Using Small MUSes to Explain How to Solve Pen and Paper Puzzles

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Pen and paper puzzles like Sudoku are popular among users. There are numerous AI methods to solve them; nevertheless, they do not use the same techniques as human players. That means that current solvers cannot explain their reasoning in a meaningful way.

A recent research paper suggests a general technique, based on MUSes (Minimal Unsatisfiable Subsets), to solve pen and paper puzzles and explain the solutions.

Solving sudoku. Image credit: Pxhere, CC0 Public Domain

As standard MUS finding algorithms are inefficient for large puzzles such as Sudoku, the researchers propose a way to find individual small MUSes. The efficiency of the proposed algorithm is verified experimentally. It outperforms previous approaches in terms of generality and the quality of explanations. It is shown that the suggested explanations closely match the ones used by real players on a variety of puzzles and tutorials.

Pen and paper puzzles like Sudoku, Futoshiki and Skyscrapers are hugely popular. Solving such puzzles can be a trivial task for modern AI systems. However, most AI systems solve problems using a form of backtracking, while people try to avoid backtracking as much as possible. This means that existing AI systems do not output explanations about their reasoning that are meaningful to people. We present Demystify, a tool which allows puzzles to be expressed in a high-level constraint programming language and uses MUSes to allow us to produce descriptions of steps in the puzzle solving. We give several improvements to the existing techniques for solving puzzles with MUSes, which allow us to solve a range of significantly more complex puzzles and give higher quality explanations. We demonstrate the effectiveness and generality of Demystify by comparing its results to documented strategies for solving a range of pen and paper puzzles by hand, showing that our technique can find many of the same explanations.

Research paper: Espasa, J., Gent, I. P., Hoffmann, R., Jefferson, C., and Lynch, A. M., “Using Small MUSes to Explain How to Solve Pen and Paper Puzzles”, 2021. Link:

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