Sudoku Puzzle Design Algorithms Computer Science Essay

Sudoku Puzzle Design Algorithms Computer acquirement EssayOur project is called generating, rating and firmness Sudoku stir up. The aim of the project is to design and larn Sudoku ingest solving program by r apiece the work over with five differences difficulty rating level. The program is fitted to earn the Sudoku puzzle by use some Artificial Intelligent algorithmic program.The final year project organic evolution period is given two trimester and we divided the development answer into two part. The first part is to develop the problem specification, design, generating and study on some Sudoku solving algorithm which to be through with(p) by trimester one. go ab reveal 2 of the project is developing the accomplish Sudoku solving program and this part allow for be done by trimester 2.Chapter 1 Introduction1.1 IntroductionSudoku is logic based second-placement puzzle. It designs with 99 control grids so that the digits 1-9 come egress of the clo mass just now once in severally wrangle, separately columns, and each 33 hero sandwich grids ( excessively called as blocks) that compose the grid. all(prenominal) Sudoku puzzle progress tod with a unique solution, which typically means that there is only one correct mode to fill in the 81 grids.The aim of the project is to design and develop Sudoku puzzle solving program using the artificial intelligence algorithm. First at all the Sudoku configuration program would be qualified to hold the puzzle with five differences difficulty rating level, which be very voiced, easy, medium, difficult and expert. The program also able to earn the Sudoku puzzle as well as providing the chances of having tips while they achieve the specify develop which could help the player to decided the puzzle when they get stuck.An interesting Sudoku puzzle is the one with the miniskirtmal hints given. The puzzle should be generated with a single solution. So, knowledge for the rules of Sudoku is very import ant which the digit 1-9 request to be pop out on the button once in each rows, each columns, and each 33 sub-grid. Then a 99 empty grid need to be constituted and filled in with the digit 1-9 by following the rules to create a complete puzzle.The contiguous step is to begin removing digits from the completed Sudoku puzzle randomly to create the game. To generate a symmetrical puzzle, we need to randomly remove the singleton from the row, column, or block, following by cutting pairs, naked triples, hidden pairs, hidden triples, and pointing pairs tallyly. The proficiencys engrossd to remove the prison cells decided the difficulty level of the Sudoku puzzle. Lastly, the program is able to process the Sudoku game by using the AI algorithm like simulated annealing.1.2 Problem statement community nowadays always busy for working and cope for the challenges on their life. With the increasing busyness of the life, stress plow a common issue that faced by the society. A long term of facing a particular problem will make a mass to feel pressure and stress themselves badly. Therefore, they need some break to take their mind off from working materials before they continue to work. Some measures, a challenges mini game like Sudoku can helps to exercise their brain and reduce stress.Besides, with the fast growing of the technologies today, most people be able to obtain the advance electronic device like PDA, laptop, and so on. So the Sudoku puzzle game is not the game that is only published in the hardcopy such as newspaper or Sudoku book. Therefore Sudoku game can be installed in their electronic device so that they can bring it anywhere.1.2 ScopeScope of the project is generating the Sudoku puzzle with different difficulty level and solves the Sudoku game. The scope coversAllow exploiter to select the difficulty level of the Sudoku game.Generate a complete Sudoku puzzle by following the Sudoku rules which are the digits 1-9 appear just now once in each row , each columns, and each 33 sub grids. rent cells to create a valid Sudoku games concord to the difficulty level that the user selected.Allow user to stimulant the digit 1-9 into the empty cells of the puzzle.Highlight the incorrect input of digit with red colour.Solve the Sudoku games.1.2 ObjectivesUnderstanding the algorithm behind Sudoku puzzle.Identify an algorithm to generate a complete Sudoku puzzle with the rules that the digit 1-9 appear exactly once in each row, each columns, and each 33 sub grids.Identify the ways of removing cells from the complete Sudoku puzzle to create a valid game.Generate Sudoku with different difficulty level.Identify the algorithm to solve the Sudoku puzzle.1.3 Project ScheduleProject Schedule Management Gantt ChartTasksActivityStart date land up dateDuration(days)Task 1Research and study relevant material7/6/201027/6/201021Task 2Planning development process28/6/20104/7/20107Task 3Implement Prototype for Generating5/7/20108/8/201035Task 4Test Pr ogram and Fix error9/8/201014/8/20107Task 5Amends report and prototype15/8/201029/8/201014Task 6Validate interim report and Submission30/8/20101/9/20101Chapter 2 Literature Review2.1 Nature of the problemSudoku puzzle is a famous game among the worldwide. Most of the newspaper has a long history of publishing the Sudoku games such as The Times, Sin Chew Daily, and so on. The games in hardcopy required user to use pen or pencil to finish on the paper and the chip of games are limited. So, in this project, a program is created to generate the Sudoku puzzle games. User does not need to use an extra writing material like pen or pencil to complete the Sudoku games. They just need a mouse to interact with the game.Other than that, the program is able to solve the Sudoku puzzle as well as identify the puzzle is solvable and tier a unique solution. User also can enter the Sudoku games that are not generated by the program and solve it by using this program.2.2 BackgroundThe origin of Sud oku is from Switzerland (By a Mathematician Leonhard Euler) and then travels to Japan by way of America. Su is the Nipponese character with the meaning of subprogram and doku means single. In another word, it called single depend which means that for each column, each row, and each sub grid, it can only contain the digit 1-9 without repetition.In this project, we need to knock over some important things like the woo that we need, the purpose of this project, the target user of this program and so on. monetary value that we need to use is to complete the project in 24 week period. The purpose of this project is to create a program which can generate the Sudoku games with different difficulty level as well as able to solve the Sudoku games.Sudoku is a game that suitable for all generation of the people. The program is able to generate the game in 5 different difficulty levels which is very easy, easy, medium, hard, and expert. Level very easy and easy are suitable for the childre n, medium is suitable for the adults, hard and expert are suitable for those who are undergo in the Sudoku games.2.3 Literature Review2.3.1 The Science behind SudokuSudoku is a game of numbers, but it does not have anything to do with mathematics such as addition or multiplication. A Sudoku grid is a peculiar(a) kind of Latin square that named by an 18th century mathematician-Leonhard Euler. Latin square is an n x n matrices that are filled in with n symbols in a way that the same symbols appear exactly once in same row and column.The number of valid Sudoku grids is 6,670,903,752,021,072,936,960 which proved by the use of logic and computers. The minimum numbers of hints that a 9 x 9 Sudoku puzzle can start with and the solution can still remain unique seem to be 17. This mean for any Sudoku which is having the hints that little than 17 will make it cannot guarantee a unique solution.CUsersVinZDesktop1.pngCUsersVinZDesktop2.pngFigure 1.An example of the token(prenominal) hints f or a Sudoku puzzles and still guarantee with the unique solution.CUsersVinZDesktop3.pngCUsersVinZDesktop4.pngFigure 2.An example of a Sudoku game with 16 hints provided and leads to two solutions.To solve a Sudoku puzzle, the most common way is backtracking. First, the program places a number 1 in the first empty cell of the Sudoku puzzle. If the number is compatible with the alive hints, it moves to the second empty cell and place a 1 again. When it encounters a conflict or clash with other existing hints, it erases the number which just placed and enters number 2. If it is invalid again then proceed with 3 or the next legal number. After the legal number placed, it moves to the next cell and start again with number 1.If the number that placed and it is still not compatible with the existing hints is 9 (9 is the largest number in Sudoku puzzle), the program will backtrack and increase the precious cells number by one. After that it moves forward until it is clash with other existi ng hints and backtracks again. Sometimes the program needs to backtrack for several times before it can move forward.Backtracking technique is good and fast when apply in a program or machine, but a valet player need to spend more time to finish the game using this technique. So they utilise smarter ways for solving Sudoku and generally turn to method trial and error method at death of the solving part.2.3.2 A Pencil-and-Paper Algorithm for Solving Sudoku PuzzleTo solve a Sudoku puzzle, it involve with the widely know concept of matching numbers across cells. In another mathematical point of view, we name it as pre-emptive sets. The most important technique to solve the Sudoku puzzles is based on two things, which are the (i) definition of pre-emptive sets and (ii) the tenancy theorem. A mark up puzzle can help to solve the Sudoku puzzle with more expeditiously by writing down the possible number that can be filled into the empty cell.CUsersVinZDesktopmarkup.pngFigure 3.Example of a mark-up Sudoku puzzle.Pre-emptive setsA pre-emptive set composed of the digits 1-9 and is a set of size m, where2 n1, n2, , nm, c(i1,j1), c(i2,j2), , c(im, jm) Where 1=CUsersVinZDesktop852.png345945962347812348234579245792345Figure 2.Pre-emptive setFor example, the skeletal system above consist of a pre-emptive set3, 4, 5, 9, c(7, 1), c(7, 2), c(8, 3), c(9, 3) with size 4.Occupancy theoremSuppose that x is a number that listed in the pre-emptive set and it lies completely in one column, or row, or a sub grid, so there can be no occurrence of x for the column, or row, or sub grid that outside of the pre-emptive set.Firstly, the algorithm use method 1 and method 2 to complete the puzzle until it cannot continue further. Method 1 is to figure out the Force cell which is the cell that can be determine by eliminate the impossible number (number that already existed in the same column, row and sub grid) that occupy the cell. Method 2 is to figure out the possible number for the e mpty cell that can be determined by checking the same number which occupies the cells in the column and row of the sub grid.After that we look for the pre-emptive set from each column, each row, and each sub grid. Try to break it into several smaller pre-emptive set and cross out the number according to the occupancy theorem. For example in Figure 2, the pre-emptive set is 3, 4, 5, 9, c(7, 1), c(7, 2), c(8, 3), c(9, 3) with size 4, it cross out the number that are not inside the pre-emptive set. In figure 2, digits 8 and 2 in cell c(8, 1) and c(9, 1) can be determined directly after the number had been crossed out. So apply again method 1 and method 2 to figure out other possible number.Clearly then, repeat again the process untill it cannot find out any pre-emptive tes ot until the sudoku puzzle is solved.if the result seem to be violated, then the sudoku game is a unsolable game.Metaheuristics can solve Sudoku puzzlesBefore the simulated annealing can straightforward, representati on, neighbourhood wheeler dealer and evaluation of scene solution need to be defined. Representation used to assign each empty cell a value randomly, but every sub gird contain the digit 1-9 exactly once.During the process, neighbourhood operator choose two different non-fixed cells (the cell which filled in by randomly) in the sub grid and swap them. Two of the non fixed cells to be swap must be in the same sub grid to make sure the criterion of each sub grid contain the digit 1-9 exactly once.Cost function used to evaluate the candidate solution by looking at each row and column, and then calculate the number of values that are not present. For example in figure 3, 1 and 2 had repeated two times in the first row, so the score of that row is 2. Note that the complete solution of the Sudoku puzzle will have zero of cost.CUsersVinZDesktop854.pngFigure 3Here, simulated annealing start with a candidate solution s and a neighbour s. s is accepted if s is better than s (according to co st function) or with the probabilityExp (-/t), which = proposed change in the cost function,t = control parameter (as known as temperature).t, temperature is very important to expend the Simulated annealing and lead to the success. The initial temperature should allow majority if the moves to be accepted, approximately 80% (according to Van Laaehoven Aart (1987)). During the process, temperature is reduced according to the simple geometric engine cooling system schedule. A formula, = use to make sure the temperature decrease according to the cooling rate. Note that , so the largest value of is less than 1, such as 0.9999. For example, value of is 0.9999 will cause the temperature overtop slowly, and the value of is 0.5 will make it drop faster.Furthermore, the use of homogenous Simulated Annealing schema takes the form of a sequence of Markov Chains that generated at the fixed value of t. The t is then changed in-between incidental chains (Van Laarhoven and Aart, 1987). The value of t here is also very important as it decided the time to explore of the search tree and also the run times.The size of the problem-instance defined by the non-fixed number in the grid whereml =ml = size of the problem-instance.= the function that indentify the number of cells in the are non-fixed.By using this formula, the value can allow each pair of the non-fixed cells have a good chance to considered at least once per Markov chain.