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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 after each individual grid has at least been partially solved.

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 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 modern **Sudoku** was most likely designed anonymously by Howard Garns, a 74-year-old retired architect and freelance puzzle constructor from Connersville, Indiana, and first published in 1979 by Dell Magazines as Number Place (the earliest known examples of modern Sudoku). Garns's name was always present on the list of contributors in issues of Dell Pencil Puzzles and Word Games that included Number Place, and was always absent from issues that did not. He died in 1989 before getting a chance to see his creation as a worldwide phenomenon. Whether or not Garns was familiar with any of the French newspapers listed above is unclear.

The Killer **Sudoku** variant combines elements of **Sudoku** and Kakuro.

The cognitive scientist, Jeremy Grabbe, found that **Sudoku** involved an area of cognition called working memory. A subsequent experiment by Grabbe showed that routine **Sudoku** playing could improve working memory in older people.

The puzzle was introduced in Japan by Nikoli in the paper Monthly Nikolist in April 1984 as Sūji wa dokushin ni kagiru (数字は独身に限る), which also can be translated as "the digits must be single" or "the digits are limited to one occurrence" (In Japanese, dokushin means an "unmarried person"). At a later date, the name was abbreviated to **Sudoku** (数独) by Maki Kaji (鍜治 真起), taking only the first kanji of compound words to form a shorter version. "Sudoku" is a registered trademark in Japan and the puzzle is generally referred to as Number Place (ナンバープレース) or, more informally, a portmanteau of the two words, Num(ber) Pla(ce) (ナンプレ). In 1986, Nikoli introduced two innovations: the number of givens was restricted to no more than 32, and puzzles became "symmetrical" (meaning the givens were distributed in rotationally symmetric cells). It is now published in mainstream Japanese periodicals, such as the Asahi Shimbun.

"Quadratum latinum" is a **Sudoku** variation with Roman numerals (I, 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.

The number of classic 9×9 **Sudoku** solution grids is 6,670,903,752,021,072,936,960, or around 6.67. This is roughly 1.2 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.

The general problem of solving **Sudoku** puzzles on n 2 ×n 2 grids of n×n blocks is known to be NP-complete. Many computer algorithms, such as backtracking and dancing 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.

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 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.

Glossary of **Sudoku** - **Sudoku** variants

The classic 9×9 **Sudoku** format can be generalized to an :N×N row-column grid partitioned into N regions, where each of the N rows, columns and regions have N cells and each of the N digits occur once in each row, column or region.

Mathematics of **Sudoku** - **Sudoku** bands

The band counts for problems whose full **Sudoku** grid-count is unknown are listed below. As in the previous section, "Dimensions" are those of the regions.

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.

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 about (with 0.065% relative error) 3.10 × 10 37 minimal puzzles and 2.55 × 10 25 nonessentially equivalent minimal puzzles exist.

Glossary of **Sudoku** - **Sudoku** variants

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 cover the grid can be relaxed.

Mathematics of **Sudoku** - Ordinary **Sudoku**

A **Sudoku** with 24 clues, dihedral symmetry (symmetry on both orthogonal axis, 90° rotational symmetry, 180° rotational symmetry, and diagonal symmetry) is known to exist, and is also automorphic. Again here, it is not known if 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.

the average rate of a standard **Sudoku** is

Mathematics of **Sudoku** - **Sudoku** bands

For large (R,C), the method of Kevin Kilfoil (generalised method: ) is used to estimate the number of grid completions. The method asserts that the **Sudoku** row and column constraints are, to first approximation, conditionally independent given the box constraint. Omitting a little algebra, this gives the Kilfoil-Silver-Pettersen formula: :where b R,C is the number of ways of completing a **Sudoku** band of R horizontally adjacent R×C boxes. Petersen's algorithm, as implemented by Silver, is currently the fastest known technique for exact evaluation of these b R,C.

Glossary of **Sudoku** - **Sudoku** variants

Other variations include additional value placement constraints, alternate symbols (e.g. letters), alternate mechanism for expressing the clues, and compositions with overlapping grids. See **Sudoku** – Variants for details and additional variants.

Mathematics of **Sudoku** - Ordinary **Sudoku**

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 Royle's database of 17-clue puzzles. **Sudoku** puzzles with 18 clues have been found with 180° rotational symmetry, and others with orthogonal symmetry, although it is not known if this number of clues is minimal in either case. **Sudoku** puzzles with 19 clues have been found with two-way orthogonal symmetry, and again it is unknown if this number of clues is minimal for this case.

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