A completed Sudoku grid is a special type of Latin square with the additional property of no repeated values in any of the 9 blocks of contiguous 3×3 cells. The relationship between the two theories is now completely known, after it was proven that a first-order formula that does not mention blocks (also called boxes or regions) is valid for Sudoku if and only if it is valid for Latin Squares (this property is trivially true for the axioms and it can be extended to any formula).
The number of classic 9×9 Sudoku solution grids is 6,670,903,752,021,072,936,960 (sequence A107739 in OEIS), or approximately 6.67×1021. This is roughly 1.2×10−6 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 (sequence A109741 in OEIS).
The maximum number of givens 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 which occupy 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—was recently proven to be 17. A number of valid puzzles with 17 givens have been found for the standard variation without a symmetry constraint, by Japanese puzzle enthusiasts, and 18 with the givens in rotationally symmetric cells. Over 48,000 examples of Sudoku puzzles with 17 givens resulting in a unique solution are known.
In 2010 mathematicians of the University of Southern California showed that the arrangement of numbers in Sudoku puzzles have greater Shannon entropy than the number arrangements in randomly generated 9×9 matrices. This is because the rules of Sudoku exclude some random arrangements that have an innate symmetry.
 Recent popularityIn 1997, New Zealander and retired Hong Kong judge Wayne Gould, then in his early 50s, saw a partly completed puzzle in a Japanese bookshop. Over six 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 November 12, 2004 (calling it Su Doku). The first letter to The Times regarding Su Doku was published the following day on November 13 from Ian Payn of Brentford, complaining that the puzzle had caused him to miss his stop on the tube.
The rapid rise of Sudoku in Britain from relative obscurity to a front-page feature in national newspapers attracted commentary in the media and parody (such as when The Guardian's G2 section advertised itself as the first newspaper supplement with a Sudoku grid on every page). Recognizing the different psychological appeals of easy and difficult puzzles, The Times introduced both side by side on June 20, 2005. From July 2005, Channel 4 included a daily Sudoku game in their Teletext service. On August 2, the BBC's program guide Radio Times featured a weekly Super Sudoku which features a 16×16 grid.
In the United States, the first newspaper to publish a Sudoku puzzle by Wayne Gould was The Conway Daily Sun (New Hampshire), in 2004.
puzzle contest first broadcast on July 1, 2005 on Sky One. It was presented by Carol Vorderman. Nine teams of nine players (with one celebrity in each team) representing geographical regions competed to solve a puzzle. Each player had a hand-held device for entering numbers corresponding to answers for four cells. Phil Kollin of Winchelsea, England was the series grand prize winner taking home over £23,000 over a series of games. The audience at home was in a separate interactive competition, which was won by Hannah Withey of Cheshire.
Later in 2005, the BBC launched SUDO-Q, a game show that combines Sudoku with general knowledge. However, it uses only 4×4 and 6×6 puzzles. Four seasons were produced, before the show ended in 2007.
In 2006, a Sudoku website published songwriter Peter Levy's Sudoku tribute song, but quickly had to take down the mp3 due to heavy traffic. British and Australian radio picked up the song, which is to feature in a British-made Sudoku documentary. The Japanese Embassy also nominated the song for an award, with Levy doing talks with Sony in Japan to release the song as a single.
Sudoku software is very popular on PCs, websites, and mobile phones. It comes with many distributions of Linux. Software has also been released on video game consoles, such as the Nintendo DS, PlayStation Portable, the Game Boy Advance, Xbox Live Arcade, the Nook e-book reader, Kindle Fire tablet, several iPod models, and the iPhone. In fact, just two weeks after Apple Inc. debuted the online App Store within its iTunes Store on July 11, 2008, there were already nearly 30 different Sudoku games, created by various software developers, specifically for the iPhone and iPod Touch. One of the most popular video games featuring Sudoku is Brain Age: Train Your Brain in Minutes a Day!. Critically and commercially well-received, it generated particular praise for its Sudoku implementation and sold more than 8 million copies worldwide. Due to its popularity, Nintendo made a second Brain Age game titled Brain Age2, 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.
- The first World Sudoku Championship was held in Lucca, Italy, from March 10–12, 2006. The winner was Jana Tylová of the Czech Republic. The competition included numerous variants.
- The second World Sudoku Championship was held in Prague from March 28 – April 1, 2007. The individual champion was Thomas Snyder of the USA. The team champion was Japan.
- The third World Sudoku Championship was held in Goa, India, from April 14–16, 2008. Thomas Snyder repeated as the individual overall champion, and also won the first ever Classic Trophy (a subset of the competition counting only classic Sudoku). The Czech Republic won the team competition.
- The fourth World Sudoku Championship was held in Žilina, Slovakia, from April 24–27, 2009. After past champion Thomas Snyder of USA won the general qualification, Jan Mrozowski of Poland emerged from a 36-competitor playoff to become the new World Sudoku Champion. Host nation Slovakia emerged as the top team in a separate competition of three-membered squads.
- The fifth World Sudoku Championship was held in Philadelphia, USA from April 29 – May 2, 2010. Jan Mrozowski of Poland successfully defended his world title in the individual competition while Germany won a separate team event. The puzzles were written by Thomas Snyder and Wei-Hwa Huang, both past US Sudoku champions.
- In the United States, The Philadelphia Inquirer Sudoku National Championship has been held three times, each time offering a $10,000 prize to the advanced division winner and a spot on the U.S. National Sudoku Team traveling to the world championships. The winners of the event were Thomas Snyder (2007), Wei-Hwa Huang (2008), and Tammy McLeod (2009). In the most recent event, the third place finalist in the advanced division, Eugene Varshavsky, performed quite poorly onstage after setting a very fast qualifying time on paper, which caught the attention of organizers and competitors including past champion Thomas Snyder who requested organizers reconsider his results due to a suspicion of cheating. Following an investigation and a retest of Varshavsky, the organizers disqualified him and awarded Chris Narrikkattu third place.