on various sizes.

Contribution

It presents a novel probabilistic approach using Markov chains to solve Sudoku and generate magic squares, extending methods to larger grid sizes.

Findings

Markov chain sampling effectively finds Sudoku solutions.

Probabilistic models successfully generate magic squares of size 8x8 and 10x10.

Method extends to other combinatorial puzzles.

Abstract

The sudoku puzzles have a long history, with variations going back more than a hundred years, but its current and perhaps surprising world-wide prominence goes back to certain initiatives and then puzzle-generating computer programmes from just after 2000. To solve a sudoko puzzle, a statistician can put up a probabilitymodel on the enormous space of $9\times9$ matrix possibilities, constructed to favour `good attempts', and then engineer a Markov chain to sample a long enough chain of sudoku table realisations from that model, until the solution is found. The methods work also for other types of puzzles, like constructing `magic squares' with wished-for properties (sums of rows, columns, diagonals equal, etc.), as is also illustrated in this article; via magic models and equally magic Markov chains I find impressively magic $8\times8$ and $10\times10$ squares.