Automated Prediction of Sudoku Puzzle Difficulty Using Convolutional Neural Networks

Abstract

Predicting the difficulty of mathematical games is a complex task that can be achieved by combining combinatorial analysis with machine learning. In this study, we propose a convolutional neural network-based approach to predict the difficulty level of Sudoku games by considering both the structural attributes of the puzzles and their meta-features. The analysis is conducted over a dataset that consists of about 4 million puzzle grids, each labeled with its solution, number of clues, and difficulty level, which is annotated from 1 (very easy) to 5 (very hard). The puzzle grids are represented as 9x9 numerical arrays, which contain the digits from 0 to 9, where 0s represent the empty positions. On the other hand, the number of clues, which may vary from 17 to 80, is scaled to [0,1] and processed through a small dense branch. In order to make regression-based predictions, we have trained the model with mean absolute error (MAE) as the loss function. The experimental results reached a validation MAE of 0.112, which indicates highly accurate predictions, since the deviation from the true difficulty labels is minimal. This framework that effectively combines the structure of the puzzles and their meta-information for automated difficulty prediction can contribute to game design and game classification in the development of educational tools and adaptive game challenges. Additionally, we have demonstrated that by choosing a regression approach over classification, we have been tracking the closeness of the difficulty levels without ignoring the distance between them.