Sudoku Strategy

Scanning

Scanning is performed at the outset and throughout the solution. Scans need be performed only once between analyses. Scanning consists of two techniques:

Cross-hatching: The scanning of rows to identify which line in a region may contain a certain numeral by a process of elimination. The process is repeated with the columns. It is important to perform this process systematically, checking all of the digits 1–9.
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, particularly in tougher puzzles, that the best way to ascertain the value of a cell is to count in reverse—that is, by scanning the cell's region, row, and column for values it cannot be, in order to see what remains. By doing this it may be possible to reduce a cell's options to a single entry which adds meaning to the term Sudoku.

Advanced solvers look for "contingencies" while scanning, narrowing a numeral's location within a row, column, or region to two or three cells. When those cells lie within the same row and region, they can be used for elimination during cross-hatching and counting. Puzzles solved by scanning alone without requiring the detection of contingencies are classified as "easy"; more difficult puzzles are not readily solved by basic scanning alone. A method for marking likely numerals in a single cell with dots. To reduce marking, this would wait until as many numbers as possible have been added via scanning. Dots are erased as the numerals are eliminated as candidates. A method for marking likely numerals in a single cell with dots. To reduce marking, this would wait until as many numbers as possible have been added via scanning. Dots are erased as the numerals are eliminated as candidates. The bottom middle sub-square needs a 3, 5, and 6 in the top row. This creates a contingency which, although unresolved, reveals that the green square must be a 4. The bottom middle sub-square needs a 3, 5, and 6 in the top row. This creates a contingency which, although unresolved, reveals that the green square must be a 4.

Marking up

Scanning stops when no further numerals can be discovered, making it necessary to engage in logical analysis. One method to guide the analysis is to mark candidate numerals in the blank cells.

Subscript notation

In subscript notation, the candidate numerals are written in subscript in the cells. Because puzzles printed in a newspaper are too small to accommodate more than a few subscript digits of normal handwriting, solvers may create a larger copy of the puzzle. Using two colors, or mixing pencil and pen marks can be helpful.

Dot notation

The dot notation uses a pattern of dots in each square, where the dot position indicates a number from 1 to 9. The dot notation can be used on the original puzzle. Dexterity is required in placing the dots, since misplaced dots or inadvertent marks lead to confusion and may be difficult to erase. However, confusion can arise as to whether a number has been excluded from a cell, or whether a number has been analyzed in the first place. Because at the start of the puzzle all numbers have not yet been excluded, the correct way to begin a puzzle using this method would be to dot all 9 numbers in each cell.

Another technique is to mark the numerals that a cell cannot be. The cell starts empty and as more constraints become known, it slowly fills until only one mark is missing. Assuming no mistakes are made and the marks can be overwritten with the value of a cell, there is no longer a need for any erasures.

The most efficient means to record analysis is to record both numbers that have been excluded, and numbers that are in the solution locus. This can be accommodated by using circles to document numbers in the solution locus, and an 'x' to document excluded numbers. The position of the character indicates a number from 1 to 9 (similar to Dot Notation), and an X can easily be written over a circle once a number has been excluded from the solution locus. (click to enlarge) An analysis in superscript notation, with all possible values written in. There are three squares which contain only three values: 4, 6, and 8. If these numbers were written in any square where they are red, it would be impossible to complete the squares where they are blue. Therefore, the numbers in red can be erased.

Analysis

The two main approaches to analysis are "candidate elimination" and "what-if".

Candidate elimination

In "candidate elimination", progress is made by successively eliminating candidate numerals to leave one choice for a given cell. After each answer is found, another scan may be performed—usually checking to see the effect on contingencies. In general, if entering a numeral prevents completion of other empty cells, then the numeral can be eliminated as a candidate.

One method of candidate elimination works by identifying "matched cell groups". For instance, if precisely two cells within a scope (a particular row, column, or region) contain only the same two candidate numerals (p,q), or if precisely three cells within a scope contain only the same three candidate numerals (p,q,r), these cells are said to be matched. The placement of those candidate numerals anywhere else within the same scope would make a solution impossible, allowing the numbers to be eliminated as candidates from those other cells.

What-if

In the "what-if" approach (also called "guess-and-check", "bifurcation", "backtracking" and "Ariadne's thread"), a cell with two candidate numerals is selected, and a guess is made. The results are followed until a duplication is found or a cell is left without a candidate, in which case the alternative must have been the solution. For each cell's candidate, the question is posed: 'will entering a particular numeral prevent completion of the other placements of that numeral?' If 'yes', then that candidate can be eliminated. If the "what-if" exercises show that either candidate is possible, then another pair should be tried. Alternatively, if the "what-if" exercises for both candidates imply an identical result, then that result is known. The what-if approach is essentially a more difficult approach, because it requires remembering only the specific moves made after the decision to conduct a "what-if," so a pencil and eraser is handy. Also, using this method can lead to finding alternative solutions, which can temporarily lead to confusion as to whether what has been imputed is correct.

There are three kind of conflicts, which can appear during puzzle solving:

basic conflicts - there are only N-1 different candidates in N cell in the area
fish conflicts - when eliminating number from N rows/columns, it will disappear also from N+1 columns/rows.
unique conflicts - this pattern means multiple solutions, all numbers in the pattern exist exactly two times in every area, row and column. If there is only one candidate in the cell, any virtual candidate can be added.

Encountering any of those would indicate that the puzzle is not uniquely solvable. Encountering any of them as a consequence of "what-if" indicates that an untried alternative is correct.

Computer solutions

There are three general approaches taken in the creation of serious Sudoku-solving programs: human solving methods, rapid-style methods, and pure brute-force algorithms. Human-style solvers will typically operate by maintaining a mark-up matrix, and search for contingencies, matched cells, and other elements that a human solver can use to determine and exclude cell values.

Many rapid-style solvers employ backtracking searches, with various pruning techniques in order to help reduce the size of the search tree. The term rapid-style may be misleading: Most human-style solvers run considerably faster than a rapid-style solver, although the latter takes less time to write and is more easily adapted to larger grids. A purely brute-force algorithm is very simple and finds a solution to a puzzle essentially by "counting" upward until a string of eighty-one digits is constructed which satisfies the row, column, and box constraints of the puzzle.

Rapid solvers are preferred for trial-and-error puzzle-creation algorithms, which allow for testing large numbers of partial problems for validity in a short time; human-style solvers can be employed by hand-crafting puzzlesmiths for their ability to rate the difficulty of a created puzzle and show the actual solving process their target audience can be expected to follow.

Although typical Sudoku puzzles (with 9×9 grid and 3×3 regions) can be solved quickly by computer, the generalization to larger grids is known to be NP-complete. Various optimization methods have been proposed for large grids.

Difficulty ratings

The difficulty of a puzzle is based on the relevance and the positioning of the given numbers rather than their quantity. Surprisingly, the number of givens do not reflect a puzzle's difficulty. Computer solvers can estimate the difficulty for a human to find the solution based on the complexity of the solving techniques required. Some online versions offer several difficulty levels.

Another approach is to rely on the experience of a group of human test solvers. Puzzles can be published with a median solving time rather than an algorithmically defined difficulty level.

Many publications sort their Sudoku puzzles into three or five rating levels. An easy puzzle can be solved using only scanning; an intermediate puzzle may take markup to solve; a hard puzzle will usually take analysis.

Difficulty is a very complex topic because it may depend on the concepts and visual representations one is ready to use.

Construction

Building a Sudoku puzzle can be performed by predetermining the locations of the givens and assigning them values only as needed to make deductive progress. This technique gives the constructor greater control over the flow of puzzle solving, leading the solver along the same path the compiler used in building the puzzle. Great caution is required, however, as failing to recognize where a number can be logically deduced at any point in construction—regardless of how torturous that logic may be—can result in an unsolvable puzzle when defining a future given contradicts what has already been built. Building a Sudoku with symmetrical givens is a simple matter of placing the undefined givens in a symmetrical pattern to begin with.

Nikoli Sudoku are hand-constructed, with the author being credited; the givens are always found in a symmetrical pattern.[9] Dell Number Place Challenger (see Variants below) puzzles also list authors. The Sudoku puzzles printed in most UK newspapers are apparently computer-generated but employ symmetrical givens; The Guardian famously claimed that because they were hand-constructed, their puzzles would contain "imperceptible witticisms" that would be very unlikely in computer-generated Sudoku.

Construction algorithm. Construction of a sudoku is fairly straightforward when a fast solver program is available: Start with an empty grid. Fill in candidate clues one by one, either randomly or according to some design rule. If a candidate leads to an illegal puzzle, discard it and try a different candidate. When the set of clues (or givens) define a valid puzzle, that is a puzzle with exactly one solution, the process is finished. This process will always result in a valid puzzle. However, the puzzles found may not have the desired qualities, where quality means anything like difficulty level, pattern symmetry, number of given clues, minimality, etc. In order to achieve a certain quality, it is often possible to improve the puzzle. Alternatively, just make new ones until the quality is satisfactory.

Although the 9×9 grid with 3×3 regions is by far the most common, 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 featured a 6×6 grid with 2×3 regions and a 7×7 grid with six heptomino regions and a disjoint region. Larger grids are also possible. The Times offers a 12×12-grid Dodeka sudoku with 12 regions of 4×3 squares. 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). Nikoli offers 25×25 Sudoku the Giant behemoths.

Another common variant is to add restrictions 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 to require the numbers in the main diagonals of the grid also 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.

A variant named "Mini Sudoku" appears in the American newspaper USA Today, which is played on a 6x6 grid with 3x2 regions. The object is the same as standard Sudoku, but the puzzle only uses the numbers 1 through 6.

Another variant is the combination of Sudoku with Kakuro on a 9 x 9 grid, called Cross Sums Sudoku, in which clues are given in terms of cross sums. The clues can also be given by cryptic alphametics in which each letter represents a single digit from 0 to 9. An excellent example is NUMBER+NUMBER=KAKURO which has a unique solution 186925+186925=373850. Another example is SUDOKU=IS*FUNNY whose solution is 426972=34*12558.

Many newspapers include the popular Hypersudoku such as The Age. The layout is identical to a normal Sudoku, but with additional interior areas defined in which the numbers 1 to 9 must appear. The solving algorithm is slightly different from the normal Sudoku puzzles because of the leverage on the overlapping squares. This overlap gives you more information to logically reduce the possibilities in the remaining squares. The approach to playing is still similar to sudoku but with possibly more emphasis on scanning the squares and overlap rather than columns and rows.

Puzzles constructed from multiple Sudoku grids are common. Five 9×9 grids which overlap at the corner regions in the shape of a quincunx is known in Japan as Gattai 5 (five merged) Sudoku. In The Times, The Age and The Sydney Morning Herald this form of puzzle is known as Samurai SuDoku. The Baltimore Sun and the Toronto Star publish a puzzle of this variant (titled High Five) in their Sunday edition. Often, no givens are to be found in overlapping regions. Sequential grids, as opposed to overlapping, are also published, with values in specific locations in grids needing to be transferred to others.

Alphabetical variations have emerged; there is no functional difference in the puzzle unless the letters spell something. Some variants, such as in the TV Guide, include a word reading along a main diagonal, row, or column once solved; determining the word in advance can be viewed as a solving aid. A looser variant on the sudoku concept is seen in Squaro, wherein circles at the vertices of a grid are filled in to meet the requirements of numbers in that grid in a combination of sudoku and minesweeper.

A tabletop version of Sudoku can be played with a standard 81-card Set deck (see Set game). A three-dimensional Sudoku puzzle was invented by Dion Church and published in the Daily Telegraph in May 2005. There is a Sudoku version of the Rubik's Cube named Sudokube.

The 2005 U.S. Puzzle Championship included 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. This version has also appeared in GAMES magazine.

One more variant of Sudoku is Greater Than Sudoku (GT Sudoku). In this a 3x3 grid of the Sudoku is given with 12 symbols of Greater Than (>) or Less Than (<) on the common line of the two adjacent numbers. Depending on difficulty this type of Sudoku may or may not be given with numbers.

Mathematics of Sudoku

A completed Sudoku grid is a special type of Latin square with the additional property of no repeated values in any partition of the 9×9 block into contiguous 3×3 blocks. The relationship between the two theories is now completely known, after Denis Berthier has proven in his recent book, "The Hidden Logic of Sudoku", 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 first known calculation of the number of classic 9×9 Sudoku solution grids was posted on a USENET newsgroup rec.puzzles in September of 2003[10] and is 6,670,903,752,021,072,936,960 (sequence A107739 in OEIS). This is roughly 0.00012% the number of 9×9 Latin squares. A detailed calculation of this figure was provided by Bertram Felgenhauer and Frazer Jarvis in 2005[11]. 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 and relabelling are taken into account, was shown by Ed Russell and Frazer Jarvis 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; 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,[13][14] and 18 with the givens in rotationally symmetric cells. Over 47,000 examples of Sudokus with 17 givens resulting in a unique solution are known.