"Sudoku solving algorithms A standard Sudoku contains 81 cells, in a 9×9 grid, and has 9 boxes, each box being the intersection of the first, middle, or last 3 rows, and the first, middle, or last 3 columns. Each cell may contain a number from one to nine, and each number can only occur once in each row, column, and box. A Sudoku starts with some cells containing numbers (""clues""), and the goal is to solve the remaining cells. Proper Sudokus have one solution. Players and investigators may use a wide range of computer algorithms to solve Sudokus, study their"
"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"
"vertices that are joined by an edge do not have the same integer assigned to them. A Sudoku solution grid is also a Latin square. There are significantly fewer Sudoku grids than Latin squares because Sudoku imposes the additional regional constraint. Nonetheless, the number of Sudoku grids was calculated by Bertram Felgenhauer and Frazer Jarvis in 2005 to be 6,670,903,752,021,072,936,960. The number computed by Felgenhauer and Jarvis confirmed a result first identified by QSCGZ in September 2003. This number is equal to 9! × 72 × 2 × 27,704,267,971, the last factor of which is prime. The result was derived"
"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"
"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"
"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"
"by Jarvis / Russell and verified by Pettersen. Similar methods have been applied to the 2x3 case, where Jarvis / Russell showed that there are 49 essentially different grids (see also the article by Bailey, Cameron and Connelly), to the 2x4 case, where Russell showed that there are 1,673,187 essentially different grids (verified by Pettersen), and to the 2x5 case where Pettersen showed that there are 4,743,933,602,050,718 essentially different grids (not verified). Ordinary Sudokus (""proper"" puzzles) have a unique solution. A ""minimal"" Sudoku is a Sudoku from which no clue can be removed leaving it a proper Sudoku. Different minimal"
"a rectangle of 30 cells (a 5 x 6 rectangular area). One example is a Sudoku with 22 clues (second image). The largest total number of empty groups (rows, columns, and boxes) in a Sudoku is believed to be nine. One example is a Sudoku with 3 empty rows, 3 empty columns, and 3 empty boxes (third image). 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. One example of a grid which is automorphic would"
"symbols. If the symbols of the Sudoku are sent in packets the channel can be described as a packet erasure channel model. A sudoku is a formula_5 number-placement puzzle. It is filled in a way, that in each column, row and sub-grid N distinct symbols occur exactly once. Typical alphabet is the set of the integers formula_6. The size of the sub-grids limit the size of SUDOKUs to formula_7 with formula_8. Every solved sudoku and every sub-grid of it is a Latin square, meaning every symbol occurs exactly once in each row and column. At the starting point (in this"
"this number of clues is minimal for this class of Sudoku. The fewest clues in a Sudoku with two-way diagonal symmetry is believed to be 18, and in at least one case such a Sudoku also exhibits automorphism. Additional constraints (here, on 3×3 Sudokus) lead to a smaller minimum number of clues. ""Du-sum-oh"" (a.k.a. ""geometry number place"") puzzles replace the 3×3 (or ""R""×""C"") regions of Sudoku with irregular shapes of a fixed size. Bob Harris has proved that it is always possible to create (""N"" − 1)-clue du-sum-ohs on an ""N""×""N"" grid, and has constructed several examples. Johan de Ruiter"
"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 –"
"of these is ""Greater Than Sudoku"". In this, a 3×3 grid of the Sudoku is given with 12 symbols of Greater Than (>) or Less Than (<) on the common line of the two adjacent numbers. Another variant on the logic of solution is ""Clueless Sudoku"", in which nine 9×9 Sudoku grids are each placed in a 3×3 array. The center cell in each 3×3 grid of all nine puzzles is left blank and form a tenth Sudoku puzzle without any cell completed; hence, ""clueless"". Examples and other variants can be found in the Glossary of Sudoku. This section refers"
"about (with 0.065% relative error) 3.10 × 10 minimal puzzles and 2.55 × 10 nonessentially equivalent minimal puzzles exist. 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"
"to classic Sudoku, disregarding jigsaw, hyper, and other variants. A completed Sudoku grid is a special type of Latin square with the additional property of no repeated values in any of the nine blocks (or ""boxes"" of 3×3 cells). The relationship between the two theories is known, after it was proven that a first-order formula that does not mention blocks is valid for Sudoku if and only if it is valid for Latin squares. The general problem of solving Sudoku puzzles on ""n""×""n"" grids of ""n""×""n"" blocks is known to be NP-complete. Many computer algorithms, such as backtracking and dancing"
"through logic and brute force computation. Russell and Jarvis also showed that when symmetries were taken into account, there were 5,472,730,538 solutions . The number of grids for 16×16 Sudoku is not known. The number of minimal Sudokus (Sudokus in which no clue can be deleted without losing uniqueness of the solution) is not precisely known. However, statistical techniques combined with a generator (), show that there are approximately (with 0.065% relative error): Other authors used faster methods and calculated additional precise distribution statistics. As in the case of Latin squares the (addition- or) multiplication tables (Cayley tables) of finite"
"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"
"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"
"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"
"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"
"and sold more than 8 million copies worldwide. Due to its popularity, Nintendo made a second ""Brain Age"" game titled ""Brain Age"", which has over 100 new Sudoku puzzles and other activities. In June 2008, an Australian drugs-related jury trial costing over A$ 1 million was aborted when it was discovered that five of the twelve jurors had been playing Sudoku instead of listening to evidence. Although the 9×9 grid with 3×3 regions is by far the most common, many other variations exist. Sample puzzles can be 4×4 grids with 2×2 regions; 5×5 grids with ""pentomino"" regions have been published"
"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"
"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"
"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"
"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"
"(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"
"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"""
"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 properties of (1) completed grids and (2) puzzles. Also studied are computer algorithms"
"is then the number of orbits, which can be computed using Burnside's lemma. The Burnside ""fixed points"" are solutions that differ only by relabeling. Using this technique, Jarvis/Russell computed the number of essentially different (symmetrically distinct) solutions as 5,472,730,538. Enumeration results for many Sudoku variants have been calculated: these are summarised below. In the table, ""Dimensions"" are those of the regions (e.g. 3x3 in normal Sudoku). The ""Rel Err"" column indicates how a simple approximation, using the generalised method of Kevin Kilfoil, compares to the true grid count: it is an underestimate in all cases evaluated so far. For large"
"Sudokus can have a different number of clues. This section discusses the minimum number of givens for proper puzzles. Many Sudokus have been found with 17 clues, although finding them is not a trivial task. A paper by Gary McGuire, Bastian Tugemann, and Gilles Civario, released on 1 January 2012, explains how it was proved through an exhaustive computer search that the minimum number of clues in any proper Sudoku is 17, and this was independently confirmed in September 2013. A few 17-clue puzzles with diagonal symmetry were provided by Ed Russell, after a search through equivalence transformations of Gordon"
"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"
"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"
"(a) + (b) = 10. Since these Sudokus are automorphic, so too their solutions grids are automorphic. Furthermore, every cell which is solved has a symmetrical partner which is solved with the same technique (and the pair would take the form a + b = 10). Notice that in the second example, the Sudouku also exhibits translational (or repetition) symmetry; clues are clustered in groups, with the clues in each group ordered sequentially (i.e., n, n+1, n+2, and n+3). The third image is the ""Most Canonical"" solution grid. This grid has 648 automorphisms and contributes to all ~ solution grids"
"completions of the lower two bands were constructed and counted for each equivalence class. Summing completions over the equivalence classes, weighted by class size, gives the total number of solutions as 6,670,903,752,021,072,936,960, confirming the value obtained by QSCGZ. The value was subsequently confirmed numerous times independently. A second enumeration technique based on band generation was later developed that is significantly less computationally intensive. Two valid grids are ""essentially"" the same if one can be derived from the other. The following operations are Sudoku preserving symmetries and always translate a valid grid into another valid grid: These operations define a symmetry"