Sudoku Problem Research Articles

A primordial DNA store and compute engine.

Any modern information system is expected to feature a set of primordial features and functions: a substrate stably carrying data; the ability to repeatedly write, read, erase, reload and compute on specific data from that substrate; and the overall ability to execute such functions in a seamless and programmable manner. For nascent molecular information technologies, proof-of-principle realization of this set of primordial capabilities would advance the vision for their continued development. Here we present a DNA-based store and compute engine that captures these primordial capabilities. This system comprises multiple image files encoded into DNA and adsorbed onto ~50-μm-diameter, highly porous, hierarchically branched, colloidal substrate particles comprised of naturally abundant cellulose acetate. Their surface areas are over 200 cm2 mg-1 with binding capacities of over 1012 DNA oligos mg-1, 10 TB mg-1 or 104 TB cm-3. This 'dendricolloid' stably holds DNA files better than bare DNA with an extrapolated ability to be repeatedly lyophilized and rehydrated over 170 times compared with 60 times, respectively. Accelerated ageing studies project half-lives of ~6,000 and 2 million years at 4 °C and -18 °C, respectively. The data can also be erased and replaced, and non-destructive file access is achieved through transcribing from distinct synthetic promoters. The resultant RNA molecules can be directly read via nanopore sequencing and can also be enzymatically computed to solve simplified 3 × 3 chess and sudoku problems. Our study establishes a feasible route for utilizing the high information density and parallel computational advantages of nucleic acids.

Analysis of the Concept of Solving the Sudoku Game Using Backtracking and Brute Force Algorithms

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.

Puzzle—Integer Linear Programming: Spreadsheet Solver Excellence Without Excel

Because of the limitations on problem size with the built-in optimization solver in Microsoft Excel, other software applications are required to illustrate the power of integer linear programming in spreadsheets. The built-in linear optimization solver in LibreOffice Calc allows much larger problems. In this paper, we build a spreadsheet in LibreOffice Calc to solve sudoku problems and a simple generalization that further illustrates the power of integer linear programming in spreadsheets. The generalization we consider, n-doku, also introduces an interesting puzzle that includes, as a special case, a union of multiple sudoku puzzles. Besides further demonstrating the power of integer linear programming in spreadsheets, we use the n-doku solutions to develop optimization intuition and to, more generally, foster critical thinking. We use this spreadsheet in the classroom to transition from continuous to categorical decisions, emphasizing the idea of one-hot encoding. We also illustrate multiple optima and add preferences to sudoku puzzles as a way introduce multiobjective optimization. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ited.2022.0068 .

A Binary Integer Programming Approach to find a Sudoku Matrix

In the basic Sudoku game, players must fill the entries of a n × n matrix, except some given entries, under the conditions that each row, each column, and each m × m sub matrix contains integers 1 through n exactly once. In this research paper, in addition to the basic game, conditions and solution process for three advanced versions, such as Sudoku X, Four Square Sudoku and Four Pyramid Sudoku were also studied. In this study, the above-mentioned conditions are converted into appropriate mathematical forms as constraints of the Integer Linear Programming Problem using Pascal Programming Language, such a way that the objective function and the constraints suit as an input mathematical model for the LINGO mathematical optimization software. Here, an unlimited version of the LINGO software has been used with the built-in Branch and Bound algorithm as the number of constraints is very high. Finally, solutions to the given Sudoku problems obtained from the solutions acquired through LINGO software.

Formulation of Sudoku Puzzle Using Binary Integer Linear Programming and Its Implementation in Julia, Python, and Minizinc

Sudoku is a number puzzle game popular among people with various difficulty levels (easy, medium, hard, and extremely hard). Sudoku can be modeled as a linear programming problem in mathematics, particularly binary integer linear programming (BILP). Completing Sudoku using BILP is quite tricky because it requires many iterations. Therefore, this study aims to analyze the Sudoku problem using the BILP formulation and implement the problem using Julia, Python, and MiniZinc. Out of 15 cases for each difficulty level, Julia performs better than Python and MiniZinc based on computation time. Moreover, Sudoku with easy difficulty levels is solved with a longer computation time than the other three difficulty levels. The computation time for solving BILP is getting faster as the difficulty level of the Sudoku problem increases. This is because Sudoku problems with easy difficulty levels have more known values as clues and generate more constraints than other difficulty levels.

An analysis of Harmony Search for solving Sudoku puzzles

The Harmony Search metaheuristic has been used to solve many different optimization problems. Several papers examined its effectiveness for solving Sudoku puzzles. Another paper claims that it is ineffective for solving Sudoku puzzles and further that the method itself lacks novelty compared to other evolutionary algorithms. Our paper analyzes the search process in harmony search when applied to a specific Sudoku puzzle examined in earlier research. The basic harmony search procedure is re-implemented and tested to evaluate its performance and verify its applicability to the specific example. We found that the while the criticisms of the method for this problem are valid, that the performance can be improved with a rather simple modification. First, we propose a new objective function for the search procedure. This proposed objective function facilitates the search method to find a proper solution. Second, the modified version of the harmony search, where harmony search is combined with local search is introduced and analyzed for its contribution of ‘improvisation’ in harmony search procedure by comparing the performance of local search and the modified search. For a specific problem, the modified version of harmony search generates a unique solution with new objective function in favorable time. Then extended experiments were performed for various Sudoku problems. We find that while the modified search procedure produces solutions more quickly, that it suffers the same issue that the original method has in that it sometimes fails to find a feasible solution.

Neural networks and Monte-Carlo method usage in multi-agent systems for sudoku problem solving

The object of research is multi-agent systems based on Deep Reinforcement Learning algorithms and analysis of ways to establish interaction within the system, based on intelligent agents. Also, part of the material in this paper covers ways to organize the management and administration of agents at the meta-level: external controllers and tools to optimize their work, describing architectural solutions that should accelerate agents’ training. The studied full-fledged multi-agent system would be flexible to expansion and would give effective acceleration in agent training and problem-solving quality.In this paper, the following neural network models were considered: DQN, DDQN, PPO, TD (methods based on Q-Learning), an approach using a neural network with Monte-Carlo tree search. The presented models were tested on a Sudoku problem with a dataset of 5039 combinations, dimensions 2x2, 4x4, and 9x9. Several sets of agent rewards were used. The presentation of data during the learning and problem-solving process was described. Also was built a multi-agent system based on the model using a Monte-Carlo tree search.According to the study results, it was revealed that for tasks in a complex environment, the models based on Q-Learning are practically ineffective (plots support the statement). The training process for these models is quite demanding on the characteristics of the workstation hardware. It was also determined that the Monte-Carlo tree search method does a good job. Even with a small number of iterations, it shows results better than other Deep Learning methods (45–50 % accuracy for 9x9). However, a significant drawback is a complexity of training the model, and the hardware requirements are too large for this kind of research.

Application of sparse optimization in Sudoku

Sudoku is a difficult integer linear programming problem. The restriction of integer constraint is eliminated by real number coding and the integer linear programming model can be converted to the $\\ell_0$ norm minimization model. The existing algorithms a注 mostly used to solve the relaxed $\\ell_1$ norm minimization model, but by the $\\ell_1$ model some Sudoku problems cannot be solved. We prove that the $\\ell_q$ $(0<q<1)$ norm minimization model is equivalent to the $\\ell_0$ norm minimization model for a particular Sudoku problem and the $\\ell_{1/2}$ model is solved by $\\ell_{1/2}$-SLP (sequential linear programming) algori定理. In numerical experiments, $\\ell_{1/2}$-SLP algori定理 can solve mo注 Sudoku problems. The efficiency of the algori定理 is verified in terms of time and success rate.

Dynamic load balancing with learning model for Sudoku solving system

This paper proposes a dynamic load balancing with learning model for a Sudoku problem solving system that has multiple workers and multiple solvers. The objective is to minimise the total processing time of problem solving. Our load balancing with learning model distributes each Sudoku problem to an appropriate pair of worker and solver when it is received by the system. The information of the estimated solution time for a specific number of given input values, the estimated finishing time of each worker, and the idle status of each worker is used to determine the worker-solver pairs. In addition, the proposed system can estimate the waiting period for each problem. Test results show that the system has shorter processing time than conventional alternatives.

吴方法在求解数独问题中的应用

吴方法在求解数独问题中的应用

Society of Agents

The development of multiple agent systems faces many challenges, including agent coordination and collaboration on tasks. Minsky's The Society of Mind provides a conceptual view for addressing these multi-agent system problems. A new classification ontology is introduced for comparing multi-agent systems. Next, a new framework called the Society of Agents is developed from Minsky's conceptual foundation. A Society of Agents framework-based problem-solving and a Game Society is developed and applied to the domain of single player logic puzzles and two player games. The Game Society solved 100% of presented Sudoku and Kakuro problems and never lost a tic-tac-toe game. The advantage of the Society of Agents approach is the efficient re-utilization of agents across multiple independent game domain problems and a centralized problem-solving architecture with efficient cross-agent information sharing.

스도쿠 풀이에서 욕심쟁이 기법과 가지치기를 이용한 완전이진트리 생성 기법

스도쿠 풀이에서 욕심쟁이 기법과 가지치기를 이용한 완전이진트리 생성 기법

Olfactory training with older people.

Loss of olfactory function is largely found with aging. Such a reduction in olfactory function affects quality of life and enhances likelihood of depressive symptoms. Furthermore, it has been shown that reduction in olfactory function is associated with cognitive impairment and several diseases such as major depression. Because several studies suggest that discontinuous exposure to odors may improve general olfactory function, the primary aim of this study was to investigate whether such "olfactory training" has positive effects on subjective well-being and cognitive function. We performed a controlled, unblinded, longitudinal study SETTING: The study took place at an outpatients' clinic of a Department of Otorhinolaryngology at a Medical University. A total of 91 participants (age 50 to 84years) completed testing. They were randomly assigned to an olfactory training (OT) group (N=60) and a control group (N=31). The study included two appointments at the Smell and Taste Clinic. Olfactory and cognitive function as well as subjective well-being was tested using standardized tests. During the 5-month interval between sessions, the OT group completed daily olfactory exposure. During the same time, the control group completed daily Sudoku problems. Analyses show a significant improvement of olfactory function for participants in the OT group and improved verbal function and subjective well-being. In addition, results indicated a decrease of depressive symptoms. Based on the present results, OT may constitute an inexpensive, simple way to improve quality of life in older people. Copyright © 2017 John Wiley & Sons, Ltd.

EVALUASI ALGORITMA RUNUT BALIK DAN SIMULATED ANNEALING PADA PERMAINAN SUDOKU

Sudoku is one of the popular puzzle games in many countries. How to solve this game is being the Non Polynomial Completeness problem. Because of that, a lot of research has been done to find the most effective algorithm completing Sudoku. This study presents the evaluation of algorithm in completing the Sudoku 2 x 2 and 3 x 3. The most common algorithm is brute force and backtracking. There is also simulated annealing algorithm. The evaluation is done by comparing the step of each algorithm to solve the Sudoku and also their running time. The result of this evaluation shows the differences of each algorithm characteristic in solving this Sudoku problem. By comparing the result of this study, the most effective algorithm can be chosen. Based on the test, concluded that the backtracking algorithm is more effective than the simulated annealing algorithm which the value reach 50% .

A Hybrid Approach for the Sudoku Problem: Using Constraint Programming in Iterated Local Search

Sudoku is not only a popular puzzle but also an interesting and challenging constraint satisfaction problem. Therefore, automatic solving methods have been the subject of several publications in the past two decades. Although current methods provide good solutions for small-sized puzzles, larger instances remain challenging. This article introduces a new local search technique based on the min-conflicts heuristic for Sudoku. Furthermore, the authors propose an innovative hybrid search technique that exploits constraint programming as a perturbation technique within the iterated local search framework. They experimentally evaluate their methods on challenging benchmarks for Sudoku and report improvements over state-of-the-art solutions. To show the generalizability of the proposed approach, they also applied their method on another challenging scheduling problem. The results show that the proposed method is also robust in another problem domain.

Reducing the generalised Sudoku problem to the Hamiltonian cycle problem

The generalised Sudoku problem with N symbols is known to be NP-complete, and hence is equivalent to any other NP-complete problem, even for the standard restricted version where N is a perfect square. In particular, generalised Sudoku is equivalent to the, classical, Hamiltonian cycle problem. A constructive algorithm is given that reduces generalised Sudoku to the Hamiltonian cycle problem, where the resultant instance of Hamiltonian cycle problem is sparse, and has O(N3) vertices. The Hamiltonian cycle problem instance so constructed is a directed graph, and so a (known) conversion to undirected Hamiltonian cycle problem is also provided so that it can be submitted to the best heuristics. A simple algorithm for obtaining the valid Sudoku solution from the Hamiltonian cycle is provided. Techniques to reduce the size of the resultant graph are also discussed.

A new membrane algorithm using the rules of Particle Swarm Optimization incorporated within the framework of cell-like P-systems to solve Sudoku

Sudoku, of order n, is a combinatorial puzzle having partially filled n2×n2 grid consisting of sub-grids of n×n dimension. In this paper, a new membrane algorithm, namely MA_PSO_M, is presented. It uses the modified rules of Particle Swarm Optimization coupled with a carefully designed mutation operator within the framework of cell-like P-systems. Another significant contribution of this paper is the novel way in which the search space for solving the Sudoku problem is defined. Initially, the proposed algorithm is used to solve Sudoku puzzles of order 3 available in literature. On the basis of experiments performed on sample Sudoku puzzles of ‘easy’ and ‘medium’ difficulty levels it is concluded that the proposed membrane algorithm, MA_PSO_M, is very efficient and reliable. For the ‘hard’ and ‘evil’ difficultly levels, too the algorithm performs very well after incorporating an additional deterministic phase. The performance of the algorithm is further enhanced with an increased population size in a very small computational time. To further demonstrate efficiency of algorithm it is applied to Sudoku puzzles of order 4. The obtained results prove that the proposed membrane algorithm clearly dominates any of the PSO based membrane algorithm existing in the literature.

A Hybrid alldifferent-Tabu Search Algorithm for Solving Sudoku Puzzles.

The Sudoku problem is a well-known logic-based puzzle of combinatorial number-placement. It consists in filling a n 2 × n 2 grid, composed of n columns, n rows, and n subgrids, each one containing distinct integers from 1 to n 2. 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 the alldifferent global constraint from the constraint programming world. The alldifferent constraint 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.

A comparison of several enumerative algorithms for Sudoku

Sudoku is a puzzle played of an n × n grid where n is the square of a positive integer m. The most common size is n=9. The grid is partitioned into n subgrids of size m × m. The player must place exactly one number from the set N={1, …, n} in each row and each column of as well as in each subgrid. A grid is provided with some numbers already in place, called givens. In this paper, some relationships between Sudoku and several operations research problems are presented. We model the problem by means of two mathematical programming formulations. The first one consists of an integer linear programming model, while the second one is a tighter non-linear integer programming formulation. We then describe several enumerative algorithms to solve the puzzle and compare their relative efficiencies. Two basic backtracking algorithms are first described for the general Sudoku. We then solve both formulations by means of constraint programming. Computational experiments are performed to compare the efficiency and effectiveness of the proposed algorithms. Our implementation of a backtracking algorithm can solve most benchmark instances of size 9 within 0.02 s, while no such instance was solved within that time by any other method. Our implementation is also much faster than an existing alternative algorithm.

An Optimized Algorithm for Solving Combinatorial Problems using Reference Graph

In this paper a novel optimized graph referencing method is proposed to tackle combinatorial problems with greater ease. The proposed algorithm is a modification of the existing backtracking algorithm. It takes into account the advantage of backtrack algorithm while eliminating its disadvantages. Special care has been taken to reduce the time complexity in avoiding frequent unsuccessful searches and in providing accurate and fast solution to the targeted sub problem. The theoretical analysis and experimental results presented reveal the superiority of proposed algorithm over the conventional algorithms. I. Introduction A special branch of mathematics known as Combinatorics deals with the study of finite structure including counting the structure of a given kind and size, depending upon criteria to be met and analysis and constructs objectives meeting such criteria. Combinatorial problems find extensive application in fields of mathematics, machine learning, AI, Auction Theory, Software Engineering etc. Backtracking is a general algorithm for finding all or some solution to some combinatorial problems. However, Backtracking suffers inherent disadvantages in terms of conjunction of a large processing time and memory space. It is here that the proposed novel optimized graph referencing method established its superiority. The algorithm as proposed by the authors, which is a modification of conventional Backtracking algorithm, not only solves efficiently different combinatorial problems but does so in the least possible time. Tackling various combinatorial problems is a subject of research worldwide. Different efficient algorithms are being developed to this purpose. Xiao et. al. (1) designed a stepwise enumerative algorithm while another algorithm based on GA was developed by Liu et. al. (2). The Backtracking algorithm is reported to be used to solve Sudoku, a combinatorial problem, by Xu (3). The solution to the Sudoku problem was proposed by Zhang (4) who intend to use the application development language AutoLISP embedded in AutoCAD for this purpose. An altogether different algorithm to solve Sudoku was proposed by Zhao et. al. (5) while the same was tackled with the help of cultural algorithm and GA optimization as reported by Mantere and Kolijonen (6-