Journal articles: 'Sudoku puzzle'

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Mat Diah, Norizan, Syahirul Riza, Suzana Ahmad, Norzilah Musa, and Shakirah Hashim. "Sudoku solutions: a comparative analysis of breadth-first search, depth-first search, and human approaches." Journal of Education and Learning (EduLearn) 19, no. 1 (2025): 561–69. http://dx.doi.org/10.11591/edulearn.v19i1.21214.

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Sudoku is a puzzle that has a unique solution. No matter how many methods are used, the result will always be the same. The player thought that the number of givens or clues, the initial value on the Sudoku puzzles, would significantly determine the difficulty level, which is not necessarily correct. This research uses two search algorithms, breadth-first search (BFS) and depth-first search (DFS), to solve a set of Sudoku puzzles based on the number of givens. The Sudoku puzzles are chosen based on the number of givens between 32 and 35. In cases where Sudoku puzzles are considered medium or intermediate difficulty, the solutions generated by both algorithms will be compared with the human-solving approach. The research aims to determine whether humans tend to solve Sudoku puzzles with solutions resembling those generated by BFS or DFS. Furthermore, if all three approaches-human, BFS, DFS-yield comparable solutions, the Sudoku puzzle has only one unique solution.

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Chen, Kaiqi. "Exploring A Better Way to Constraint Propagation Using Naked Pair." ITM Web of Conferences 70 (2025): 04025. https://doi.org/10.1051/itmconf/20257004025.

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Sudoku is an NP-complete problem therefore developing various efficient algorithms is crucial. This paper presents an enhanced approach to solving sudoku puzzles by improving on recursive backtracking with constraint propagation and bitmask, focusing on the implementation of the naked pair technique. This research aims to add constraints to reduce the backtracking steps in solving sudoku using naked pairs, which is a technique that eliminates candidates of a cell from other cells in the same row, column or sub-grid when two cells share the same candidate sets. This research uses a dataset of 1 million sudoku puzzlers from Kaggle to evaluate the proposed solver’s performance. The solver first processes the puzzle into bitmask vectors then uses the singles and naked pairs technique to minimize the candidates, and then uses depth-first search to backtrack and solve the puzzle. As a result, the naked pair strategy slightly increases the total steps, and it significantly reduces the computation of depth-first search steps, making the solver potentially more efficient in solving difficult puzzles.

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G M, Ananya, and Ankit Kumar Singh. "Augmented Reality Sudoku Solver." ECS Transactions 107, no. 1 (2022): 2085–93. http://dx.doi.org/10.1149/10701.2085ecst.

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Solving Sudoku puzzles is now made easy through a few mobile applications. However, these applications fail to provide a fast and accurate solution. Computer Vision and Deep Learning have advanced to a level where another dimension for solving such puzzles has been added. The paper aims to propose a novel method to generate the solution of Sudoku puzzles and to display it using Augmented Reality. The Sudoku puzzle is recognized over several image processing phases, followed by recognition of letters using Deep Learning. The captured puzzle is then solved by a solver written in Rust. Eventually, the resultant mask of the solution is merged with the original image by computer vision and uses Augmented Reality to render the solution. The end-to-end pipeline takes milliseconds before an accurate final solution is obtained. The solver is extremely reliable in terms of precision and speed.

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Umar Khan, Muhammad, Muaaz Mulla, and Dr M. Upendra Kumar. "Optimized real-time sudoku puzzle solving and solution overlay on images and videos using advanced computer vision and TensorFlow." International Journal of Multidisciplinary Research and Growth Evaluation 5, no. 1 (2024): 66–70. http://dx.doi.org/10.54660/.ijmrge.2024.5.1.66-70.

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This paper presents a novel approach to solving Sudoku puzzles using computer vision techniques. Sudoku, a popular logic-based number placement game, poses an interesting challenge for automated solutions. The proposed Sudoku solver employs image processing and computer vision algorithms to extract the puzzle grid from a given image. The system then employs optical character recognition (OCR) to recognize and digitize the numbers within the grid. The image preprocessing involves techniques such as edge detection, contour identification, and perspective transformation to enhance the accuracy of grid extraction. Subsequently, OCR is applied to recognize the digits within each cell of the Sudoku grid. The solver then utilizes a backtracking algorithm to systematically fill in the missing numbers while adhering to the rules of Sudoku. The computer vision-based Sudoku solver offers advantages in terms of versatility, as it can handle puzzles of varying difficulty levels and appearances. The system's performance is evaluated on a diverse set of Sudoku puzzles, demonstrating its effectiveness in accurately solving puzzles with different grid sizes and layouts. The results indicate that the proposed approach achieves competitive solving times while maintaining a high level of accuracy. The presented Sudoku solver serves as a testament to the potential of computer vision in solving complex puzzles, opening avenues for further research in the integration of artificial intelligence and image processing for recreational problem-solving applications.

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Saecker, Mary. "Chemistry Sudoku Puzzle." Journal of Chemical Education 84, no. 4 (2007): 577. http://dx.doi.org/10.1021/ed084p577.

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Burgess, David L. "Lessons from Sudoku." EDFA Technical Articles 8, no. 1 (2006): 25–28. http://dx.doi.org/10.31399/asm.edfa.2006-1.p025.

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Abstract Failure analysis requires a working knowledge of physics, technology, materials, manufacturing practices, analytical techniques, and more. It is difficult to imagine what failure analysis would be like if all the laws of physics were perfectly apparent and all the challenges of sample preparation and testing were removed. Eliminating those considerations, failure analysis would be more like Sudoku. Sudoku is a simple number puzzle enjoyed by both children and adults. The simplicity of Sudoku compared to failure analysis is the point. Although Sudoku is starkly simple, the process is challenging to the best of us. Good and bad problem-solving practices are clearly exposed by the experience of solving a few Sudoku puzzles. Some insights that are brilliantly clear in Sudoku are equally true in failure analysis.

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Manisha. "Sudoku Solver – A Novel Approach Using Recognition of Printed Text Image." Journal of Computer Science and Technology Studies 6, no. 4 (2024): 38–46. http://dx.doi.org/10.32996/jcsts.2024.6.4.6.

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Detecting and analyzing text in multimedia is required for recognition of characters which can be used for performing multi-objectives in numerous research areas and applications. Characters or numbers in images can be hand-written, printed document, typewritten or artificially added for the purpose of searching or indexing, editing or image understanding. Various applications such as License Plate Detection, Handwritten documents recognition, Sudoku Solving, multi-oriented recognition focuses on character recognition. Image Sudoku solver works on Sudoku puzzle images. In this paper, the printed text Sudoku image is used as the input. The image is enhanced by filling the horizontal and vertical gaps in the grids that may occur due to shadow or distortion of the scanned image such that all the 81 grids are extracted. Secondly, the numbers and grids in the enhanced image are extracted and a dancing link approach using Algorithm X is adopted to solve the Sudoku. The proposed work is compared with a backtracking approach and dancing link approach without enhancing the image and its computation time for various ranks of difficulty of a puzzle taken. The Sudoku puzzle can be used to secure information stored in an image as the number of solutions to solve a Sudoku puzzle is very large, allowing it to be used as a security tool resistant to brute force attacks.

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Pu-Sheng Tsai, Pu-Sheng Tsai, Ter-Feng Wu Pu-Sheng Tsai, Jen-Yang Chen Ter-Feng Wu, and Jen-Feng Huang Jen-Yang Chen. "Integrating of Image Processing and Number Recognition in Sudoku Puzzle Cards Digitation." 網際網路技術學刊 23, no. 7 (2022): 1573–84. http://dx.doi.org/10.53106/160792642022122307012.

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The aim of Sudoku puzzle is to fill in the blank cells in a square matrix consisting of nine 3x3 blocks with the numbers 1-9 in a way that no number appears more than once in any row, column, or block. We combined image processing, a convolutional neural network (CNN), and a Sudoku game algorithm to automatically place the numbers 1-9 in the blank cells of a Sudoku square matrix. An image of the Sudoku square matrix is first captured using a camera, and then the vertical lines, horizontal lines, and outer frame of the Sudoku square matrix are detected using Hough transform (HT). Based on the OpenCV module, we proposed an image processing algorithm that captures the numbers in the image and calculates the location coordinates of the numbers in the image. We trained the CNN using the MNIST handwritten digit dataset to recognize the numbers in the Sudoku square matrix. Finally, we used the Python programming language to design a Sudoku puzzle backtrace algorithm that automatically deduces and fills in the blank cells in the square matrix. This study provides further understanding of the critical operating principles of CNNs and lays down a foundation for future research. &nbsp;

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Hoffman, Dean. "Super Sudoku Squares." Communications on Number Theory and Combinatorial Theory 1, no. 1 (2020): 1–8. http://dx.doi.org/10.70013/wpbj2rt4.

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We refer to a completed Sudoku puzzle as a Sudoku Square of order 3. We define a more restricted, but natural, extension --- a Super Sudoku Square of order n, and determine for which orders n one exists.

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Soto, Ricardo, Broderick Crawford, Cristian Galleguillos, Fernando Paredes, and Enrique Norero. "A Hybridalldifferent-Tabu Search Algorithm for Solving Sudoku Puzzles." Computational Intelligence and Neuroscience 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/286354.

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The Sudoku problem is a well-known logic-based puzzle of combinatorial number-placement. It consists in filling an2 × n2grid, composed ofncolumns,nrows, andnsubgrids, each one containing distinct integers from 1 ton2. Such a puzzle belongs to the NP-complete collection of problems, to which there exist diverse exact and approximate methods able to solve it. In this paper, we propose a new hybrid algorithm that smartly combines a classic tabu search procedure with thealldifferentglobal constraint from the constraint programming world. Thealldifferentconstraint is known to be efficient for domain filtering in the presence of constraints that must be pairwise different, which are exactly the kind of constraints that Sudokus own. This ability clearly alleviates the work of the tabu search, resulting in a faster and more robust approach for solving Sudokus. We illustrate interesting experimental results where our proposed algorithm outperforms the best results previously reported by hybrids and approximate methods.

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Abudayah, Mohammad, Omar Alomari, and Torsten Sander. "Spectrum of free-form Sudoku graphs." Open Mathematics 16, no. 1 (2018): 1445–54. http://dx.doi.org/10.1515/math-2018-0125.

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AbstractA free-form Sudoku puzzle is a square arrangement ofm×mcells such that the cells are partitioned intomsubsets (called blocks) of equal cardinality. The goal of the puzzle is to place integers 1, . . ,min the cells such that the numbers in every row, column and block are distinct. Represent each cell by a vertex and add edges between two vertices exactly when the corresponding cells, according to the rules, must contain different numbers. This yields the associated free-form Sudoku graph. This article studies the eigenvalues of free-form Sudoku graphs, most notably integrality. Further, we analyze the evolution of eigenvalues and eigenspaces of such graphs when the associated puzzle is subjected to a ‘blow up’ operation, which scales the cell grid including its block partition.

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Nishikawa, Kohei, and Takahisa Toda. "Exact Method for Generating Strategy-Solvable Sudoku Clues." Algorithms 13, no. 7 (2020): 171. http://dx.doi.org/10.3390/a13070171.

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A Sudoku puzzle often has a regular pattern in the arrangement of initial digits and it is typically made solvable with known solving techniques called strategies. In this paper, we consider the problem of generating such Sudoku instances. We introduce a rigorous framework to discuss solvability for Sudoku instances with respect to strategies. This allows us to handle not only known strategies but also general strategies under a few reasonable assumptions. We propose an exact method for determining Sudoku clues for a given set of clue positions that is solvable with a given set of strategies. This is the first exact method except for a trivial brute-force search. Besides the clue generation, we present an application of our method to the problem of determining the minimum number of strategy-solvable Sudoku clues. We conduct experiments to evaluate our method, varying the position and the number of clues at random. Our method terminates within 1 min for many grids. However, as the number of clues gets closer to 20, the running time rapidly increases and exceeds the time limit set to 600 s. We also evaluate our method for several instances with 17 clue positions taken from known minimum Sudokus to see the efficiency for deciding unsolvability.

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Jones, Sian K. "Quantum Distribution of a Sudoku Key." Recreational Mathematics Magazine 3, no. 6 (2016): 87–94. http://dx.doi.org/10.1515/rmm-2016-0009.

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Abstract Sudoku grids are often cited as being useful in cryptography as a key for some encryption process. Historically transporting keys over an alternate channel has been very difficult. This article describes how a Sudoku grid key can be secretly transported using quantum key distribution methods whereby partial grid (or puzzle) can be received and the full key can be recreated by solving the puzzle.

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Indriyono, Bonifacius, Natalinda Pamungkas, Zudha Pratama, Ery Mintorini, Imelda Dimentieva, and Pita Mellati. "Comparative Analysis of the Performance Testing Results of the Backtracking and Genetics Algorithm in Solving Sudoku Games." International Journal of Artificial Intelligence & Robotics (IJAIR) 5, no. 1 (2023): 29–35. http://dx.doi.org/10.25139/ijair.v5i1.6501.

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Games that hone thinking skills and logical accuracy have recently been very popular. One of them is the game Sudoku. Sudoku is a game that hones logic through puzzles arranged in rows and columns. Sudoku is also defined as a puzzle game that aims to arrange several numbers in a grid from one to nine on a grid consisting of 9x9 squares. The concept of this Sudoku game is to enter numbers into the rows and columns provided. The rule of this game is that the numbers arranged on the board cannot be the same in every row, column, and 3x3 square in the grid. In another sense, each number entered must appear once in each row and column. When running Sudoku, several numbers are already instructions for players to fill in the next boxes. The number of clues at the beginning of the game determines the difficulty level players face. The fewer clues, the more difficult the Sudoku is to solve. This study aims to compare how to solve Sudoku using genetic algorithms, backtracking, and the completion time needed. The tests' results show that the genetic and backtracking algorithms can solve Sudoku games quickly. Still, the backtracking algorithm has the advantage of being relatively shorter, and the process is not so complicated that the backtracking algorithm can be an alternative solution to solving Sudoku logic games.

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Jonas Reycian B. Saraosos and Michael Anthony Jay B. Regis. "Development of an Android-Based Visual Sudoku Solver Using Contour Finding and Backtracking Algorithm." Journal of Science, Engineering and Technology (JSET) 6, no. 1 (2018): 208–17. http://dx.doi.org/10.61569/pw8s6r19.

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This study developed an Android application that automatically solves Sudoku puzzle. An image of the Sudoku puzzle is either captured by the built-in camera or loaded directly from the media gallery. Contour finding algorithm and Hough transform would detect the game board. The givens (i.e. number that exist in a Sudoku board) are acquired by dividing the entire game board by 81 cells with the contour finding algorithm extracting the particular given. Identification is done using the Tesseract – Optical Character Recognition (OCR). Backtracking algorithm calculated the solution and displayed it back in the form of an image. After testing 30 Sudoku images, a 74.47% detection percentage and 13.09% branching factor with a Receiver Operating Characteristic - Area Under the Curve (AUC) of 0.684 was obtained when identifying the givens indicating that the system is effective.

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Louis Lee, N. Y., Geoffrey P. Goodwin, and P. N. Johnson-Laird. "The psychological puzzle of Sudoku." Thinking & Reasoning 14, no. 4 (2008): 342–64. http://dx.doi.org/10.1080/13546780802236308.

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Bukhori, Iksan, Jason Felix, and Saddam Ali. "Using Genetic Algorithm to Solve Puzzle Games: A Review." Journal of Computer Networks, Architecture and High Performance Computing 6, no. 1 (2024): 201–11. http://dx.doi.org/10.47709/cnahpc.v6i1.3348.

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Puzzles have been recognized for their development as a popular form of entertainment due to their ability to intricately challenge the mind while engendering creativity in the player. The development of puzzle games has given rise to a new generation of puzzle games characterized by diverse sequences and different image variations. With the rapid development of puzzle games, we looked at solving approaches using Genetic Algorithms (GA). In this paper, we try to analyze several puzzle games such as Sliding Blocks, Sudoku, Tic-Tac-Toe, and Jigsaw that can be solved using GA. We found that 120 papers have examined the use of GA for puzzle games, and eliminated into 14 papers. We evaluated these 14 papers for each puzzle game we selected by comparing the chromosome representation, GA operator, GA parameters, and the results. Based on the discussion, the application of GA to solve puzzle games can be effectively executed with a high degree of accuracy. Puzzle games that use measurement methods such as Sliding Block, Sudoku, and Jigsaw run in a similar pattern. What is common to all of them is that the chromosomes are represented as matrices or arrays in all cases, and standard genetic operators such as selection, crossover, and mutation are used. The population size is large, often 1000 chromosomes, and parameters such as mutation rate are kept low, around 5%. On the other hand, the performance of GA for solving Tetris and Tic-Tac-Toe from each publication cannot be compared due to different measurement methods and metrics.

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Nikita, Ms. "Sudoku Solving Algorithm and Grid Based Models for Digit Recognition." International Journal for Research in Applied Science and Engineering Technology 11, no. 5 (2023): 6251–58. http://dx.doi.org/10.22214/ijraset.2023.52828.

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Abstract: In recent information technology advances, computer vision has been exalted. Due to its broad range of uses and features, it has gained attention for resolving difficult problems in the real world that are beyond the capacity of human intellect. This marvel of artificial intelligence becomes practical and convenient when it is applied to everyday life. Sudoku has been a common source of entertainment in many articles in newspapers, magazines, and applications. The conclusion of this research is to use computer vision techniques to solve the Sudoku puzzle. We used some well-known tools, including Open CV, OCR, and Tensorflow. The strategy we utilised to employ these tools to solve the Sudoku puzzle is described in the paper. The paper's technique includes a number of stages, including.

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Naji, Areen, and Saleh Salous. "Sudoku Puzzle Difficulty Rating based on Fuzzy Logic." Journal of Computer Science and Technology Studies 6, no. 2 (2024): 86–91. http://dx.doi.org/10.32996/jcsts.2024.6.2.10.

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In this document, we propose a new fuzzy logic-based rating technique for Sudoku difficulty, which takes into account Sudoku instance parameters such as the number of filled cells as well as parameters relating to the distribution of filled numbers on the cells. This new technique is validated using historical data from a certificate paper [Mantere, 2008], which includes 45 Sudoku instances of all rank levels, three of each level, and the average/max time consumed in 100 runs using different algorithms for each instance. First and foremost, these instances were analyzed and parameterized, and their parameters were quantitatively analyzed to be considered in fuzzy logic. The instance parameters' correlation with their solving time is studied, and dimensionality reduction was performed on these as variables to ensure that no unnecessary variable was included in the study. As solving time parameters, the number of filled cells in the instance, the minimum number of filled cells in rows and columns, and the number of empty sub-squares (3*3) in the instance are all accepted. Because there should be a functional relationship between the Sudoku rank and the time required to solve it, a linear regression model was performed on the historical data between the old rank and the solving time, and the same regression model was performed on the new rank to validate it. As a result, a new clear and simple ranking technique that outputs more correlated ranks with the time required to solve Sudoku puzzles is validated.

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Indriyono, Bonifacius Vicky, Ratna Wardani, Titin Susanti, et al. "Analysis of the Concept of Solving the Sudoku Game Using Backtracking and Brute Force Algorithms." International Journal of Artificial Intelligence & Robotics (IJAIR) 6, no. 1 (2024): 40–47. http://dx.doi.org/10.25139/ijair.v6i1.8377.

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Sudoku is a puzzle game that arranges numbers from one to nine on a board consisting of 9x9 squares. This game has a rule: the numbers arranged on a board cannot be the same in each row or column. The main goal of this Sudoku game is to complete the board and win by entering numbers into each empty box. Therefore, in this Sudoku game, there is only one valid solution. At the start of the game, several numbers are provided, which serve as clues for players to fill in the next box. The number of clues at the start of the game determines whether the Sudoku game level is difficult. The fewer the clues given, the more difficult it is to solve Sudoku. Solving Sudoku problems cannot be done manually, so several algorithms are needed, which require a lot of repetition and search functions. In this research, we will compare concepts in solving Sudoku using the Backtracking algorithm and the backtracking algorithm using the brute force technique. This comparison is seen from the effectiveness of solving Sudoku. The results of this research conclude that to solve Sudoku logic quickly and efficiently, the backtracking algorithm can be used compared to the Brute Force algorithm. This is because the backtracking algorithm has better complexity, and the solution steps are fast and “smart”. In contrast, the brute algorithm takes longer to complete because the brute force algorithm has to search for all the possibilities. The average time required to complete 10 9 x 9 puzzle tests with backtracking is 10910.96 ms, while the brute force algorithm takes longer, around 15871.90 ms.

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Mishra, Priyanka, and Rakesh Ahuja. "A Novel Image Watermarking Method Against Crop Attack Using Two-Step Sudoku Puzzle." ECS Transactions 107, no. 1 (2022): 8351–60. http://dx.doi.org/10.1149/10701.8351ecst.

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Digital image watermarking is used to secure the digital images. There are several methods for securing the images. But there is probability to destroy the information by an attacker. Attacker can damage even a single segment of watermark image. There are many techniques to overcome from several attacks through which the cropped section of an image can be recover and there is no need to detect watermark. A robust image watermarking method against various cropping attack by the use of Sudoku puzzle is suggested in this paper. This method is build upon the attributes of permutation of Sudoku matrix and it gives Sudoku table has 9x9 formats, i.e. uniformly divided into multiple sample of images in the sections of the original image. Justified Sudoku result is explained for the embedded and extraction of the watermark image. The standard format of Sudoku having 9x9 matrix is used and the method expressed the hardiness up to great extent against crop attack.

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Job, Dhanya, and Varghese Paul. "Recursive Backtracking for Solving 9*9 Sudoku Puzzle." Bonfring International Journal of Data Mining 6, no. 1 (2016): 07–09. http://dx.doi.org/10.9756/bijdm.8128.

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BIRD, RICHARD. "FUNCTIONAL PEARL: A program to solve Sudoku." Journal of Functional Programming 16, no. 6 (2006): 671–79. http://dx.doi.org/10.1017/s0956796806006058.

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Wnuk, Emil, and Edyta Łukasik. "Analiza porównawcza algorytmów rozwiązujących Sudoku." Journal of Computer Sciences Institute 2 (December 30, 2016): 140–43. http://dx.doi.org/10.35784/jcsi.130.

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Artykuł przedstawia analizę porównawczą wybranych algorytmów, służących do rozwiązywania Sudoku. Porównywane są puzzle różnej wielkości, posiadające różny poziom trudności. Sprawdzany jest wpływ tych czynników na czas działania algorytmów. Na potrzeby badań stworzono specjalną aplikację, która zawiera wszystkie potrzebne algorytmy.

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Cheewaprakobkit, Pimpa, Timothy K. Shih, Timothy Lau, Yu-Cheng Lin, and Chih-Yang Lin. "Rule-based explaining module: Enhancing the interpretability of recurrent relational network in Sudoku solving." Machine Graphics and Vision 32, no. 3/4 (2023): 125–45. http://dx.doi.org/10.22630/mgv.2023.32.3.7.

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Computer vision has gained significant attention in the field of information technology due to its widespread application that addresses real-world challenges, surpassing human intelligence in tasks such as image recognition, classification, natural language processing, and even game playing. Sudoku, a challenging puzzle that has captivated many people, exhibits a complexity that has attracted researchers to leverage deep learning techniques for its solution. However, the reliance on black-box neural networks has raised concerns about transparency and explainability. In response to this challenge, we present the Rule-based Explaining Module (REM), which is designed to provide explanations of the decision-making processes using Recurrent Relational Networks (RRN). Our proposed methodology is to bridge the gap between complex RRN models and human understanding by unveiling the specific rules applied by the model at each stage of the Sudoku puzzle solving process. Evaluating REM on the Minimum Sudoku dataset, we achieved an accuracy of over 98.00%.

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Mishra, Pallavi, D. K. Gupta, and Rakesh P. Badoni. "A new algorithm for enumerating all possible Sudoku squares." Discrete Mathematics, Algorithms and Applications 08, no. 02 (2016): 1650026. http://dx.doi.org/10.1142/s1793830916500269.

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Sudoku is an interesting number placement puzzle with some simple rules. For some positive integer [Formula: see text], the puzzle involves an [Formula: see text] grid partitioned into [Formula: see text] distinct blocks each of size [Formula: see text] such that the integers [Formula: see text] through [Formula: see text] appear exactly once in each row, each column and each block of the grid. Often some static numbers known as givens are placed according to the difficulty rating in the puzzle and then the rule is to place the numbers from [Formula: see text] to [Formula: see text] in the partially filled grid such that the conditions of the puzzle are satisfied. The aim of this paper is to propose a new algorithm for enumerating all possible Sudoku squares of size [Formula: see text]. It is based on the concepts of permutations derived from [Formula: see text] S-permutation matrices termed as S-permutations. There is a one-to-one correspondence between Sudoku squares and the set of those S-permutation matrices which are mutually disjoint to each other. The proposed algorithm uses the set of all S-permutations generated by some permutation generation algorithm as input and then enumerates all the subsets of cardinality [Formula: see text] of all S-permutations which are mutually disjoint to each other. The correctness of the algorithm is established. Its worst-case time complexity is O[Formula: see text], where [Formula: see text]. A mathematical formula involving the total number of the sets of [Formula: see text] mutually disjoint S-permutations is also derived. Experimentally, the proposed algorithm is verified for the Sudoku squares of size [Formula: see text]. It is observed that our algorithm is more systematic and better in terms of computational efficiency.

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Johnson, Mark R. "Casual Games Before Casual Games: Historicizing Paper Puzzle Games in an Era of Digital Play." Games and Culture 14, no. 2 (2018): 119–38. http://dx.doi.org/10.1177/1555412018790423.

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This article examines “paper puzzle games”—crosswords, Sudoku, Kakuro, word searches, and so forth—in order to historicize and contextualize “casual games,” complicate our notions of “casual” play, and open up paper puzzle games to game studies consideration for the first time. The article begins by identifying the dearth of literature on paper puzzle games and offers an initial examination of these games through the lens of casual games, play, and players. It focuses on six traits in casual game design: appealing themes, ease of access, ease of learning, minimal required expertise, fast rewards, and temporal flexibility. It demonstrates that—from a perspective of mechanics, demographics, and contexts of play—paper puzzle games are excellent examples of casual games and therefore important to fully study. It also shows the complexity of paper puzzles as a topic in their own right, opening them up for future examination.

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Moraglio, Alberto, Cecilia Di Chio, Julian Togelius, and Riccardo Poli. "Geometric Particle Swarm Optimization." Journal of Artificial Evolution and Applications 2008 (February 21, 2008): 1–14. http://dx.doi.org/10.1155/2008/143624.

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Using a geometric framework for the interpretation of crossover of recent introduction, we show an intimate connection between particle swarm optimisation (PSO) and evolutionary algorithms. This connection enables us to generalise PSO to virtually any solution representation in a natural and straightforward way. The new Geometric PSO (GPSO) applies naturally to both continuous and combinatorial spaces. We demonstrate this for the cases of Euclidean, Manhattan and Hamming spaces and report extensive experimental results. We also demonstrate the applicability of GPSO to more challenging combinatorial spaces. The Sudoku puzzle is a perfect candidate to test new algorithmic ideas because it is entertaining and instructive as well as being a nontrivial constrained combinatorial problem. We apply GPSO to solve the Sudoku puzzle.

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Rai, Deepika. "POLYNOMIAL 3-SAT REDUCTION OF SUDOKU PUZZLE." International Journal of Advanced Research in Computer Science 9, no. 3 (2018): 194–97. http://dx.doi.org/10.26483/ijarcs.v9i3.6096.

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Sidiakina, A., and I. Ovcharuk. "SIMULATION OF THE GAME OF SUDOKU: MATHEMATICAL APPROACHES, STRATEGIES." POLISH JOURNAL OF SCIENCE, no. 64 (July 17, 2023): 57–59. https://doi.org/10.5281/zenodo.8153558.

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The article highlights some mathematical approaches and strategies used for modeling games of mathematical direction. In this case, the considered methods are used for the game "Sudoku" of different levels of complexity.

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Eleftheriou, Maria, та Nikoletta Christodoulou. "Η Επίδραση της Γνωστικής Ανάπτυξης στη Συνεργατική Επίλυση Προβλήματος με τη χρήση του γρίφου Sudoku σε Μαθητές Στ΄ Δημοτικού". Preschool and Primary Education 6, № 1 (2018): 35. http://dx.doi.org/10.12681/ppej.14109.

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Collaborative learning is a subject that has occupied many researchers throughout the world. Many researchers have maintained that when students of all school stages, from kindergarten to higher education work in teams, this leads to high performance (Johnson, Skon, & Johnson, 1980), but also has a positive impact emotionally and psychologically (Schmitz & Winskel, 2008). However, there have been studies claiming that teamwork does not actually result in any substantive improvement in students’ efficiency (Samuelsson, 2010), or that only some students gain from this method of learning (Sears & Reagin, 2013). The present study explores how cognitive development relates to problem solving in pairs or individually, in students of the sixth grade in school using the numerical puzzle Sudoku. More specifically, the study explored four different types of pairs of two, according to the level of cognitive development (high – high, high – low, average – average and low – low) and three different types of units (high, average, low), all taken from sixth graders of a public elementary school. Additionally, the research studied whether solving Sudoku was more effective in pairs rather than individually. The sample of this research included thirteen sixth grade classes from a public elementary school in a city in Cyprus. The city was intentionally selected to facilitate the research. Two hundred twenty students completed the tool of cognitive development Group Assessment of Logical Thinking (GALT). Then, 60 students individually solved the Sudoku puzzle, while 160 students solved the puzzle in pairs. The results of the qualitative study showed that there were differences between the four types of pairs in terms of how they collaborated. Although students showed interest or even enthusiasm in solving the Sudoku puzzle, the majority of them did not work together so as to improve their performance. The results of the quantitative study confirmed that students failed to cooperate. The performance of the Individual Problem Solving Ability was not different among the four types of pairs. The majority of high-level cognitive development pairs started to solve the puzzle competitively. However, in the process they worked together to solve the puzzle correctly. They also behaved in an intensely self-centred way. In non-homogeneous pairs, low-performing students had a passive role in the group. Additionally, it seemed that the level of cooperation was related to high performance. The research demonstrated that students had difficulty in developing combinational thinking. This was the reason they could not solve the puzzle. Pairs did not cooperate, despite the fact that they had clear guidelines to do that. This finding should be a concern for teachers and the educational system of Cyprus, in general. The role of the teacher should be supportive in helping students overcome their difficulties, considering the theory of Vygotsky (2012) on systematic facilitating, development, and the Zone of Proximal Development.

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XIAO, Hua-yong, Hai-jiao CHENG, and Yue-xing WANG. "Research of equation solution algorithm about Sudoku puzzle." Journal of Computer Applications 32, no. 10 (2013): 2907–10. http://dx.doi.org/10.3724/sp.j.1087.2012.02907.

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Masadeh, Shadi R., Hamza Abbass Al Sewadi, and Mohammad Abbas Fadhil Al Husainy. "A message encryption scheme inspired by Sudoku puzzle." International Journal of Information and Computer Security 21, no. 3/4 (2023): 399–413. http://dx.doi.org/10.1504/ijics.2023.132739.

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Angeli, Alvin, and Tukino Tukino. "RANCANG BANGUN GAME SUDOKU ANDROID BERBASIS FLUTTER." Computer Based Information System Journal 12, no. 1 (2024): 52–62. http://dx.doi.org/10.33884/cbis.v12i1.8249.

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Perkembangan teknologi, khususnya dalam penggunaan gadget, telah menjadi bagian integral dalam kehidupan sehari-hari, terutama di tengah era pandemi Covid-19. Penggunaan gadget di Indonesia telah meningkat pesat, tercapai 50,92% pada tahun 2018. Namun, kecanduan game online, yang berdampak negatif pada perkembangan anak-anak dan orang dewasa, menjadi salah satu dampak negatif dari penggunaan gadget yang berlebihan. Oleh karena itu, perlu adanya alternatif hiburan yang juga memiliki nilai edukatif, seperti permainan puzzle Sudoku. Penelitian ini bertujuan untuk merancang dan mengembangkan aplikasi game Sudoku berbasis Flutter yang memiliki antarmuka pengguna intuitif, kustomisasi permainan yang bervariasi, dan fitur edukatif. Metode pengembangan perangkat lunak yang digunakan adalah model Waterfall, dengan tahapan meliputi perencanaan, analisis, perancangan, implementasi, pengujian, dan evaluasi. Aplikasi Sudoku ini diharapkan dapat memberikan hiburan, meningkatkan kemampuan logika dan konsentrasi, serta menjadi referensi bagi pengembang aplikasi lain yang ingin mengintegrasikan Flutter dalam pengembangan game.

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Alfany, Fauzan Maulana, Christy Atika Sari, Cahaya Jatmoko, Deddy Award Widya Laksana, Candra Irawan, and Solichul Huda. "Completing Sudoku Games Using the Depth First Search Algorithm." Journal of Applied Intelligent System 9, no. 1 (2025): 88–100. https://doi.org/10.62411/jais.v9i1.10017.

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Sudoku is a digital game that is included in the type of logic-based puzzle game where the goal is to fill in the puzzle with random numbers. Therefore, in this research it is proposed to use Artificial Intelligence which contains the Depth First Search Algorithm to track the number of possible solutions that lead to only one so that it becomes efficient. This game has different levels of difficulty such as easy, medium and difficult. The time and complexity of execution will vary depending on the difficulty so it is proposed to use Android Studio software. The experimental results prove that there is an increase in playing the Sudoku game quickly and accurately by applying the Depth First Search Algorithm method. This is proven by the ability to complete this game using the Depth First Search Algorithm using the Android Studio programming language. The average time at the easy level is 11:04 minutes, at the normal level is 10:52 minutes, at the hard level is 25:46 minutes, and at the extreme level is 38 minutes.

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Binsasi, Eva, Elinora Naikteas Bano, and Cecilia Novianti Salsinha. "Permainan Matematika Sudoku di Sekolah Menangah Atas Katolik (SMAK) Warta Bhakti Kefamenanu." Dedication : Jurnal Pengabdian Masyarakat 3, no. 1 (2019): 1–6. http://dx.doi.org/10.31537/dedication.v3i1.179.

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Permainan matematika merupakan salah satu cara untuk membangun minat siswa dan merupakan langkah yang baik agar siswa tidak merasa bosan dalam belajar matematika. Salah satu permainan matematika yang cukup populer saat ini adalah pemainan matematika Sudoku. Puzzle Sudoku identik dengan angka dan cukup rumit jika belum paham aturan permainannya. Tujuan dari teka-teki ini adalah untuk menempatkan angka-angka pada baris dan kolom yang tersedia. Kegiatan ini bertujuan untuk membangkitkan minat belajar matematika siswa dengan metode permainan. Kegiatan ini dilaksanakan selama tiga (3) hari di SMAK Warta Bhakti Kefamenanu kelas X1 IPA 1 dengan jumlah siswa sebanyak 30 orang. Hasil yang dicapai dari kegiatan pengabdian ini adalah rata-rata hasil pre-test siswa sebelum diberikan permainan adalah 48 sedangkan rata-rata hasil post-test siswa setelah diberikan permainan adalah 87.59 . Hal ini menunjukkan bahwa setelah siswa diberikan pemahaman tentang cara bermain Sudoku, hasil yang diperoleh lebih memuaskan dibanding sebelum diberikan pemahaman mengenai permainan ini. Melalui kegiatan ini pula minat belajar matematika siswa dapat dikatakan meningkat.

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Al Husainy, Mohammed Abbas Fadhil, Hamza Abbass Al Sewadi, and Shadi Masadeh. "A Message Encryption Scheme inspired by Sudoku Puzzle (MESP)." International Journal of Information and Computer Security 1, no. 1 (2022): 1. http://dx.doi.org/10.1504/ijics.2022.10047993.

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Forbes, Tamsin, and Tony Forbes. "103.37 A sudoku puzzle with a big rectangular hole." Mathematical Gazette 103, no. 558 (2019): 517. http://dx.doi.org/10.1017/mag.2019.119.

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Ijeri, Sanmitra, Shivananda Pujeri, Shrikant B, and Usha B A. "Image Steganography using Sudoku Puzzle for Secured Data Transmission." International Journal of Computer Applications 48, no. 17 (2012): 31–35. http://dx.doi.org/10.5120/7443-0460.

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Isra, Natheer Alkallak, and Zedan Shaban Ruqaya. "Establishing a cyclic schedule for nurse in the health unit." International Journal of Electrical and Computer Engineering (IJECE) 12, no. 3 (2022): 2876–84. https://doi.org/10.11591/ijece.v12i3.pp2876-2884.

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This research presents the interleaving two approaches. These are intelligence's ideas as well as heuristic technique as 8-puzzle and sudoku grid to solve the nurse rostering. The research proposed algorithm to assign shifts cyclically. It is considered by three shifts in one day to 9 nurses, each nurse has 8 days work with a holiday in the cyclically scheduling. The task appeared the allocation of nursing staff in health unit management theoretically. There are three shifts which cover 24 hours. The shifts are early, late, and night. This algorithm simulated the shifts through the directions of blank’s move in 8-puzzle with the methodology of sudoku grid with hard constraints should be met at all times. In our solution do on two goals first, we create a schedule that meets all the tough constraints and guarantees fairness. The second objective is to try to verify as many of the soft constraints as possible, by shifting and rotating while maintaining the soft constraints. The approach was implemented as a simulation, and a satisfactory result was demonstrated. experimental effects are extremely convenient and versatile to find appropriate nursing rostering schedule, rather than using manual techniques. The code developed to simulate it in MATLAB.

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Tomaszewski, Radosław, Wojciech Walendziuk, Roman Trochimczuk, Adam Wolniakowski, Kacper Jaroszewicz, and Zdenek Slanina. "Mechatronics system for solving SuDoKu puzzle based on Cartesian robot." IFAC-PapersOnLine 58, no. 9 (2024): 149–53. http://dx.doi.org/10.1016/j.ifacol.2024.07.387.

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Khovanova, Tanya, and Wayne Zhao. "Mathematics of a Sudo-Kurve." Recreational Mathematics Magazine 5, no. 10 (2018): 5–28. http://dx.doi.org/10.2478/rmm-2018-0005.

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Abstract We investigate a type of a Sudoku variant called Sudo-Kurve, which allows bent rows and columns, and develop a new, yet equivalent, variant we call a Sudo-Cube. We examine the total number of distinct solution grids for this type with or without symmetry. We study other mathematical aspects of this puzzle along with the minimum number of clues needed and the number of ways to place individual symbols.

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Natheer Alkallak, Isra, and Ruqaya Zedan Shaban. "Establishing a cyclic schedule for nurse in the health unit." International Journal of Electrical and Computer Engineering (IJECE) 12, no. 3 (2022): 2876. http://dx.doi.org/10.11591/ijece.v12i3.pp2876-2884.

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This research presents the interleaving two approaches. These are intelligence's ideas as well as heuristic ‎technique as 8-puzzle and sudoku grid to solve the nurse rostering. The research proposed algorithm ‎to assign shifts cyclically. It is considered by three shifts in one day to 9 nurses, each nurse has 8 days work with ‎a holiday in the cyclically scheduling. The task appeared the allocation of nursing staff in health unit management ‎theoretically. There are three shifts which cover 24 hours. The shifts are early, late, and night. This algorithm ‎simulated the shifts through the directions of blank’s move in 8-puzzle with the methodology of sudoku grid with ‎hard constraints should be met at all times. In our solution do on two goals first, we create a ‎schedule that meets all the tough constraints and guarantees fairness. The second objective is to try to verify as many ‎of the soft constraints as possible, by shifting and rotating while maintaining the soft constraints. The approach ‎was implemented as a simulation, and a satisfactory result was demonstrated. experimental effects are extremely ‎convenient and versatile to find appropriate nursing rostering schedule, rather than using manual techniques. The code developed to simulate it in MATLAB.‎

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B, Pooja. "Intelligent System to Secure the Information based on Innovative Games." INTERANTIONAL JOURNAL OF SCIENTIFIC RESEARCH IN ENGINEERING AND MANAGEMENT 08, no. 05 (2024): 1–5. http://dx.doi.org/10.55041/ijsrem33862.

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Cryptography, while essential for data security, faces challenges such as interception during transmission, potentially exposing information to unauthorized parties. To combat this, game-based cryptography techniques have emerged, offering innovative solutions to safeguard sensitive data. This project focuses on developing security models inspired by popular games like Sudoku puzzles. These models aim to enhance data security while engaging users through interactive gameplay. As encryption gains increasing significance amidst rising data breaches and cyber threats, traditional methods often lack user engagement and accessibility. The proposed game-based encryption approach seeks to address these shortcomings by providing a more intuitive and interactive means of securing data across various complexity levels. By integrating puzzle elements into the encryption process, the project aims to bridge the gap between encryption experts and general users, making data security more understandable and engaging for everyone. Keywords : Cryptography, Encryption, Information, Games

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Maji, Arnab Kumar, Sunanda Jana, and Rajat Kumar Pal. "An Algorithm for Generating only Desired Permutations for Solving Sudoku Puzzle." Procedia Technology 10 (2013): 392–99. http://dx.doi.org/10.1016/j.protcy.2013.12.375.

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Tukaram, Moger, and Sai Krishna G. "An imperative SuDoKu reconfiguration method for photovoltaic array under partial shading conditions." Journal of India Chemical Society Vol. 97, No. 10b, Oct 2020 (2020): 1892–900. https://doi.org/10.5281/zenodo.5969293.

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Department of Electrical Engineering, National Institute of Technology Karnataka, Surathkal-575 025, Karnataka, India E-mail: saikrishna240@gmail.com Manuscript received online 21 May 2020, revised and accepted 26 August 2020 Partial shading condition (PSCs) is one of the major affecting problems that downwards the PV array performance. In this context, array reconfiguration is one of the useful techniques that maximize the PV array performance under PSCs. Reconfiguration is an approach to spread the shading effects uniformly over the array. This paper proposed an imperative SuDoKu puzzle reconfiguration connection to reduce the shading impacts on Total-Cross-Tied (TCT) PV array. In this approach, the actual locations of PV modules in the TCT array are modified based on the imperative SuDoKu, thereby the power output of the TCT can be improved. This method is tested on 9x9 array under various grouping shading conditions by obtained performance index parameters. Also, the proposed method validated with the other existed reconfiguration methods under PSCs.

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Chaker, Abderrahmen, and Hachemi Khalid. "Sudoku puzzle approach for the drugs assignment in an automated dispensing cabinets." Supply Chain Forum: An International Journal 21, no. 2 (2020): 103–16. http://dx.doi.org/10.1080/16258312.2020.1803023.

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Herimanto, P. Sitorus, and E. M. Zamzami. "An Implementation of Backtracking Algorithm for Solving A Sudoku-Puzzle Based on Android." Journal of Physics: Conference Series 1566 (June 2020): 012038. http://dx.doi.org/10.1088/1742-6596/1566/1/012038.

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Larios-García, Raúl, Roberto López-Duarte, Francisco Javier Ríos-Suriano, and Herminio Suárez-Quiroz. "A RANDOMIZED ALGORITHM FOR SOLVING $n$-DIMENSIONAL SUDOKU PUZZLE WITH CONTINUOUS DIFFICULTY LEVEL." Far East Journal of Mathematical Education 16, no. 1 (2016): 1–11. http://dx.doi.org/10.17654/me016010001.

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Smith, Adam. "Answer Set Programming in Proofdoku." Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment 13, no. 2 (2021): 118–24. http://dx.doi.org/10.1609/aiide.v13i2.12972.

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Proofdoku is an AI-based game design that extends Sudoku. In addition to playing by the rules of the traditional logic puzzle, players must explain their reasoning. An AI system checks this reasoning and provides hints that guide the player to discover new reasoning patterns for themselves. Co-developing the game design and the AI system, implemented using the technology of Answer Set Programming (ASP), guided us to include features that depend on high-complexity combinatorial search and optimization. We developed Proofdoku to better understand the implications of designing and deploying game systems that depend on ASP for live interaction. This paper offers design tradeoffs and makes suggestions for future deployments of ASP-backed game designs.

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Journal articles on the topic 'Sudoku puzzle'

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