Main Board
00:00
Difficulty:
The mathematical foundation of Sudoku traces back to the work of Leonhard Euler (1707–1783), the prolific Swiss mathematician. Euler published his formal study of Latin squares in 1782 in a paper titled Recherches sur une nouvelle espèce de quarrés magiques. A Latin square is an arrangement of symbols in a grid such that each symbol appears exactly once in every row and exactly once in every column. Euler coined the name "Latin square" because he used Latin letters as symbols in his original work. While Latin squares provided the theoretical backbone for Sudoku, Euler's grids did not include the 3x3 subgrid constraint that defines the modern puzzle. [Wikipedia, Latin Square]
The modern Sudoku puzzle first appeared in May 1979 under the name "Number Place" in the publication Dell Pencil Puzzles and Word Games. Howard Garns (March 2, 1905 – October 6, 1989), an architect and freelance puzzle designer from Connersville, Indiana who worked at the Daggett architecture firm in Indianapolis, created the puzzle. Garns was 74 years old when his puzzle was first published, and Dell did not credit him by name during his lifetime. His key innovation was adding the 3x3 subgrid constraint to the Latin square concept, producing the 9x9 grid of 81 cells that players know today. Garns passed away in 1989, years before his creation became a worldwide phenomenon. [Wikipedia, Howard Garns] [Wikipedia, Sudoku History]
The puzzle remained largely obscure in the United States throughout the early 1980s. That changed when Maki Kaji, founder and president of the Japanese puzzle publisher Nikoli, encountered Number Place in an American magazine and introduced it to Japanese audiences. Nikoli published the puzzle in April 1984 in its magazine Monthly Nikolist under the full Japanese name "Suji wa dokushin ni kagiru," a phrase that translates loosely to "the digits must be single." Kaji later abbreviated this to "Sudoku," combining the Japanese words su (number) and doku (single). Nikoli also refined the puzzle's conventions, establishing that givens should be arranged symmetrically and that each puzzle must have exactly one valid solution. Kaji became widely known as the Godfather of Sudoku, championing the puzzle internationally until his death in August 2021. [Wikipedia, Maki Kaji] [Nikoli, Sudoku]
Sudoku exploded onto the global stage through the efforts of Wayne Gould, a retired Hong Kong judge born in New Zealand. In 1997, Gould discovered a Sudoku book in a Tokyo bookshop and spent the next six years developing a computer program capable of generating puzzles at varying difficulty levels. In 2004, he approached The Times of London and offered to supply puzzles at no charge. The first Times Sudoku was published on November 12, 2004, and the reader response was immediate and overwhelming. Within weeks, nearly every major British newspaper including The Daily Telegraph, The Guardian, The Independent, and The Daily Mirror had launched its own daily Sudoku feature. By early 2005, the craze had spread to the United States, Australia, Europe, and beyond. The puzzle's universal appeal stems from the fact that it requires no language skills and no arithmetic knowledge, only pure logical reasoning, making it accessible to solvers in every country and culture. The first World Sudoku Championship was hosted in Italy in 2006, and annual championships have been held ever since. [Wikipedia, Wayne Gould] [SudokuPulse, History of Sudoku]
Modern Sudoku applications use computer algorithms to generate an essentially unlimited supply of puzzles across a range of difficulty levels. The typical process begins with the construction of a fully solved, valid grid. Numbers are then removed one at a time while a solver algorithm verifies at each step that the puzzle retains a unique solution. This guarantees that every generated puzzle can be solved through logical deduction alone, without guessing. The number of valid completed Sudoku grids is astronomically large, ensuring that puzzle generation never runs short of novel arrangements. [AMS, Sudoku Mathematics]
Sudoku is played on a 9x9 grid divided into nine 3x3 subgrids called boxes or regions. The puzzle begins with a set of cells already filled in with digits, known as givens or clues. The player must fill every remaining empty cell so that each row, each column, and each 3x3 box contains every digit from 1 to 9 exactly once. No digit may repeat within any row, column, or box. Importantly, Sudoku is a logic puzzle, not a math puzzle. The digits 1 through 9 serve purely as nine distinct symbols and could just as easily be replaced by letters, shapes, or colors without changing the nature of the challenge. No addition, multiplication, or arithmetic of any kind is required. [Wikipedia, Sudoku]
Sudoku is formally classified as a constraint satisfaction problem. A completed Sudoku grid is a special type of Latin square with the additional 3x3 box constraint layered on top. The total number of valid completed 9x9 Sudoku grids is exactly 6,670,903,752,021,072,936,960, which is approximately 6.67 sextillion (6.67 x 10^21). When equivalent grids produced by symmetry operations such as rotating the board, reflecting it, or relabeling digits are counted as one, the number of essentially different Sudoku grids reduces to 5,472,730,538. Research has also established that the minimum number of givens required to produce a puzzle with a unique solution is 17, meaning no valid Sudoku puzzle can be constructed with 16 or fewer starting clues. [Wikipedia, Mathematics of Sudoku]
Sudoku difficulty is determined not simply by how many givens are provided, but by the solving techniques a player must apply to reach the solution. Easy puzzles can be completed using only the most basic techniques, specifically naked singles and candidate elimination. Medium puzzles require hidden singles and simple subset analysis. Hard puzzles introduce pattern-based techniques such as X Wings and pointing pairs. Very hard puzzles may demand Swordfish patterns, forcing chains, or other advanced deductive methods. The difficulty levels in this game correspond to the solving techniques each tier requires, giving players a genuinely escalating challenge. [Sudoku.com, How to Play]
This game uses the open source sudoku.js library by Rob McGuire to generate puzzles. The library defines difficulty by the number of given squares provided to the player at the start: Easy puzzles provide 62 givens, Medium puzzles provide 53, Hard puzzles provide 44, and Very Hard puzzles provide 35. The generator begins by constructing a randomized valid solution grid and then systematically removes cells down to the target number of givens. The library also incorporates a solver to confirm solvability throughout this removal process, producing well-formed puzzles. [GitHub, sudoku.js by robatron]
Naked Singles: The simplest possible move. When an empty cell has only one remaining candidate after all givens in its row, column, and box are accounted for, that candidate must be the correct digit. Scanning systematically for naked singles is always the first step in solving any puzzle, and on Easy difficulty it is often sufficient to complete the entire grid without any additional technique.
Candidate Elimination (Cross Hatching): For each empty cell, mentally list every digit from 1 to 9 and eliminate any that already appear in the same row, column, or box. The digits that survive this elimination are the cell's candidates. Repeating this process across the grid progressively narrows down possibilities and often reveals naked singles that were not immediately visible. [Sudoku of the Day, Naked Singles]
Hidden Singles: A hidden single occurs when a particular digit can only be placed in one cell within a given row, column, or box, even though that cell appears to have multiple candidates. By scanning each unit for digits that appear as a candidate in only one location, a solver can place that digit with certainty. Hidden singles solve a large portion of Medium difficulty puzzles. [Sudoku of the Day, Hidden Singles]
Hidden Pairs and Triples: When exactly two digits appear as candidates in only the same two cells within a row, column, or box, those two cells must contain those two digits in some order, and all other candidates can be removed from those cells. Hidden triples extend this concept to three digits confined to three cells. Spotting these patterns requires careful candidate bookkeeping but can unlock progress when simpler techniques stall. [Sudoku of the Day, Hidden Pairs]
Naked Pairs and Triples: When two cells in the same row, column, or box each contain exactly the same two candidates and no others, those two digits are locked into those two cells. Every other cell in that unit can have those two digits removed from its candidate list. Naked triples work identically but involve three cells and three candidates. [Sudoku of the Day, Naked Pairs]
X Wing: An X Wing forms when a particular candidate digit appears in exactly two cells in each of two different rows, and both rows share the same two columns for those placements. Because the digit must occupy one of those two cells in each row, it can be safely eliminated from every other cell in those two columns. The pattern resembles four corners of a rectangle, hence the name. X Wings appear frequently in Hard and Very Hard puzzles and are one of the most recognizable advanced patterns. [Sudoku of the Day, X Wing]
Swordfish: Swordfish is a three-row extension of the X Wing pattern. When a candidate digit appears in two or three cells across three different rows, and all of those occurrences fall within the same three columns, the digit can be eliminated from every other cell in those three columns. Swordfish patterns are harder to spot than X Wings but are essential tools for solving Very Hard puzzles without resorting to trial and error. [Sudoku of the Day, Swordfish]
Forcing Chains: Forcing chains are among the most powerful purely logical techniques available before resorting to brute force guessing. The solver temporarily assumes that a particular candidate in a particular cell is either true or false, then follows the chain of logical consequences that assumption creates across the grid. If the assumption leads to a contradiction (a row, column, or box with no valid placement for some digit), the assumption must be false, and the opposite value is confirmed. A strong forcing chain analysis can resolve puzzles that resist every pattern-based technique. [SudokuWiki, Forcing Chains]