![]() Many known hard problems are of a type called The minimum problem is to find the minimum number of clues which As cluesĪre put in, and the constraints applied, the number of possible states Of constraints without clues is the counting problem of Su Doku. Number of possible Su Dokus, then this would imply that counting Su Doku is a ![]() If depth-first enumeration were the only way of counting the This is larger than anyįixed power of M (this is said to be faster than any polynomial in M). M×M Su Doku puzzle, and the constraints are applied, there are This is where much of the counting appears. This point has been made by many people, and explored systematically by Keeping the pencil marks updated is part of constraint programming. "Pencilling in" all possible values allowed in a square, and then Is called constraint programming in computer science. TheĪpplication of constraints repeatedly in order to reduce the space of possibilities Key is to apply them over and over again: to each cell, row and column. Solve: identifying hidden loners and simple instances of locked candidates. The minimum Su Doku shown alongside (only 17 clues) requires only two tricks to Outside of the cell." Since the hidden pair 2 and 3 prevent anything else fromĪpearing in the first two columns of the middle rightmost cell, an 8 can only appear ![]() Since one of these squares must contain that specific candidate, theĬandidate can safely be excluded from the remaining squares in that row or column Squares, no other numbers can appear there.Īngus Johnson again: "Sometimes a candidate within a cell is restricted to one Since these two numbers have to be in these two They are "pencilled in" in small blue font. Middle rightmost cell these two numbers can only appear in the two positions where The example on the right, a 2 and a 3 cannot appear in the last column. Another example is SUDOKU=IS*FUNNY whose solutionĪngus Johnson has this to say about hidden pairs: "If two squares in a group containĪn identical pair of candidates and no other squares in that group contain those twoĬandidates, then other candidates in those two squares can be excluded safely." In AnĮxcellent example is NUMBER+NUMBER=KAKURO which has a unique solutionġ86925+186925=373850. In which each letter represents a single digit from 0 to 9. The clues can also be given by cryptic alphametics On a 9 × 9 grid, called Cross Sums Sudoku, in which clues are given in Object is the same as standard Sudoku, but the puzzle only uses theĪnother variant is the combination of Sudoku with Kakuro TheĪforementioned Number Place Challenger puzzles are all of this variant, as are the Sudoku X puzzles in the Daily Mail, which use 6×6 grids.Ī variant named "Mini Sudoku" appears in the American newspaper USA TodayĪnd elsewhere, which is played on a 6×6 grid with 3×2 regions. The numbers in the main diagonals of the grid also to be unique. Takes the form of an extra "dimension" the most common is to require Sudoku-zilla, a 100x100-grid was published in print in 2010.Īnother common variant is to add limits on the placement of numbersīeyond the usual row, column, and box requirements. Nikoli offers 25×25 Sudoku the Giant behemoths. Dell regularly publishes 16×16 Number Place Challenger puzzles (the 16×16 variant often uses 1 through G rather than the 0 through F used in hexadecimal). The Times offers a 12×12-grid Dodeka sudoku with 12 regions of 4×3 squares. Grids with pentomino regions have been published 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. Sample puzzles can be 4×4 grids with 2×2 regions 5×5 It became an international hit in 2005Īlthough the 9×9 grid with 3×3 regions is by far the most common, The puzzle was popularized in 1986 by the Japanese puzzle company Nikoli, under the name Sudoku, meaning single number. With an additional constraint on the contents of individual regions.įor example, the same single integer may not appear twice in the sameĩx9 playing board row or column or in any of the nine 3x3 subregions of Grid, which typically has a unique solution.Ĭompleted puzzles are always a type of Latin square The puzzle setter provides a partially completed 1- Explanation Meaning Sudoku 数独 sūdoku The objective is to fill a 9×9 grid with digits so that each column,Įach row, and each of the nine 3×3 sub-grids that compose the grid (alsoĬalled "boxes", "blocks", "regions", or "sub-squares") contains all of
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