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Another common variant is for additional restrictions to be enforced on the placement of numbers beyond the usual row, column, and region requirements. Often the restriction takes the form of an extra "dimension"; the most common is for the numbers in the main diagonals of the grid to also be required to be unique. The aforementioned Number Place Challenger puzzles are all of this variant, as are the Sudoku X puzzles in the Daily Mail, which use 6×6 grids. The Daily Mail also features Super Sudoku X in its Weekend magazine: an 8×8 grid in which rows, columns, main diagonals, 2×4 blocks and 4×2 blocks contain each number once. Another dimension in use is digits with the same relative location within their respective regions; such puzzles are usually printed in colour, with each disjoint group sharing one colour for clarity. Also found is the Circular Sudoku, also known as Target Sudoku, invented by Essex mathematician Peter Higgins. [3] [4] In this variant, all the numbers must appear in all the concentric rings as well as in all pairs of adjacent wedges. Yoshimitsu Kanai published his computerized puzzle generator under the name Single Number for the Apple Macintosh [15] in 1995 in Japanese and English, for the Palm (PDA) [16] in 1996, and for the Mac OS-X [17] in 2005. The level of difficulty of the puzzles can be selected to suit the audience. The puzzles are often available free from published sources and may also be custom-generated using software. Counting 1-9 in regions, rows, and columns to identify missing numerals. Counting based upon the last numeral discovered may speed up the search. It also can be the case (typically in tougher puzzles) that the easiest way to ascertain the value of an individual cell is by counting in reverse—that is, by scanning the cell's region, row, and column for values it cannot be, in order to see which is left. Ideally one needs to find a combination of techniques which avoids some of the drawbacks of the above elements. The counting of regions, rows, and columns can feel boring. Writing candidate numerals into empty cells can be time-consuming. The what-if approach can be confusing unless you are well organised. The proverbial Holy Grail is to find a technique which minimizes counting, marking up, and rubbing out. The numerals in Sudoku puzzles are used for convenience; arithmetic relationships between numerals are absolutely irrelevant. Any set of distinct symbols will do; letters, shapes, or colours may be used without altering the rules In "candidate elimination", progress is made by successively eliminating candidate numerals from one or more cells to leave just one choice. After each answer has been achieved, another scan may be performed—usually checking to see the effect of the contingencies. This is a column, 9 cells tall. A filled-in column must have one of each digit. That means that each digit appears only once in the column. There are 9 columns in the grid, and the same applies to each of them.

The name Sudoku is the Japanese abbreviation of a longer phrase, "suuji wa dokushin ni kagiru (????????)," meaning "the digits must remain single"; it is a trademark of puzzle publisher Nikoli Co. Ltd in Japan. The 2005 U.S. Puzzle Championship includes a variant called Digital Number Place: rather than givens, most cells contain a partial given—a segment of a number, with the numbers drawn as if part of a seven-segment display. In 1997, retired Hong Kong judge Wayne Gould, 59, a New Zealander, saw a partly completed puzzle in a Japanese bookshop. Over 6 years he developed a computer program to produce puzzles quickly. Knowing that British newspapers have a long history of publishing crosswords and other puzzles, he promoted Sudoku to The Times in Britain, which launched it on 12 November 2004 (calling it su doku). The puzzles by Pappocom, Gould's software house, have been printed daily in the Times ever since. The name Sudoku is the Japanese abbreviation of a longer phrase, "suuji wa dokushin ni kagiru (????????)," meaning "the digits must remain single"; it is a trademark of puzzle publisher Nikoli Co. Ltd in Japan. Yoshimitsu Kanai published his computerized puzzle generator under the name Single Number for the Apple Macintosh [15] in 1995 in Japanese and English, for the Palm (PDA) [16] in 1996, and for the Mac OS-X [17] in 2005. You solve the puzzle with reasoning and logic. In "candidate elimination", progress is made by successively eliminating candidate numerals from one or more cells to leave just one choice. After each answer has been achieved, another scan may be performed—usually checking to see the effect of the contingencies. Here are some of the more notable single-instance variations:

Sudoku puzzle game and solver by MuddyFunksters

This principle also works with candidate numeral subsets, that is, if three cells have candidates (p,q,r), (p,q), and (q,r) or even just (p,r), (q,r), and (p,q), all of the set (p,q,r) elsewhere within that same scope can be deleted. The principle is true for all quantities of candidate numerals. Most publications sort their Sudoku puzzles into four rating levels, although the actual cut-off points of the levels and indeed the names of the levels themselves can vary widely. Typically, however, the titles are some set of synonyms of "easy", "intermediate", "hard", and "challenging". The first principle is based on cells where only matched numerals appear. The second is based on numerals that appear only in matched cells. The validity of either principle is demonstrated by posing the question, 'Would entering the eliminated numeral prevent completion of the other necessary placements?' If the answer to the question is 'Yes,' then the candidate numeral in question can be eliminated. Advanced techniques carry these concepts further to include multiple rows, columns, and regions. In the "what-if" approach, a cell with only two candidate numerals is selected, and a guess is made. The steps above are repeated unless a duplication is found or a cell is left with no possible candidate, in which case the alternative candidate is the solution. In logical terms, this is known as reductio ad absurdum. Nishio is a limited form of this approach: for each candidate for a cell, the question is posed: will entering a particular numeral prevent completion of the other placements of that numeral? If the answer is yes, then that candidate can be eliminated. The what-if approach requires a pencil and eraser. This approach may be frowned on by logical purists as trial and error (and most published puzzles are built to ensure that it will never be necessary to resort to this tactic) but it can arrive at solutions fairly rapidly. Although the 9×9 grid with 3×3 regions is by far the most common, numerous variations abound: sample puzzles can be 4×4 grids with 2×2 regions; 5×5 grids with pentomino regions have been published under the name Logi-5; the World Puzzle Championship has previously featured a 6×6 grid with 2×3 regions and a 7×7 grid with six heptomino regions and a disjoint region; Daily SuDoku features new 4×4, 6×6, and simpler 9×9 grids every day as Daily SuDoku for Kids. [1] Even the 9×9 grid is not always standard, with Ebb regularly publishing some of those with nonomino regions (also known as a jigsaw variation); the 2005 U.S. Puzzle Championship had a Sudoku with parallelogram regions that wrapped around the outer border of the puzzle, as if the grid were toroidal. Larger grids are also possible, with Daily SuDoku's 12×12-grid Monster SuDoku [2], the Times likewise offers a 12×12-grid Dodeka sudoku with 12 regions each being 4×3, Dell regularly publishing 16×16 Number Place Challenger puzzles (the 16×16 variant often uses 1 through G rather than the 0 through F used in hexadecimal), and Nikoli proffering 25×25 Sudoku the Giant behemoths. Every puzzle has just one correct solution. Dr. House was clearly seen working on a Sudoku puzzle on his office computer in one scene of the December 13, 2005 episode of House, M. D.; Sudoku is supposedly now banned on the studio set due to the cast constantly playing it. Some cells already contain numerals, known as "givens" (or sometimes as "clues"). The goal is to fill in the empty cells, one numeral in each, so that each column, row, and region contains the numerals 1–9 exactly once.

Although the 9×9 grid with 3×3 regions is by far the most common, numerous variations abound: sample puzzles can be 4×4 grids with 2×2 regions; 5×5 grids with pentomino regions have been published under the name Logi-5; the World Puzzle Championship has previously featured a 6×6 grid with 2×3 regions and a 7×7 grid with six heptomino regions and a disjoint region; Daily SuDoku features new 4×4, 6×6, and simpler 9×9 grids every day as Daily SuDoku for Kids. [1] Even the 9×9 grid is not always standard, with Ebb regularly publishing some of those with nonomino regions (also known as a jigsaw variation); the 2005 U.S. Puzzle Championship had a Sudoku with parallelogram regions that wrapped around the outer border of the puzzle, as if the grid were toroidal. Larger grids are also possible, with Daily SuDoku's 12×12-grid Monster SuDoku [2], the Times likewise offers a 12×12-grid Dodeka sudoku with 12 regions each being 4×3, Dell regularly publishing 16×16 Number Place Challenger puzzles (the 16×16 variant often uses 1 through G rather than the 0 through F used in hexadecimal), and Nikoli proffering 25×25 Sudoku the Giant behemoths. The numerals in Sudoku puzzles are used for convenience; arithmetic relationships between numerals are absolutely irrelevant. Any set of distinct symbols will do; letters, shapes, or colours may be used without altering the rules Although for standard Sudoku problems highly optimized and sophisticated backtracking programs are fastest, another popular way of solving such constraint problems is Donald Knuth's Dancing Links Algorithm for solving the exact matrix cover problem, of which the Sudoku problems are a special case. Knuth's algorithm can be applied by converting the Sudoku puzzle to a matrix cover problem, solve this problem instead, and convert the solution obtained back to a completed Sudoku grid. This method is now preferred by many Sudoku programmers, by virtue of its execution speed, simplicity and ease of implementation and the availability of documentation and reference source code. Scanning is performed at the outset and throughout the solution. Scans only have to be performed one time in between analysis periods. Scanning consists of two basic techniques: The general problem of solving Sudoku puzzles on n2 x n2 boards of n x n blocks is known to be NP-complete [9]. This gives some indication of why Sudoku is difficult to solve, although on boards of finite size the problem is finite and can be solved by a deterministic finite automaton that knows the entire game tree. In Japanese, the word is pronounced [s??dok?]; in English, it is usually spoken with an Anglicised pronunciation, [s?'d??ku?] (BrE) [s?'do?ku?] (AmE) or ['su?d??ku] (BrE) ['su?do?ku] (AmE) (See IPA, International Phonetic Alphabet for notation usage.) The maximum number of givens that can be provided while still not rendering the solution unique is four short of a full grid; if two instances of two numbers each are missing and the cells they are to occupy form the corners of an orthogonal rectangle, and exactly two of these cells are within one region, there are two ways the numbers can be assigned. Since this applies to Latin squares in general, most variants of Sudoku have the same maximum. The inverse problem—the fewest givens that render a solution unique—is unsolved, although the lowest number yet found for the standard variation without a symmetry constraint is 17, a number of which have been found by Japanese puzzle enthusiasts [12] [13], and 18 with the givens in rotationally symmetric cells.