"Sudoku Completed games are always a type of Latin square with an additional constraint on the contents of individual regions. For example, the same single integer may not appear twice in the same row, column, or any of the nine 3×3 subregions of the 9x9 playing board. French newspapers featured variations of the puzzles in the 19th century, and the puzzle has appeared since 1979 in puzzle books under the name Number Place. However, the modern Sudoku only started to become mainstream in 1986 by the Japanese puzzle company Nikoli, under the name Sudoku, meaning ""single number"". It first appeared"
"at least one Sudoku that has 18 clues, exhibits two-way diagonal symmetry and is automorphic. The maximum number of clues that can be provided while still not rendering a unique solution is four short of a full grid (77); if two instances of two numbers each are missing from cells that occupy the corners of an orthogonal rectangle, and exactly two of these cells are within one region, the numbers can be assigned two ways. Since this applies to Latin squares in general, most variants of Sudoku have the same maximum. The number of classic 9×9 Sudoku solution grids is"
"object is the same as that of standard Sudoku, but the puzzle only uses the numbers 1 through 6. A similar form, for younger solvers of puzzles, called ""The Junior Sudoku"", has appeared in some newspapers, such as some editions of ""The Daily Mail"". Another common variant is to add limits on the placement of numbers beyond the usual row, column, and box requirements. Often, the limit takes the form of an extra ""dimension""; the most common is to require the numbers in the main diagonals of the grid to also be unique. The aforementioned ""Number Place Challenger"" puzzles are"
"links can solve most 9×9 puzzles efficiently, but combinatorial explosion occurs as ""n"" increases, creating limits to the properties of Sudokus that can be constructed, analyzed, and solved as ""n"" increases. A Sudoku puzzle can be expressed as a graph coloring problem. The aim is to construct a 9-coloring of a particular graph, given a partial 9-coloring. The fewest clues possible for a proper Sudoku is 17 (proven January 2012, and confirmed September 2013). Over 49,000 Sudokus with 17 clues have been found, many by Japanese enthusiasts. Sudokus with 18 clues and rotational symmetry have been found, and there is"
"order of 100,000,000, while the number of Band1 permutations covered by each class however varies from 4 – 3240. Within this wide size range, there are clearly two clusters. Ranked by size, the lower 33 classes average ~400 permutations/class, while the upper 11 average ~2100. The disparity in consistency between the distributions for size and number of completions or the separation into two clusters by size is yet to be examined. Sudoku regions are polyominoes. Although the classic 9×9 Sudoku is made of square nonominoes, it is possible to apply the rules of Sudoku to puzzles of other sizes –"
"a Sudoku mode, which features more than 100 puzzles across three different modes – Beginner, Intermediate, and Advanced. Sudoku involves a 9×9 grid with numbers in every square. Some of these numbers are visible, while others are not. The objective is to fill in the hidden numbers using the visible numbers as hints. Each row, column, and 3×3 grid has nine squares in it, and each must contain each number in the range from 1 to 9. Many neurologists recommend the game for prevention of dementia/Alzheimer's. Nintendo of America has refused to support any scientific claims to the benefits of"
"a ""1"" in that cell. When checking for violations, if it is discovered that the ""1"" is not allowed, the value is advanced to ""2"". If a cell is discovered where none of the 9 digits is allowed, then the algorithm leaves that cell blank and moves back to the previous cell. The value in that cell is then incremented by one. This is repeated until the allowed value in the last (81st) cell is discovered. The animation shows how a Sudoku is solved with this method. The puzzle's clues (red numbers) remain fixed while the algorithm tests each unsolved"
"""neither"" 1 nor 3. The following tables list the possible combinations for various sums. 1: 1 3: 12 6: 123 10: 1234 15: 12345 28: 1234567 36: 12345678 Killer sudoku Killer sudoku (also killer su doku, sumdoku, sum doku, sumoku, addoku, or samunamupure) is a puzzle that combines elements of sudoku and kakuro. Despite the name, the simpler killer sudokus can be easier to solve than regular sudokus, depending on the solver's skill at mental arithmetic; the hardest ones, however, can take hours to crack. A typical problem is shown on the right, using colors to define the groups of"
"Sujiko Sujiko is a logic-based, combinatorial number-placement puzzle created by Jai Gomer of Kobayaashi Studios. The puzzle takes place on a 3x3 grid with four circled number clues at the centre of each quadrant which indicate the sum of the four numbers in that quadrant. The numbers 1-9 must be placed in the grid, in accordance with the circled clues, to complete the puzzle. Sujiko is featured in UK newspapers including ""The Times"", ""The Telegraph"", and ""The Sun"". The puzzle has a mathematical property, often ignored by puzzle setters, that can sometimes make supposedly-difficult examples somewhat trivial, as follows: The"
"to solve Sudokus, and to develop (or search for) new Sudokus. Analysis has largely focused on enumerating solutions, with results first appearing in 2004. There are many Sudoku variants, partially characterized by size (""N""), and the shape of their ""regions"". Unless noted, discussion in this article assumes classic Sudoku, i.e. ""N""=9 (a 9×9 grid and 3×3 regions). A rectangular Sudoku uses rectangular regions of row-column dimension ""R""×""C"". Other variants include those with irregularly-shaped regions or with additional constraints (hypercube) or different constraint types (Samunamupure). A ""puzzle"" is a partially completed ""grid"", and the initial values are ""givens"" or ""clues"". ""Regions"""
"off with increasing N at sufficient speed to have people believe that ; the largest number now known not to be a sum of four cubes is 7373170279850, and the authors give reasonable arguments there that this may be the largest possible. The upper bound is due to Linnik in 1943. (All nonnegative integers require at most 9 cubes, and the largest integers requiring 9, 8, 7, 6 and 5 cubes are conjectured to be 239, 454, 8042, 1290740 and 7373170279850, respectively.) 13792 is the largest number to require seventeen fourth powers (Deshouillers, Hennecart and Landreau showed in 2000 that"
"also occur in some puzzles played on a grid, such as Sudoku. A Sudoku is a type of Latin square with the additional property that each element occurs exactly once in sub-sections of size (called ""boxes""). Combinatorial explosion occurs as increases, creating limits to the properties of Sudokus that can be constructed, analyzed, and solved, as illustrated in the following table. One example in a game where combinatorial complexity leads to a solvability limit is in solving chess (a game with 64 squares and 32 pieces). Chess is not a solved game. In 2005 all chess game endings with six"
"the 2×2 and 4×4 square puzzles, for example. Only ""N""×""N"" Sudoku puzzles can be tiled with square polyominoes. Another popular variant is made of rectangular regions – for example, 2×3 hexominoes tiled in a 6×6 grid. The following notation is used for discussing this variant. Puzzles of size ""N""×""N"", where ""N"" is prime can only be tiled with irregular ""N""-ominoes. Each ""N""×""N"" grid can be tiled multiple ways with ""N""-ominoes. Before enumerating the number of solutions to a Sudoku grid of size ""N""x""N"", it is necessary to determine how many ""N""-omino tilings exist for a given size (including the standard"
"another branch. Although it has been established that approximately 6.67 x 10 final grids exist, a brute force algorithm can be a practical method to solve Sudoku puzzles. A brute force algorithm visits the empty cells in some order, filling in digits sequentially, or backtracking when the number is found to be not valid. Briefly, a program would solve a puzzle by placing the digit ""1"" in the first cell and checking if it is allowed to be there. If there are no violations (checking row, column, and box constraints) then the algorithm advances to the next cell, and places"
"solution. Rows, columns and regions can be collectively referred to as groups, of which the grid has 27. The One Rule encapsulates the three prime rules, i.e. each digit (or number) can occur only once in each row, column, and box; and can be compactly stated as: ""Each digit appears once in each group."" The classic 9×9 Sudoku format can be generalized to an This accommodates variants by region size and shape, e.g. 6-cell rectangular regions. (""N""×""N"" Sudoku is square). For prime N, polyomino-shaped regions can be used and the requirement to use equal-sized regions, or have the regions entirely"
"(i.e., is hit by) exactly one selected possibility. In the standard 9×9 Sudoku variant, there are four kinds of constraints sets corresponding to the four kinds of constraints: Since there are 9 rows, 9 columns, 9 boxes and 9 numbers, there are 9×9=81 row-column constraint sets, 9×9=81 row-number constraint sets, 9×9=81 column-number constraint sets, and 9×9=81 box-number constraint sets: 81+81+81+81=324 constraint sets in all. In brief, the standard 9×9 Sudoku variant is an exact hitting set problem with 729 possibilities and 324 constraint sets. Thus the problem can be represented by a 729×324 matrix. Although it is difficult to present"
"colours exactly once"". The Sudoku Cube is similar to the Nine-Colour Cube in that the objective of the Sudoku Cube is to arrange it such that each face contains each ""number"" (1 to 9) exactly once. There are a total of 26 pieces (called cubies) on the puzzle. Out of the nine colours, one colour (green in the picture) only appears on 2 cubies; both of these are corner pieces. The remaining eight colours all appear on 3 cubies each; two of those (red and purple in the picture) appear on three edge pieces, while the other six are on"
"II, III, IV, ..., IX) proposed by ""Hebdomada aenigmatum"", a monthly magazine of Latin puzzles and crosswords. Like the Wordoku, it presents no functional difference from a normal Sudoku, but adds the visual difficulty of using Roman numerals. Hyper Sudoku uses the classic 9×9 grid with 3×3 regions, but defines four additional interior 3×3 regions in which the numbers 1–9 must appear exactly once. It was invented by ""Peter Ritmeester"" and first published by him in Dutch Newspaper ""NRC Handelsblad"" in October 2005, and since April 2007 on a daily basis in ""The International New York Times"" (International Herald Tribune)."
"the first column to the border column, so we have the four cases shown above. Without the restriction that a is the identity there are 16 cases: Of the 576 Latin squares, 288 are solutions of the 2×2 version of Sudoku, sometimes called Shi Doku . With first row abcd and first column acbd this reduces to 2. For order 5 there are 161,280 Latin squares with symbols a, b, c, d, and e. There are 280 loops: 5 times the number of reduced Latin squares. 30 of these are groups, all are versions of the cyclic group of order"
"still ""not"" rendering a unique solution, it is four short of a full grid (77). If two instances of two numbers each are missing and the cells they are to occupy are the corners of an orthogonal rectangle, and exactly two of these cells are within one region, there are two ways the last digits can be added (two solutions). It has been conjectured that no proper Sudoku can have clues limited to the range of positions in the pattern above (first image). The largest rectangular orthogonal ""hole"" (region with no clues) in a proper Sudoku is believed to be"
"the next sudoku, and the ""World Record"" round, in which solvers competed to set a Guinness World Record for fastest sudoku solution. Of the 13 championships held so far, Kota Morinishi of Japan (2014, 2015, 2017 and 2018) has been the most successful winners with four individual titles each, succeeded to Thomas Snyder of United States (2007, 2008 and 2011), Jan Mrozowski of Poland (2009, 2010 and 2012) who have won three. From 2007 there has also been a team competition. Japan is the most successful team, having won the title in 2007, 2012, 2014, 2015 and 2018; Czech Republic"
"formula_38 and formula_39 are already used in the sub-grid. The last row contains no information at all. Adding all information up one gets formula_40 bits. The rate in this example is formula_41. The exact number of possible Sudoku grids according to Mathematics of Sudoku is formula_42. With the total information of formula_43 the average rate of a standard Sudoku is formula_44. The average number of possible entries for a cell is formula_45 or formula_46 of information per Sudoku cell. Note that the rate may vary between codewords. The minimum number of given entries that render a unique solution was proven"
"+ e Q is trivially obvious, by symmetry (or can be similarly proved from first principles). </poem> Sujiko Sujiko is a logic-based, combinatorial number-placement puzzle created by Jai Gomer of Kobayaashi Studios. The puzzle takes place on a 3x3 grid with four circled number clues at the centre of each quadrant which indicate the sum of the four numbers in that quadrant. The numbers 1-9 must be placed in the grid, in accordance with the circled clues, to complete the puzzle. Sujiko is featured in UK newspapers including ""The Times"", ""The Telegraph"", and ""The Sun"". The puzzle has a mathematical"
"6,670,903,752,021,072,936,960 , or around . This is roughly times the number of 9×9 Latin squares. Various other grid sizes have also been enumerated—see the main article for details. The number of essentially different solutions, when symmetries such as rotation, reflection, permutation, and relabelling are taken into account, was shown to be just 5,472,730,538 . Unlike the number of complete Sudoku grids, the number of minimal 9×9 Sudoku puzzles is not precisely known. (A minimal puzzle is one in which no clue can be deleted without losing uniqueness of the solution.) However, statistical techniques combined with a puzzle generator show that"
"after each individual grid has at least been partially solved. Puzzles constructed from more than two grids are also common. Five 9×9 grids that overlap at the corner regions in the shape of a ""quincunx"" is known in Japan as ""Gattai"" 5 (five merged) Sudoku. In ""The Times"", ""The Age"", and ""The Sydney Morning Herald"", this form of puzzle is known as Samurai SuDoku. ""The Baltimore Sun"" and the ""Toronto Star"" publish a puzzle of this variant (titled High Five) in their Sunday edition. Often, no givens are placed in the overlapping regions. Sequential grids, as opposed to overlapping, are"
"under the name Logi-5; the World Puzzle Championship has featured a 6×6 grid with 2×3 regions and a 7×7 grid with six ""heptomino"" regions and a disjoint region. Larger grids are also possible. ""The Times"" offers a 12×12-grid ""Dodeka Sudoku"" with 12 regions of 4×3 squares. Dell Magazines regularly publishes 16×16 ""Number Place Challenger"" puzzles (using the numbers 1–16 or the letters A-P). Nikoli offers 25×25 ""Sudoku the Giant"" behemoths. A 100×100-grid puzzle dubbed Sudoku-zilla was published in 2010. Under the name ""Mini Sudoku"", a 6×6 variant with 3×2 regions appears in the American newspaper ""USA Today"" and elsewhere. The"
"The first time it was called Hyper Sudoku was in ""Will Shortz's Favorite Sudoku Variations"" (February 2006). It is also known as Windoku because with the grid's four interior regions shaded, it resembles a window with glazing bars. In Twin Sudoku two regular grids share a 3×3 box. This is one of many possible types of overlapping grids. The rules for each individual grid are the same as in normal Sudoku, but the digits in the overlapping section are shared by each half. In some compositions neither individual grid can be solved alone – the complete solution is only possible"
"of ""units""). See the Glossary of Sudoku for an expanded list of variants. Mathematics of Sudoku The class of Sudoku puzzles consists of a partially completed row-column grid of cells partitioned into ""N"" regions each of size ""N"" cells, to be filled in (""solved"") using a prescribed set of ""N"" distinct symbols (typically the numbers {1, ..., ""N""}), so that each row, column and region contains exactly one of each element of the set. The properties of Sudoku puzzles and their solutions can be investigated using mathematics and algorithms. The analysis of Sudoku falls into two main areas: analyzing the"
"number to a particular cell is a possibility (or candidate). When Sudoku is played with pencil and paper, possibilities are often called pencil marks. In the standard 9×9 Sudoku variant, in which each of 9×9 cells is assigned one of 9 numbers, there are 9×9×9=729 possibilities. Using obvious notation for rows, columns and numbers, the possibilities can be labeled The fact that each kind of constraint involves exactly one of something is what makes Sudoku an exact hitting set problem. The constraints can be represented by constraint sets. The problem is to select possibilities such that each constraint set contains"
"the top row to be 3 2 5 4 1 and the bottom again to be 1 2 3 4 5, which would lead to the same contradiction. Only a 5 is permissible in the top left corner, from which we deduce 5 4 3 2 1 at top and 1 2 5 4 3 at bottom. The remainder of the solution is simple elimination. A solved futoshiki puzzle is a Latin square. As with sudoku case, more difficult futoshiki puzzles require the use of various types of chain patterns. A futoshiki puzzle is published in the following UK newspapers:"
"cells and so on are eliminated. Sudoku players use this method of logic excluding to solve most sudoku puzzles. The aim of error-correcting codes is to encode data in a way such that it is more resilient to errors in the transmission process. The encoder has to map data formula_21 to a valid sudoku grid from which the codeword formula_22 can we taken e.g. in row-scan order. formula_23 shows the necessary steps. A standard formula_11 sudoku has about 72.5 bits of information as calculated in the next section. Information after Shannon is the degree of randomness in a set of"
"two different sets of ten numbered squares. Numbers one through ten on top contained pictures. Numbers eleven through twenty on the bottom contained scrambled words related to the pictures. Each square also concealed part of a ""super word"", a word where each letter was separated by a space. The pictures, scrambled words, and ""super word"" all related to the toss-up identity. Starting with the child who won the toss-up identity, each child chose one number revealing a picture from the top half of the puzzle board, and one number revealing a scrambled word from the bottom half. In order to"
"also published, with values in specific locations in grids needing to be transferred to others. A tabletop version of Sudoku can be played with a standard 81-card Set deck (see Set game). A three-dimensional Sudoku puzzle was published in ""The Daily Telegraph"" in May 2005. ""The Times"" also publishes a three-dimensional version under the name Tredoku. Also, a Sudoku version of the Rubik's Cube is named Sudoku Cube. Many other variants have been developed. Some are different shapes in the arrangement of overlapping 9×9 grids, such as butterfly, windmill, or flower. Others vary the logic for solving the grid. One"