Text. Solving sudoku as an Integer Programming problem 5-4 The 1st equation below corresponds to the constraint on columns, the 2nd one refers to the constraint on rows and the 3rd one to the constraint on the 3 ×3 squares. Important note. Input: m, a 9x9 matrix of values 0 thru 9. This puzzle idea can accommodate games of other sizes. Therefore, we can define an N × N Sudoku matrix with N = n 2 as shown in Definition 1. a An example of the modified Sudoku matrix. This is a perfect example of a recursion and backtracking algorithm. Here is an example of a Sudoku puzzle and its solution: Presently, Sudoku puzzles are getting progressively famous among individuals everywhere on the world. All you need to do is to learn the rules . For the second assumption, note that all puzzles from WebSudoku.com can be solved without guessing (or so they claim). If you are looking for Sudoku free games, search no more. The darker green cells form the swordfish. Those red numbers in the puzzle have been automatically added to the paper by the algorithm we're about to create. Numbers just happen to be easier. Basée sur le princi. 1. Given an incomplete Sudoku configuration in terms of a 9 x 9 2-D square matrix (grid[][]), the task to find a solved Sudoku. . ; Each column must contain the digits 1-9 without repetition. Then click on the cells where you want to place the selected digit. # # The Sudoku board is a 9x9 grid, which is further divided into a 3x3 grid # of 3x3 grids. The portion of the exact cover matrix corresponding to these constraints (with zeroes replaced by blanks for clarity) is: The four 729x81 matrices can be horizontally concatenated to form the entire 729x324 Sudoku exact cover matrix. The directory on MacOS X is ~/Library/Sudoku. Thus, in n-1 columns, there exists at most n-1 truths. In general, any n × n game can be created, where n = m2 and m is any positive integer. Values 1 thru 9 occur in each row, column, and the nine non-overlapping 3x3 matrices starting at the top left corner. For example, if eight out of nine positions in a sudoku column already have a value, the nine position must contain the digit not already mentioned in the column. You may enter more than one digit in a cell as a note by activating the pencil mark option near the . If there is no solution for the matrix we will print The given sudoku can't be solved. C++ Program to Solve Sudoku Problem. In this section, we will try to solve the famous number maze problem called Sudoku. A list of lists of the same length is regarded as a matrix in Mathematica, so we input for the puzzle and then show it in matrix form. These are the top rated real world Python examples of sudoku_matrix.SudokuMatrix extracted from open source projects. This puzzle idea can accommodate games of other sizes. equals at least one. Today's example will walk through using image processing and deep learning to automatically solve a Sudoku puzzle. We know that Sudoku is a 9 x 9 number grid, and the whole grid are also divided into 3 x 3 boxes There are some rules to solve the Sudoku. Categories Matrix Interview Questions Tags Amazon, Apple, Facebook, Google . We have to use digits 1 to 9 for solving this problem. We are provided with a partially filled 9x9 matrix and have to fill every remaining cell in it. If the solve () function returns false, it means that the digit found in this step (digit 4) can't be used at the given position (Grid [0]). 3. Then we will look upon each 3×3 square if the numbers are not repeated. ; Each of the nine 3 x 3 sub-boxes of the grid must contain the digits 1-9 without repetition. Each row will have items from 1 to 9. Quick Start. First, we will solve by checking if the rows and columns don't have repeated numbers. Algorithm: Start. X9 i=1 x ijk =1 for j,k=1 to 9 . Sudoku Solver. So, remove the value of 4 from the Grid [0] and decrement the number of resolved fields. 2 shows examples of large-size Sudoku matrices. Moreover, After the numbers are not checked for repetition, we will now fill the numbers in the correct order. For example, on my machine, resolving the example Sudoku takes anywhere between 1 and 3 milliseconds while creating a thread and switching to it takes at least 0.5 millisecond. The matrix which is the result of the . For everyone who doesn't know what Sudoku is. digit 5). In this article, I will be demonstrating the backtracking strategy by solving a popular problem known as the Sudoku Solver. It can return true for unsolvable sudoku. Also, mention time and space complexity to validate . I want to write a code in python to solve a sudoku puzzle. In Sudoku, there are several constraints. Sudoku Generator. Therefore, there are a total of 729 different combinations or variables C Program source code to solve a Sudoku. Example sudoku and data structures. Implement an efficient sudoku solver that takes in an incomplete board and solves it. Most clues gives you five random numbers from each nine cell horizontal row. The first assumption restricts the dimensions of the Sudoku puzzles that will be consid-ered. We will be starting the algorithm by dividing it into parts. Fig. Example Input: 8.6..3.9. Given a partially filled 9 x 9 sudoku board, you need to output a solution to it. You can rate examples to help us improve the quality of examples. X9 i=1 x ijk =1 for j,k=1 to 9 . I propose that we handle this puzzle as a two-dimensional Python array where empty boxes are represented with zeros and other boxes with corresponding numbers. There is a 9x9 grid which consists of nine blocks which have a size of 3x3. It is noticeable that one can easily extend the conception of a 9 × 9 Sudoku matrix to other squared sizes, e.g. In this article, we will learn to implement a Sudoku Solver using C++. Then, we convert our 9 by 9 Sudoku board into the exact cover matrix. Valid sudoku has the following properties. . Sudoku Solving algorithms. A Sage matrix over \(\ZZ\) containing the first solution found, otherwise None. Each cell in the grid must take a value from 0 to 9. Automorphic Sudokus. N Sudoku puzzles follow. This matrix contains all the possible values for each position . According to the rules of Sudoku, at least one number cannot be repeated at least two times on a row, a column, or in a n 1 2 × n 1 2 square, where n is the dimension of the Sudoku. Also, suma.m now terminates when TimeLimits (1) is met (instead of trying again). . Sudoku uses numbers, but no mathematics is needed, and that is why, it is so popular. Once matrix is fully filled, remove K elements randomly to complete game. (a) An example of 3D Sudoku 4 × 4 × 4 matrix and (b) its corresponding visual aid using four colors of cubes. The matrix will have 9^3 rows, i.e., one row for every single possible position (9 rows x 9 columns) of every possible number (9 numbers). . Keep Hashmap for the row, column and boxes. This code will try to produce the hardest Sudoku it can. matrix of G is an n x n Symmetrix binary matrix defined over the ring of integers such that The first assumption restricts the dimensions of the Sudoku puzzles that will be consid-ered. If this directory doesn't exist, the program will create it automatically and place the default puzzles there. 3. Please remember them all. Sudoku Puzzle. For each column 2 through n, there exists at most one truth. One digit cannot be repeated in one row, one column or . We have to use digits 1 to 9 for solving this problem. Each combination will have 9 binary variables. EXAMPLES: An example that was used in previous doctests. Valid Sudoku Checker in Python. In fact, the 25×25 puzzle is sometimes called Samurai Sudoku, because it is much . Each 3*3 cell (9 elements) will have items from 1 to 9. Code. In general, any n × n game can be created, where n = m2 and m is any positive integer. Try Smart Solver now. Formula of Sudoku Puzzle (1) the constraint on columns For simplicity, you may assume that there will be only one unique solution. There is a recursive function to solve the sudoku. Each row must contain the digits 1-9 without repetition. Fill all the diagonal 3x3 matrices. Each region surrounded by thick lines contains each digit of 1-4 exactly once. Sudoku involves filling empty spaces with the number 1-9, without repeating any numbers within each row, column or square. The program is supplied with a directory of example puzzles to try. To solve a Sudoku puzzle, download the two files, enter the Sudoku matrix that you want the algorithm to solve at the top of solve_sudoku.m (an example for the formatting is included in the file), save, and run solve_sudoku.m. b An example of 16 × 16 matrix SEM derived from (a) from publication . Since rand ('state',seed) is deprecated in newer versions of MATLAB, suma.m now uses a try/catch block that first attempts the new RandStream approach. And its solution is given below. For the second assumption, note that all puzzles from WebSudoku.com can be solved without guessing (or so they claim). Create a function that checks if the given matrix is valid sudoku or not. Within that grid, there are nine sub-regions consisting of three cells by three cells. ; Note: We should obtain the following: Suppose we have a Sudoku grid and we have to solve this famous number maze problem, Sudoku. For simplicity, you may assume that there will be only one unique solution. Sudoku Solver As a Sudoku fan myself, I was excited to dive into this problem. Solve a Standard 9x9 Sudoku. Repeat the process untill all zeroes are replaced by a number from 1-9. You can, of course, enter and save your own puzzle manually as well. Let us look at the below examples to understand the implementation of the above approaches. .4..1..68 2..87.5 1.8..5.2. 2.Fill the rest of the spaces with zeroes (0). 4. every column contains the digits 1-9 once. SUDOKU SOLVER In 'C'. Only the filled cells need to be validated according to the following rules:. Each column contains each digit of 1-4 exactly once. # No two grid cells in the same row, column, or 3x3 subgrid may take the # same value. Example 1 Input For each block of such cells, Remove the occurrences of the 'n' unique values from remaining cell's possible's list in that row / column / 3×3 grid. Have you ever tried to solve a 9×9 Sudoku puzzle? 7.5.3.9.. .21..7.4. Top view of the atomic structures of the NbMoTaW high-entropy alloy that are based on (c) the 3D Sudoku matrix and on (d) a random 3D matrix. I read somewhere in net about a algorithm which solves it by filling the whole box with all possible numbers, then inserts known values into the corresponding boxes.From the row and coloumn of known values the known value is removed.If you guys know any better algorithm . Then we assign the utility function (puzzle) to print the grid. Traditional Sudoku is a 9×9 puzzle grid made up of nine 3×3 regions. The transpose of a Sudoku matrix is still a correct, but different, Sudoku matrix (of the same dimension). 3.Produce a random number and test the number to the sudoku rules .Columns, 3x3 grids and rows should have unique numbers from 1-9. Given an incomplete Sudoku configuration in terms of a 9 x 9 2-D square matrix (grid[][]), the task to find a solved Sudoku. This same strategy of finding missing digits easily extends to rows or squares. The exact cover matrix will contain 81 columns corresponding to these box/number constraints. Over www.sudoku.name you can find thousands of online Sudoku puzzle games that you can play for free. It's a really popular puzzle. Every Sudoku puzzle we provide comes with its step-by-step solution, such as the hard example shown below. Sudoku, or Su-doku, is a Japanese fun puzzle game. . A Sudoku grid is automorphic if it can be transformed in a way that leads back to the original grid, when that same transformation would not otherwise lead back to the original grid. The number of truths is the first column is: greater than or equal to n (by matrix rows) minus at least (n-1) by matrix columns. First select a digit on the side of the grid. BUG+1 est une technique de fin de résolution d'une grille de Sudoku, simple et facile, qu'il faut impérativement connaître et maîtriser. Figure 1 is an example of a 9×9 Sudoku puzzle. If any number has a frequency greater than 1 in the hashMap return false else return true; In this article, I show you how to write a program to solve difficult Sudoku problems, using a technique I call combinatorial evolution. Note: Character '.' indicates empty cell. Given a 9x9 sudoku as a 2D matrix where empty spaces are filled with 0, we will try to solve the matrix by backtracking in C++. /*The following program is an implementation of a Sudoku Solver in C. Sudoku is a 9*9 grid in which each row,each column and each 3*3 grid contains all numbers from 1 to 9 only once. The step-by-step shows one way to solve puzzle using only logical reasoning. In fact, the 25×25 puzzle is sometimes called Samurai Sudoku, because it is much . This example highlights why deep learning and image Each region, row, and column contains nine cells each. Note: we add one taking into account the extra column headers row. readSudoku: Read a File Containing a Sudoku Grid; solveSudoku: Solve a Sudoku Puzzle; writeSudoku: Write a Sudoku Grid to a File; Browse all. The numbers shown in the example are the "givens". While Sudoku is not solved, Check if any row, column or 3×3 block has 'n' cells such that, (n>1) All possible values in those 'n' cells are exactly 'n' unique values. All the rows should be filled with numbers (1 - 9) exactly once. It is reasonable to use the 9 9 matrix as it is the most common type of the Sudoku puzzle. Naturally each puzzle can be solved in many different ways, but every way leads to the same solution. 2. Sudoku is a number puzzle game, played on a nine by nine cell grid. There are n matrix rows, thus at least n matrix truths. First, we map each row, column, value tuple into a row position in our cover matrix. All the row of the square contains each of the numbers from 1 to n only once. 4 × 4, 16 × 16, 25 × 25 etc. Sudoku is a 9 x 9 number grid, and the whole grid are also divided into 3 x 3 boxes There are some rules to solve the Sudoku. Its property is that every row, every column, and every one among the nine 3X3 blocks consist integers 1 through 9 exactly once, no repeats. Each row, column and 3*3 box of the output matrix contains unique numbers. 1. Sudoku. .3.1.5. Determine if a 9 x 9 Sudoku board is valid. Insert. */ #include . C++ Server Side Programming Programming. A 9x9 2D matrix is taken as an input and using backtracking with C++ we will give solved sudoku as output. 5's in row 4 limited to c4,i4. For every cell to be filled, we try all numbers until we find a safe number to be placed. Note: An empty Sudoku game is also valid to play. . Sudoku Solver in C++. This function matches the behavior of the prior Sudoku solver and is included only to replicate that behavior. Here i have developed a program in C to solve Sudoku for 2x2 to 9x9 matrix!the code will ask for no of rows and columns at first,it means you have to add 2 by 2,or 3x3,or .9x9,now give the inputs one by one,and after entering each i/p press enter,after finishing giving i/ps,it will show the sudoku matrix,and you can . First, enter the values of the sudoku and enter 0 for the unassigned cells. Every Sudoku puzzle we provide comes with its step-by-step solution, such as the hard example shown below. First, we need to create a matrix that will represent Sudoku puzzle as an Exact Cover problem. Fig. Steps to solve the sudoku puzzle in Python. Of course, a 4 × 4 puzzle would be easier and a 16 × 16 puzzle, harder. A list of lists of the same length is regarded as a matrix in Mathematica, so we input for the puzzle and then show it in matrix form. Python SudokuMatrix - 8 examples found. In this version, there are three levels of difficulty, which are Most clues, Medium clues, and Least clues. Now we explain each rule in detail by using the example puzzle following. for k = 0:8. row = 3*floor (k/3); col = 3*mod (k,3); for colNew = col+ (1:3) The colon operator can accept two or three inputs. A Sudoku puzzle is an example of a combinatorial optimization problem. (value = 9, row =1 , col=1) = 0 , Since the row and the column can both range from 1 to 9, there can be 81 different combinations of (row, col). Fig2. For example, a Sudoku problem is given below. In this example, the possible locations for 5's are highlighted in green. every block contains the digits 1-9 once. Download : Download full-size image; Fig. I tested my code using the following puzzle. Sample Sudoku for you to get the logic for its solution: Example 1: Where sudoku puzzles get challenging is when there are multiple open positions in a row or column. Sudoku solver Tags: Alldifferent, Integer programming, Logic programming Updated: September 16, 2016 In case you have missed out on the Sudoku hype, the goal is to fill in unspecified elements in a matrix with numbers between 1 to 9, keeping elements in all rows and columns different, and keeping all elements in the 9 3x3 blocks different. In this method for solving the sudoku puzzle, first we assign the size of the 2D matrix to variable M (M*M). Fill recursively rest of the non-diagonal matrices. In the three input case, the middle input is the step. 6.2.8.. .876.4..3 . First Steps to Solving Sudoku . Do you guys have any idea about a good algorithm for this purpose. Sudoku is a 9x9 matrix filled with numbers 1 to 9 in such a way that every row, column and sub-matrix (3x3) has each of the digits from 1 to 9. Input. Sudoku is a logic-based number-placement puzzle, The objective of the puzzle is to fill n * n grid with digits so that:-All the column of the square contains each of the numbers from 1 to n only once. Sudoku In a Nutshell. Answer (1 of 7): Here i have developed a program in C to solve Sudoku for 2x2 to 9x9 matrix!the code will ask for no of rows and columns at first,it means you have to add 2 by 2,or 3x3,or .9x9,now give the inputs one by one,and after entering each i/p press enter,after finishing giving i/ps,it . If yes, then you must be tempted to look at the algorithm behind finding the solution. For example, a sudoku with the following first three lines: For example, a sudoku with the following first three lines: 1 2 3 . For each block of such cells, Remove the occurrences of the 'n' unique values from remaining cell's possible's list in that row / column / 3×3 grid. Unspecified elements are indicated by zeros here. if element (i, j) of the sudoku matrix contains k, and 0 otherwise. Here is a sample puzzle. Image by author We can represent a sub grid as a 3x3 matrix of the numbers contained within it. In this program, we will see how to solve sudoku in C++ using a simple approach. This is achieved by the following calculation: row_position = 9 * (r * 9) + (c * 9) + v + 1. 1. Rule. Proof. One example is a Sudoku with 3 empty rows, 3 empty columns, and 3 empty boxes (third image). Scary but fascinating stuff. We need to find the given Sudoku is valid or not on the basis of the following rules: . Download scientific diagram | Two examples of the modified Sudoku matrices. Don't worry, we have got everybody . In this example, the swordfish is missing 5's at c6,d4. To solve a sudoko puzzle, a statistician can put up a probability model on the enormous space of 9 x 9 matrix possibilities, constructed to favour "good attempts", and then engineer a Markov chain to sample a long enough chain of sudoku table realisations from that model, until the solution is found. Sudoku class will assign num to the Sudoku: //github.com/pastosm/Sudoku-Solver-Dancing-Links '' > Mathematics of Sudoku puzzles by number! Open source projects ( a ) from publication ( instead of trying again ) is the & ;... And 9 by activating the pencil mark option near the excited to dive into this problem repeat the untill... So that: every row contains each digit of 1-4 exactly once find same num in the input. For j, k=1 to 9 input for this puzzle idea can sudoku matrix example games of other.! '' > sudoku.R - gurobi.com < /a > Rule ( task ) 2, where you select the next (! The unassigned cells are nine sub-regions consisting of blanks ( shown as ) or integers between 1 and 9 matrix. 9 elements ) will have items from 1 to 9 for solving this problem a function that checks if rows... Values 0 thru 9 that satisfy Sudoku rules 2.. 87.5 1.8.. 5.2 ). Cells where you want to place each number in each row will have from... Which have a Sudoku fan myself, I was excited to dive into this problem Sudoku solution without VBA continuoous! Easily extends to rows or squares cells where you select the next digit ( i.e many different ways but! Code used to test the overhead the hardest Sudoku it can * n size N=9. Once matrix is valid Sudoku - Wikipedia < /a > Sudoku puzzles get challenging is when there are sub-regions... A note by activating the pencil mark option near the created, where n is the step between elements the. Option near the do you guys sudoku matrix example any idea about a good algorithm for this problem denotes a.... And a 16 × 16, 25 × 25 etc basis of the numbers not! And that is why, it is the step between elements in the given Sudoku as matrix, row and. Keep Hashmap for the matrix we will be only one unique solution an enormous number of constraints completed! ) of the following rules: in a row position in our matrix. The row, column and boxes used to test the number of fields. Submatrix should be filled, we have to use digits 1 to 9 for solving problem. Really popular puzzle, 9 × 9 ) multiplied by the number of rows columns! By activating the pencil mark option near the 9×9 Sudoku puzzle games that you can of. All you need to be placed this puzzle idea can accommodate games of other sizes idea about good! We need to be filled, remove K elements randomly to complete game Internet try... That is why, it is reasonable to use digits 1 to 9 for solving this will! T have repeated numbers puzzle is sometimes called Samurai Sudoku, because it is the quot. And try to solve the Sudoku matrix contains unique numbers from each nine cell horizontal row t have repeated.. Continuoous < /a > example Sudoku and enter 0 for the second assumption, note all... Submatrix should be filled, we try all numbers until we find a safe number to the sudoku matrix example, column! Course, a 4 × 4 puzzle would be easier and a 16 × 16 puzzle,.... 9 that satisfy Sudoku rules you have given Sudoku is a partially filled 9x9 of. Until it is reasonable to use the 9 9 matrix as it is much non-overlapping 3x3 matrices starting at algorithm. > two examples of the modified Sudoku matrices define an n × n Sudoku contains... Online Sudoku puzzle games that you can play for free numbers until we same. And backtracking algorithm could be safely deprecated, since all of its functionality is included in the given is... And save your own puzzle manually as well Python SudokuMatrix - 8 found! Example Sudoku and data structures, is a 9x9 matrix of values 0 thru 9 occur in each,! S at c6, d4, After the numbers are not repeated repetition..., 16 × 16 puzzle, harder Sudoku Solver as a note by activating the pencil option... Function to solve the Sudoku the nine 3 x 3 sub-boxes of the shown. A number from 1-9 keep Hashmap for the matrix we will solve by checking the!, 25 × 25 etc lines with each line containing 9 space separated integers are multiple open positions a! A valid Sudoku in C++ using a simple approach givens & quot givens... By dividing it into parts have repeated numbers select a digit on the cells where you to! Most common type of the modified Sudoku matrices sub-regions consisting of blanks ( shown ). Later it will assign num to the following rules: all you need to be placed ] - <... Must take a random Sudoku puzzle side of the nine 3 x 3 sub-boxes the! Codespeedy < /a > Text swordfish, and does not effect the possible eliminations items. Fan myself, I was excited to dive into this problem why, it the! The Sudoku puzzle is sometimes called Samurai Sudoku, because it is to... Has gotten main stream now in an enormous number of resolved fields from. Incomplete spaces will be starting the algorithm by dividing it into parts must take a random number and the... Rated real world Python examples of the output matrix contains K, and that is why, it reasonable. Sudoku board that can only be completed in one way to solve using. Values of the Sudoku matrix with n = m2 and m is any positive integer with (! Sudoku rules.Columns, 3x3 grids and rows should be filled with numbers ( 1 - )... Find thousands of online Sudoku puzzle '' > GitHub - pastosm/Sudoku-Solver-Dancing-Links < /a > Sudoku simplicity... Or 3x3 subgrid may take the # same value n * n size where N=9 column boxes... # no two grid cells in the example puzzle following of nine which! One contains 9 lines with each line containing 9 space separated integers the step between elements in the row... This directory doesn & # x27 ; s example will walk through using image and... Into a row or same column or separated integers Sudoku Generator different ways but! Reasonable to use digits 1 to n only once i=1 x ijk =1 for j, k=1 to 9 solving... Today & # x27 ; s a really popular puzzle with numbers ( 1 - 9 ) once... Is no solution for the matrix we will be 0, and that is,..., k=1 to 9 you must be tempted to look at the algorithm by dividing it into.... Sudoku_Matrix.Sudokumatrix extracted from open source projects the most common type of the Sudoku. Not repeated ) to print the grid: we add one taking into the! Will be starting the algorithm by dividing it into parts to find the given Sudoku &! A list of nine lists consisting of blanks ( shown as ) or integers 1. Are multiple open positions in a cell as a 3x3 matrix of n * n where! Program is supplied with a partially filled 9x9 matrix of n * n size where N=9,... 9X9 grid which consists of nine blocks which have a Sudoku fan myself, I excited. Process untill all zeroes are replaced by a number from 1-9 extra column headers row random puzzle! Digit ( i.e not be repeated in one row, column, value tuple into row. Can only be completed in one way to solve the famous number maze problem, Sudoku output. - pastosm/Sudoku-Solver-Dancing-Links < /a > Sudoku puzzles columns, there exists at most n-1 truths examples < /a Fig!, j ) of the following rules: is solved lists consisting of three by. Of Sudoku - Wikipedia < /a > Sudoku Solver and is included in example! Values of the Sudoku class first, we try all numbers until we find same num in the order. Each column contains nine cells each are provided with a directory of puzzles. 4 puzzle would be easier and a 16 × 16 matrix SEM derived from ( a ) publication! As ) or integers between 1 and 9 position in our cover matrix 3.produce a random number and test number... Source code to solve the Sudoku rules strategy of finding missing digits easily extends to rows or squares 0 an. Every way leads to the same solution the extra column headers row cell ( 9 elements will! And is included only to replicate that behavior for the unassigned cells in this section we. We need to do is to learn the rules K, and does not effect the possible eliminations you,! Mark option near the accommodate games of other sizes are nine sub-regions consisting of blanks ( shown as or. Have items from 1 to 9 matrix with n = m2 and m is positive. At least n matrix truths space complexity to validate > sudoku.R - gurobi.com < /a Sudoku... Puzzle, harder the cells where you want to place the selected digit each of the numbers contained within... & # x27 ;. & # x27 ; t be solved the example the! Free games, search no more program source code to solve a Sudoku fan myself, I was to... Of the grid remove K elements randomly to complete game since all of its functionality is only... So that: every row contains the digits 1-9 once 4 × 4 puzzle would be and. Examples to help us improve the quality of examples x ijk =1 for j, k=1 to for. 0 sudoku matrix example and decrement the number of constraints thru 9 that satisfy Sudoku rules is common with swordfish and... Puzzle would be easier and a 16 × 16, 25 × 25 etc of nations numerous!
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