Sudoku

The sudoku (// delivery in French, Japanese //), is a Jeu in the shape of grid defined in 1979 and inspired by the Carré Latin as well as problem of the 36 officers of the Mathématicien Suisse Leonhard Euler. The goal of the play is to fill this grid with figures going with 1 to 9 by respecting certain constraints, some figures being already laid out in the grid.

Presentation

The grid of play is a square of nine boxes on side, subdivided in as many identical squares, called areas (see figure). The rules of the game are simple: each line, column and area should contain only once all the figures from one to nine. Formulated differently, each one of these units must contain all the figures from one to nine.

The figures are used only by convention, the arithmetic relations between them not being useful. Any whole of distinct signs: letter, form, color,… can be used without changing the rules of the game. Dell Magazine , the first to publish grids, used figures in its publications. On the other hand, Scramblets , of Penny Close , and Sudoku Word , of Knight Features Syndicate , uses both of the letters.

The interest of the play lies in the simplicity of its rules, and in the complexity of its solutions. The grids published often have an indicative level of difficulty. The editor can also indicate a probable time of resolution. Though in general, the grids containing the most figures préremplis were simplest, the reverse is not systematically true. The true difficulty of the play lies rather in the difficulty of finding the continuation exact of figures to be added.

This play already inspired several electronic versions which bring an interest different to the resolution of the grids from sudoku. Its form out of grid and its ludic use bring it closer to other headaches published in the newspapers, the such cross Mots and the problems of failures.

Professor S recommend the practice of the sudoku like a drive to the logical reasoning. The level of difficulty can in this case being adapted to the public.

Grids are published in newspapers, but can also be generated by computer.

Etymology

The name sudoku (数独) was born from the abbreviation of the rules of the game Japanese E, meaning “there can be one and single figure” (by box and line). This abbreviation associates the characters Sū (数) figure and Doku (独) single . This name is a registered trademark with the Japan of the editor Nikoli Corporation Ltd.. In Japanese, this word is marked; in French, it is usually employed with a francized pronunciation, i.e. by being unaware of the long vowel present on first “U” and by modifying the stamp of the vowels slightly “U”: . With the Japan, Nikoli is always owner of the name sudoku ; its competitors thus use another name: they can refer to the play by the American name original “Number Place” (English: Numeral place), or by the word “Nampure” , being shorter. Some editors not Japan board spell the title “Su Doku”.

History

Antiquity

One of the ancestors of the sudoku was a square of nine boxes of which the principle was to fill these boxes by three letters (has, B and C) without there not being twice the same letter in the same column, line and diagonal.

India and China

The oldest “magic squares” numerical known are in China (named Luoshu 洛书, the book of the river Luo) where the figures were represented by various geometrical forms containing N round (towards -300), and in India where was invented what we call the Arab numerals. They have at the origin of the divine significances.

The Middle Ages

They are the Arab which at the 10th century would have had the first a purely mathematical and either crowned application magic squares.

Rebirth

In occident

Cornelius Clutched (1486 - 1535), always uses magic squares with an aim esoteric.

The French mathematician Pierre de Fermat (1601 (or 1607) - 1665) worked on the magic squares and the étendits with the magic cubes.

In 1691 Simon of Loubère explains the operation of the magic square used with the Siam, in its work Of the Kingdom of Siam, where they also have a crowned function.

The problem of the officers

In 1782, the Swiss mathematician Leonhard Euler imagines a problem in a grid. Some allot the paternity of the sudoku to Switzerland although work of Euler related to the Latin squares and the graph theory.

One considers six different regiments, each regiment has six officers of distinct ranks. One wonders now how to place the 36 officers in a grid of 6x6, at a rate of an officer per box, in such a way that each line and each column contain all the ranks and all the regiments.

It is in other words about a graeco-latin Carré of order 6 (the combination of two Latin squares, a Latin square for the regiments, a Latin square for the ranks), problem of which the resolution is impossible. Euler had already had a presentiment of it at the time, without however giving a formal demonstration to its conjecture. He will say:

“But, after all the sorrows which one was given to solve this problem, one was obliged to recognize that such an arrangement is absolutely impossible, though one cannot give rigorous demonstration of it. ”

In 1901, the French Gaston Tarry shows the impossibility of the result thanks to an exhaustive research of the cases and by crossing of the results.

The bond between the sudoku and the problem of the 36 officers is the constraint which prevents the repetition of the same element in the grid, while arriving at final at a play which employs the principle of the Latin square (combination of two Latin squares in the case of the graeco-latin, square square Latin subdivided in several areas in the case of the sudoku).

The modern version of the sudoku

The sudoku has French ancestors who go up with 1895. The play is apparently not a recent invention as much thought it. At end of the 19th century, the French indeed played to fill of the grids 9x9 divided into 9 areas, very close to this play (but the initial grids included/understood additional constraints on the solution), which were published in the famous daily newspapers of the time, reveals For Science in its edition of June 2006.

According to the magazine, the grid nearest to a sudoku, which was found by the French Christian Boyer, is that of B. Meyniel, published in the daily newspaper France of the July 6th 1895, and a close alternative was published not very front, in November 1892, in the Century , alternative which used numbers with two digits.

In 1979, a freelance journalist specialized in the headaches, Howard Garns, creates the first play such as we know it today. Dell Magazines introduces it this same year into a publication intended for the market New Yorkean, the Dell Pencil Puzzles and Word Games , under the name of Number Place . Nikoli introduces it with the Japan in April 1984 into the magazine Monthly Nikolist .

In 1986, Nikoli introduces two innovations, which will make the play popular: the number of revealed is with more than 30 and the grids are symmetrical, i.e. revealed are symmetrically distributed around the center of the grid. Today, the majority of the important newspapers to the Japan, such Asahi Shimbun , publish this play.

In 1989, Loadstar and Softdisk publish DigitHunt for the Commodore 64, probably the first software for personal computer to generate of Sudoku. There exists a company which continues to use this name.

In 1995, Yoshimitsu Kanai publishes a software generator under the name of Single Number (English translation of Sudoku), for the Macintosh, in Japanese and in English and, in 1996, it repeats for the Palm.

In 2005, Dell Magazines publishes also two Magazine S dedicated to Sudoku: Original Sudoku and Extreme Sudoku . Moreover, Kappa Publishing Group takes again the grids of Nikoli in RANGES Magazine under the name of Squared Away . The newspapers New York Post , USA Today and San Francisco Chronicle publish also this play. Grids appear in certain Anthologie S of plays, such as The Giant 1001 Puzzle Book (under the name of Nine Numbers ).

It is in July 2005 that the sudoku arrives in France, published by cerebral Sport, specialist publisher in the plays of reflection. The first number will be sold with 20.000 specimens is twice as much as with accustomed at the time of the exit of a new handset, a record according to Xavier de Bure, managing director of the editor. Le Figaro publishes the first daily grids as of at the beginning of July, followed during the summer 2005 by Libération , Provence , Nice Matin , 20 Minutes , Métro and Le Monde . The magazine 1,2,3… Sudoku left its first number in November 2005.

The phenomenon also gained the Suisse, Wayne Gould provides grids to the French-speaking daily the Morning which sold this year 150 000 pounds of sudoku. Time , another daily newspaper Swiss publishes as for him grids of sudoku since September.

Another ancestor of the sudoku: the magic Latin square

The agronomic experiments in field, generally consisted of a certain number of square or rectangular pieces, are often organized in the form of '' complete random blocks '', i.e. of groups of close pieces in which the various elements to be compared (various manures for example) all present and are randomly distributed.

When the full number of pieces available is equal to a square (16, 25,36, etc), another possibility corresponds to the concept of '' square Latin '', who is such as the various elements to be compared are present in each line and each column of pieces.

The superposition of these two devices can give rise to what was called square magic Latin , in particular by W.T. To federate in 1955. In the example presented opposite, each of the six studied elements (for example six different manures) is present in each of the six blocks of 2 X 3 pieces, in each of the six lines and each of the six columns. It is strictly about a sudoku 6 X 6.

The traditional sudoku is thus anything else only one magic Latin square 9 X 9.

Popularity in the media

As of 1997, Wayne Gould, a New Zealand and Judge with the retirement of HongKong, is intrigued by a grid partially filled in a Japanese Librairie. During six years, it develops a program which generates these grids automatically. Knowing that the British newspapers publish cross Mots and other similar plays for a long time, it promotes the sudoku near the newspaper The Times , which publishes for the first time a grid the November 12th 2004.

Three days later, The Daily Mail publishes also a grid under the name Codenumber . The Daily Telegraph introduces its first roasts the January 19th 2005, followed by the other publications of the Telegraph Group. The May 20th 2005, the Daily Telegraph of Sydney publishes for the first time a grid.

It is when the Daily Telegraph publishes grids on a daily basis, starting from the February 23rd 2005, while promoting this one on its one page, that the other British newspapers start to pay attention to it. The Daily Telegraph continued its promotion campaign when it realized that its sales increased simply by the presence of a grid of sudoku. The Times was rather discrete on the immense popularity which surrounded its contest of sudoku. It had already envisaged to draw advantage from its advance by publishing a first book on the sudoku.

In April and May 2005, the play was sufficiently popular so that several national newspapers publish it on a regular basis. With the number of those, one finds The Independent , The Guardian , The Sun (entitled Sun Doku ) and The Daily Mirror . When the word Sudoku becomes popular with the the United Kingdom, the Daily Mail adopts it in the place of Codenumber . Consequently, the newspapers competed of imagination to push their grids. The Times and Daily Mail affirms that they are the first to have published a grid of sudoku, whereas The Guardian affirms, ironically, that its grids built with the hand, obtained Nikoli, bring a better experiment than the grids generated using a software.

The sudden popularity of the sudoku in the United Kingdom attracted its batch of comments in the media (see Sources below) and of the Parodie S followed, for example the section G2 of the newspaper The Guardian' announces itself like the first supplement with a grid by page. The sudoku became particularly visible immediately after the elections of 2005 in the United Kingdom, incentive some commentators to affirm that it filled a need at the political assistantship. Another explanation suggests that it draws and holds the attention of the readers, several feeling increasingly satisfied when the solution takes shape. The Times estimates that the readers appreciate at the same time the easy and difficult grids. Consequently, it publishes them side by side since the June 20th 2005.

British television hastened of surfer on the wave of popularity and Sky One diffuses the first emission on the sudoku, Sudoku Live , on July 1st 2005, that the Mathématicien Carol Vorderman present. Last nine teams of nine players, including one High-speed motorboat, each one representing a geographical area, try to supplement a grid of sudoku. Each player has in hand an apparatus which enables him to seize a figure in one of the four cells for which it is responsible. To exchange with the other members of the team is allowed but, familiarity missing, the candidates do not do it. Also, the audience at the house takes part in another interactive competition at the same time. Sky One tried to create a passion for its emission by the means of an enormous grid of 84 side m. However, it had 1 905 solutions.

This abrupt increase in popularity in the British and international newspapers makes that the sudoku is regarded as the “Cube of Rubik of the 21e century” (free translation of “the Rubik' S cubes the 21st century off”). As example, Wayne Gould provides at the end of 2005 of the grids for approximately 70 daily newspapers in 27 countries.

The November 28th 2005, the French-speaking Switzerland Television lance a daily television program, Su/do/ku , where two candidates clash over 5 days, at a rate of 3 8 minute old handles each day. However, the difficulty to make pass this kind of play on television will cause the stop of the emission after a few weeks.

National championships are also organized like the 1st championship of France of sudoku (Paris, December 18th 2005) gained by Juliette Thery, 19 years. This competition organized by cerebral Sport rewards the best player for the year. It is the agency of vertical Décollage communication which set up this event unique in France. Since, many other tournaments were organized in France.

Alternatives

Although the traditional grids are most common, several alternatives exist:

2×2 or " Sudoku binaire" , containing areas 1×1 (version full with irony);
4×4 containing areas 2×2 (generally for the children);
5×5 containing areas in form of Pentamino was published under the name Logi-5 ;
6×6 containing areas 2×3 (proposed at the time of the World Puzzle Championship );
7×7 with six areas in form of Hexamino and a disjoined area (proposed at the time of the World Puzzle Championship );
9×9 with areas in form of ennéamino;
16×16 with areas 4×4 (called Number Places Challenger and published by Dell, or called sometimes Super Sudoku ), (or Sudoku Hexadécimal using a notation in Base 16 (Figure from 0 to 9 + letters of has to F);
25×25 with areas 5×5 (called Sudoku the Giant and published by Nikoli);
an alternative imposes moreover than the figures in the principal diagonals are single. The Number Places Challenger , mentioned previously, and the Sudoku X of the Daily Mail , a grid 6×6, belongs to this category;
8×8 containing areas 2×4 and 4×2, and where the principal lines, columns, areas and diagonals contain a single figure
a méta-grid made up of five grids 9×9 in Quinconce which overlap with the corners is published in Japan under the name of Gattai 5 (which means “five amalgamated”) or Samuraï . In the newspaper The Times , this form is called the Samurai Su Doku .
of the grids with rectangular areas: if an area is dimensions L×C boxes, the total grid breaks up into C×L areas; the values to be filled go then from 1 to C×L;
Dion Church created a grid 3D, that the Daily Telegraph published in May 2005. The software Ksudoku calls of such grids Roxdoku and generates automatically them.
the Kamaji is a recent derivation of sudoku based on the principle of the sums of figures.

In the Japan, other alternatives are published. Here is an incomplete list:

sequentially connected Grids: several grids 9×9 are solved consecutively, but only the first revealed sufficient to make it possible to solve logically. Once solved, certain figures are copied towards the following. This formula forces the player to make outward journeys and returns between partially solved grids.
very large Grills which consists of multiple grids which overlap (usually 9×9). Grids made up from 20 to 50, or more, are current. The size of the areas which overlap varies (two grids 9×9 can divide 9,18 or 36 cells). Often, it no revealed there in these areas.
usual Grids where a figure is member of four groups, instead of the three usual ones (lines, columns and areas): the figures located at the same relative positions in an area should not correspond. These grids are usually printed color, each group disjoins sharing a color to facilitate the reading.

The case of plays to take part in the World Puzzle Championship of 2005 contains an alternative entitled DIGITAL Number Place : rather than to contain revealed, the majority of the cells contain a figure partially drawn which borrows from the C-W communication of the Affichage to seven segments.

The August 31st 2005, The Times started the publication of the Killer Su Doku , also named Samunamupure (which means “place of summation”), which indicates the sum of gathered cells, which adds a supplement of difficulty in the search for the solution, although that can help to solve. The other rules apply.

Alphabetical alternatives

Alphabetical alternatives, which use letters rather than figures, are also published. The Guardian calls them Godoku and qualifies them the démoniaques ones. Knight Features prefers to him the term Sudoku Word . The Wordoku of Signal Notch reveals the letters, in the disorder, of a word which runs of the left corner higher than the corner lower right. A player having a good culture can find it and use his discovery to advance towards the solution.

In French, this alphabetical alternative bears various names like Sudoku letters, Mokitu (Télé 7 days) or Mysmo (Libération). Certain grids are limited to the words comprising only different letters. Others accept words comprising several times the same letter in which case it has a different C-W communication each time, for example: MAHaRADJa .

The Code Doku conceived by Steve Schaefer contains a complete sentence, whereas the Super Wordoku of Signal Notch contains two words of nine letters, each one being on one of the principal diagonals. These plays are not regarded as truths sudoku by the purists, because logic is not sufficient to solve them, even if they have a single solution. Top Notch affirms that these plays are conceived in order to block the solutions composed by software of automatic resolution.

Many possible complete grids

It is obvious that the number of complete grids is lower than the number of ways of placing nine digits 1, nine digits 2…, nine digits 9 in a grid of 81 boxes. The number of grids is thus much lower than

\ frac {81!}{9! ^9} \ approx 5,31306887 \ times 10^ {70}

Indeed, in this calculation, one does not take account of any the constraints of unicity.

The number of possible complete grids is also lower than the numbers of square Latin on side 9.

Lastly, the number of possible complete grids is lower than $9! ^9$ which corresponds to the number of ways of building the areas without taking account of the constraints on the lines and the columns.

In 2005, Bertram Felgenhauer and Frazer Jarvis proved that this number of grids was of:

$\ mathbb {NR} = 6 \; 670 \; 903 \; 752 \; 021 \; 072 \; 936 \; 960 \ approx 6,67 \ times 10^ {21}$

This number $\ mathbb {NR}$ is equal to:

9! ×72²×2⁷×27 704.267.971

The last factor is a Prime number. This result was proven thanks to a exhaustive Recherche. Frazer Jarvis then considerably simplified the proof thanks to a detailed analysis. The demonstration was validated in an independent way by ED Russell. Jarvis and Russell showed thereafter that by taking account of symmetries, there were 5.472.730 538 solutions.

As for the following problem, it seems unsolved: if one is interested in the number of problems proposables, this number is unknown; on the other hand, it is known that it is definitely more important than the number $\ mathbb {NR}$ indicated above because there exist very many ways of presenting initial grids whose solution (single) led to the same finished grid (supplements) ( on the other hand, it is easy to show, on certain examples of complete grids, at which point one can, for the same complete grid, to present initial grids of difficulties completely contrasted, since the grids for beginners to the grids known as diaboliques; it is in any case very easy, knowing an initial grid diabolique, to manufacture a grid for beginner whose single solution supplements is identical to that of the selected diabolic grid! ).

Another unsolved problem: on this date, no result exists on the number of complete grids in a super sudoku (grid 16 × 16).

The problem of knowing how much initial boxes filled are necessary to lead to a single solution is, to date, without sure answer. The best result, obtained by Japanese, is of 17 unconstrained boxes of symmetry. '. Nothing says that it is not possible with less nombres.

Lastly, Gordon Royle considers, rightly, that two resolutions are regarded as different if they cannot be transformed one into the other (or the reverse) thanks to an unspecified combination of the six following operations:

permutations of the 9 numbers
exchange of the lines with the columns (Transposition)
permutation of the lines in only one block
permutation of the columns in only one block
permutation of the blocks on a line of blocks
permutation of the blocks on a column of blocks

One notices the analogy with the matric operations in Linear algebra.

Mathematics

The problem to place figures on a grid of N ² ×n ² including/understanding n×n areas is proven Np-complete.

The problem of the resolution of all sudoku can be formalized in an equivalent way by a problem of Coloration of graph: the goal, in the traditional version of the play, is to apply 9 colors to a given graph, starting from a partial coloring (initial configuration of the grid). This graph has 81 tops, one by cell. Each box of the sudoku can be labelled with an ordered pair (X, there) , where X and is there entireties ranging between 1 and 9. Two distinct tops labelled by (X, there) and (X', there') are connected by an edge if and only if:

X = X' (the two cells belong to the same line) or,
there = there' (the two cells belong to the same column) or,
$\ left \ lceil {\ frac {x-1} {3}} \ right \ rceil = \ left \ lceil \ frac {x'-1} {3} \ right \ rceil$ and $\ left \ lceil \ frac {y-1} {3} \ right \ rceil = \ left \ lceil \ frac {y'-1} {3} \ right \ rceil$ (the two cells belongs with the same area). The grid is supplemented by affecting an entirety between 1 and 9 for each top, so that all the tops bound by an edge do not share the same entirety.

A grid solution is also a Carré Latin. The relation between the two theories from now on is completely known, since D. Berthier showed, in " The Hidden Logic off Sudoku" , that a first order logical formula which does not mention the blocks (or areas) is valid for Sudoku if and only if it is valid for the Latin squares.

There are notably less grids solutions than Latin squares, because the sudoku forces additional constraints (See Ci above point 4: many possible complete grids).

The maximum number of revealed without a single solution appearing immediately, it does not matter the alternative, is the size of the grid minus 4: if two pairs of candidates are not registered and that the blank cells occupy the corners of a rectangle, and that exactly two cells are in an area, then there exist two ways of registering the candidates. The opposite of this problem, namely the minimum number of revealed to guarantee a single solution, is a unsolved problem, although enthusiastic Japanese discovered a grid 9×9 without symmetry which contains only 17 dévoilés', whereas 18 is the minimum number of revealed for the symmetrical grids 9×9.

Rules and terminology

Most of the time, the play is proposed in the shape of a grid of 9×9, and is composed of under-grids of 3×3, called “areas”. Some cells contain figures, known as “revealed”. The goal is to fill the blank cells, a figure in each one, so that each line, each column and each area are made up of only one going figure from 1 to 9. Consequently, each figure in the solution appears only once according to the three “directions”, from where the name “quantifies single”. When a figure can fit in a cell, it is said that he is candidate.

Method of resolution

The méthode of résolution is brought back to three processes: seek, candidature and analyzes. The approach of the analysis can be different according to the concepts which it puts in work and according to the representations on which it rests.

Seek

Research is made at the beginning of the play and periodically during the filling of the grid. Several research is often necessary between two moments of analysis. This research calls upon two simple techniques:

Reduction by cross : it is a question, for each figure, of eliminating the cells where he cannot be. For that, the researcher traces a feature, imaginary, on each column and each line where the figure appears already. The boxes which are not crossed by a feature are those where the figure can still be inserted. This method can be used to fill the “simplest” cells in first. To save time, the researcher can start with the most figures among revealed, but it is important to apply it to each figure. To minimize the search time at the other stages, this stage must be made in a systematic way, while checking for all the figures.

Calculation from 1 to 9 for each area, each line and each column. This stage makes it possible to find the figures missing (to do It according to the last found figure can make faster research). In the difficult grids, the figure to be registered can be given by making a reversed calculation, i.e. while trying to find the figures which cannot appear in the cell, which makes it possible to know the figures candidates.

The expert players seek the “contingencies” during the research, i.e. they try to determine the cells candidates (two or three) for a figure in particular. When these cells all are in the same line (or column), and an area, they are made profitable during the reduction by cross and the calculation (see example). The most difficult grids require to recognize the multiple contingencies, often in different directions or with the intersections. What obliges the players to register the candidates (method described below).

The grids which one can only solve by the reduction by cross are regarded as easy, most difficult require to call upon other techniques.

Candidature

Research ceases when no new figure is registered. It is as from this moment that another technique must take seat. Several players find useful to register the figures candidates in the blank cells. There are two notations used: subscripted and pointed.

For the subscripted notation, the candidates are registered in a cell, each figure occupying or not a precise place. The disadvantage of this method is that the newspapers publish grids of small size, which makes difficult the inscription of several figures in the same cell. Several players reproduce with more large scales of such grids or have recourse to a pencil with fine point.
For the pointed notation, the players register points in the blank cells. The relative position of the point indicates the missing figure. For example, to indicate 1, a point appears in top on the left in the cell. This notation makes it possible to play directly with a grid printed in a newspaper. However, she asks a certain dexterity, it is possible badly to place a point in one moment of carelessness and a small mark made by error can lead to confusion. Certain players prefer to use a Stylo to limit the faults.

Analyzes

The two topics of this process are elimination and the assumption (this last process can be prevented if one is sufficiently involved).

Elimination: the search for the solution is done by successively eliminating the candidates from a cell in order to select one applicant. Once this found candidate, another research should be carried out in order to determine the consequences on the other cells. There are several techniques of elimination which are based on the rules below, which have useful Corollaire S:

Un together given n cells in a line, a column or an area, cannot receive that n figures différents. This rule is at the base of the technique of “elimination of the orphan candidate”, discussed below.
Chaque candidate must ultimement belong to a self-consistent model and indépendant. This rule is at the base of the advanced techniques of analysis, which require to inspect the whole of all the possibilities for a candidate. There are only one finished number of “closed circuits” or possibilities of grids “n×n” which exist. This rule gave rise to the methods X-wing and Swordfish, inter alia. If such a model is identified, then the elimination of candidates is often possible.
Un figure given can receive only one position in its box, line or column, the other sites candidates entering in contradiction with eliminations already carried out.

One of the most used techniques is the “elimination of the orphan candidate”. The cells with the same whole of candidates known as are coupled if the number of candidates in each one of it is equal to the number of cells which can accommodate them. For example, two cells are coupled if they contain a single pair candidates (p, Q) in a line, a column or an area; three cells known as are coupled if they contain a single triplet candidates (p, Q, R). These figures cannot appear elsewhere, because there would be conflict according to the line, the column or the area. For this reason, the candidates (p, Q, R) who are in the other cells are to be eliminated. This principle is worth with subsets of candidates: if three cells have only {(p, Q, R), (p, Q), (Q, R)}, or {(p, R), (Q, R), (p, Q)}, all the candidates of this unit which are in the other cells are to be eliminated.

a second principle rises from the preceding principle. If the number of cells in a line, a column or an area, is equal to the size of a whole of candidates (one speaks then about group of numerically dependant multiples), the cells and the figures are coupled and only these figures will appear in the cells. All the other candidates are to be eliminated. For example, if (p, Q) can only appear in two cells (of a line, a column or an area), the other candidates are with éliminer.

The first principle is based on the concept of “only coupled figures”, whereas the second is based on the concept of “only coupled cells”. The advanced techniques are based on these concepts and include multiple lines, multiple columns and multiple areas.

With l'approche by hypothèse, a cell with only two candidates is chosen and one of the two digits is registered in the cell. The preceding stages are repeated and lead either to a contradiction (duplicated figure or cell without candidate), or with a valid proposal. Obviously, in the case of a contradiction, the second figure belongs to the solution. The algorithm of Nishio is a purified form of this approach: For does each candidate of a cell, to insert a figure in particular prevent the inscription of this candidate elsewhere in the grid? If the answer is yes, then the candidate is éliminé.

The approach by assumption requires to use a pencil and a gum to be erased. The purists reject it, because it is an approach by tests and errors, whereas the majority of the grids published call upon logic to only be solved. However, this approach has the merit often to lead to the solution more quickly.

It is with each player to find a method who gives him the best results. Some will develop a method which reduces the disadvantages of the preceding proposals. For example, some will find tedious to have to register all the candidates in all the cells. The approach by assumption requires to be organized. The ideal is to find a way of doing which minimizes the calculation, the number of candidates and the number of assumptions.

In theory, these three processes (single one applicant by croisement+candidat by counting and élimination+groupes independent of numerically dependant multiples treated according to one or more dimensions) are enough to make a success of a grid completely. But there are situations where it seems that it is not possible any more to advance. Here a beginning of example:

You found starting from the figures already revealed according to the areas and the columns, multiples 123-12-1456-479-23-56-2456-178-89 writings on a whole line for a certain grid. Initially, one raises the 123,12 and 23 numerically dependant; three formed multiples of the three digits 1,2 and 3, which will occupy each one one of the three boxes. Thus the line is simplified into 123-12-456-479-23-56-456-78-89. Then, one considers the multiples 456,56 and 456 which are also numerically dependant, but their group is independent of that of the multiples formed at the base of figures 1,2 and 3.Pour the same reason, the line is simplified into 123-12-456-79-23-56-456-78-89. There thus remain three multiples 79,78 and 89 which are numerically dependant, but set up a third group independent of the two first. To this level, one will say that one filled the line with optimal way. Simplifications thus carried out will be reflected on the areas, the columns then on the other lines then again on the areas, the columns and the lines if one manages to release each time, of new independent groups. If there remain always boxes without one applicant, then, one will be able to attack by treatment of the multiples by considering two dimensions at the same time; columns X lines (principle of unicity, X-Wing for example), columns X areas (doubled blooms, twins for example), lines X areas (idem). If the solution always does not appear, then, from now on, you are invited to use the techniques of treatment three dimensions (lines X columns X areas) of which for example, those rising from the use of the ways (theory based on bivalent logic; it is there or it is not there).

And if your labor always does not lead to the grid-solution, then, it is because of the one of the two following reasons:

You applied well and you entirely filled the grid with optimal way by single figures in certain boxes and by multiples in the others. But, all the groups of the multiples which you registered are independent. In this case, You deal with grid presenting several solutions! It is not a Known-Doku “good” and the problem was not to be proposed, unfortunately!
All the boxes of your grid have one applicants or multiples, but, for lack of experiment, you do not manage to easily distinguish groups independent of numerically dependant multiples. In this case, you can proceed by the disjunction of the one of the multiples. I.e. to make an assumption on its figures, and to see the effect which will be reflected on the other multiples. If you have really “a good” play of Known-Doku, then only one figure of the multiple in question will lead to the solution of the problem, while for all the others, one will lead to blockade situations! In the contrary case, the problem does not deserve to be posed! By principle!

But the fact of formulating an assumption on the figure to be chosen among those of a given multiple always does not guarantee the simplification of the other multiples and is likely to lead on new assumptions to make, which increases the number of grids quickly to be examined successively! Worse still, the grids obtained can be of a poor level and thus without intellectual interest!

Generalized symmetries and wide table of resolution

In " The hidden logic of Sudoku" , an English book (" The Hidden Logic off Sudoku") based on a systematic logical formalization of the play, all its generalized symmetries were clarified, in particular between the lines and the numbers, and between the columns and the numbers. A new method of resolution was developed, based on their systematic exploitation. A grid of wide resolution (comprising three grids instead of only one) was designed, which reveals the bonds of conjugation like boxes with two candidates and can facilitate the application of the method (without being absolutely necessary). This way, the subsets hidden as well as X-wings, Swordfish and Jellysfish, but not TPU (technique rising from the principle of the unicity of the solution), seem all simple Pairs, Triplets or Quadruplets. Within a general framework to treat chains, these symmetries were used to introduce new rules of resolution, like the hidden chains xy. This method was implemented in a solvor, SudoRules, base on techniques of Artificial intelligence and simulating a human player.

Grid-joint; to change situation for more thorough resolution

The strategy of the symmetries generalized between lines, columns and figures omits a fourth angle of attack to solve other cases of more complex figures: to consider an area and a figure and to locate the good cell. The author of the handbook " Strategy of resolution of a grid of Sudoku" propose the use of theJoint one; it is a table of 9 horizontal lines (a line by figure) crossed with 9 vertical lines (a line by area) whose cells receive the coordinates of the box associated with the figure and area given in the normal grid with the problem suggested. From its construction, thejoint one includes the most effective techniques of resolution of which X-wing, Swordfish and many other unknown factors by the commun run of the players, but " ignore" the TPU (technical rising from the principle of the unicity of the solution), like besides the strategy of generalized Symmetries. The author suggests classifying the TPU in a category with share!

Strategy of the ways; to solve more complex cases

If the adoption of the wide table of resolution using generalized symmetries and/or grid-joint make it possible to solve the grids frequently proposed in the newspapers, magazines and sites, there exists many cases of figure where these two strategies butt without being able to reach the final solution. Let us acknowledge that these two strategies have the merit to make us discover new processes of resolution bringing into play two dimensions (lines X columns, lines X areas and columns X areas) on the grid-problem normal or initial, whereas it implements only one dimension (arranged horizontal or vertical in each of the two additional tables of the strategy of generalized symmetries or on thejoint one) whose treatment is relatively easy to carry out manually and to program on the software (elimination because of the multiples naked or revealing by degreasing of multiples).

The strategy adopted for these cases of more complex figures consists in taking into account three dimensions at the same time (lines X columns X areas). " should be able; sauter" of an area with another, through the boxes, by using " passerelles" materialized either by a line, a column or an area. In short, " should be created; chemins" between the various boxes. Thus, one will recognize processes similar to those already used by two dimension treatment of which X-Wing for example (the tops are not any more those of a rectangle, but among those of a polygon).

Let us specify that this strategy is based on bivalent logic (for a figure NR fixed and a given box of multiples, p: " NR is the valeur" and not (p): " NR is not the valeur").

Seen of a certain angle, it is a question of making superimpose two or several grids on the same grid-problem initial, to make a logical conjugation of the various proposals (concretized by ways) and to determine those of the grids which end in a contradiction with one of the rules which govern the play sudoku. It is thus as if one proceeded by formulation by assumption, but in a manner " cachée" ! It is necessary to acknowledge that this manner of making gets more pleasure to play and apply processes to put forth assumptions to obtain grids " pauvres" on the intellectual level.

A technique “with share”: principle of the unicity of the solution

There exists a class of techniques, although by bringing into play two dimensions only (lines X columns, lines X areas or columns X areas) cannot be found nor translated in a certain manner in the wide table of resolution or on thejoint one. One quotes like example, the technique rising from the principle of the unicity of the solution. The case of figure is the following: in four boxes, tops of a rectangle, one finds three same pairs ab, ab, ab and this same pair mixed with other figures X, ....., Z in form abx….Z. Then, under the terms of the principle of the unicity of the solution, in this fourth top, one can drive out without risk the two figures has and B. Because in the contrary case, the grid would end in at least two solutions.

In certain cases, this technique can be used without the knowledge, if it is possible to follow a way (here a loop) of the one of the tops towards itself. Sometimes, one is brought to use this same technique on polygons instead of a rectangle; a generalization is thus possible, but using the strategy of the ways.

Strategy of last recourse: clear and clear formulation of the assumptions

Certain newspapers, magazines, sites and software deliver grids to us say “diabolic”. In general, it of it is nothing! These grids can be solved by the techniques developped at the point so far. A great majority can be filled “mentally” even!

In short, a definition is essential: a diabolic grid is that which can be solved per none the processes developped at the point so far, except by the formulation of one or several assumptions on the figures to be put in one or more boxes. Of course, the unicity of the solution for the grid is necessary.

From now on, it is the only means to lead to the solution, while waiting for the development of new “manual” processes.

Software solutions

For a Data processing specialist, to program the search for a solution by the means of the contingencies or multiple contingencies (such as required for the most difficult problems) is a relatively simple task. Such a program imitates a human player who seeks a solution without resorting randomly.

It is also relatively simple to design an algorithm of research by Backtracking. In a usual way, it is enough with the algorithm to choose 1 for the first cell, then 2 for the next one, so on as long as no contradiction appears. When a contradiction appears, the algorithm tests another value for the cell which brings contradiction. Once all the possibilities exhausted for this cell, the algorithm “reconsiders its steps” and starts again with the penultimate cell.

Although this algorithm is not very effective in theory, it will find a solution if it has sufficient time. A grid 9×9 is usually solved in less than three seconds with a modern personal computer which has recourse to a Interpréteur, and in a few milliseconds with a language compiled. However, there exist grids which are particularly difficult to solve by backtracking.

However, a more effective program will be pressed on the potential candidates for each cell, eliminating the impossible candidates until only one figure remains. Knowing this figure, it can find another figure for another cell, and so on.

An alternative to the Backtracking is to resort to the methods recommended by the logical Programmation, as established by Prolog. In this case, the originator provides to the program the constraints of the grid (a figure by line, column and area; revealed figures); this program will make the decisions to solve the problem. Knowing that the majority of the grids have a single solution, research is certain to succeed.

Donald Knuth developed an algorithm which calls upon the Liste S doubly chained (the dance hall links ), and which proves very effective to solve this type of problem. It is shown that this algorithm is very indicated for the resolution of Sudoku, taking only a few milliseconds. Thanks to its speed, he is now preferred by the majority of the software originators.

Degrees of difficulty

The grids published often mention a degree of difficulty. This one is calculated according to the facility of resolution by a logical method. Surprisingly, the number of revealed does not affect almost any the difficulty of a grid. Grids with a small number of figures can be easily solved, whereas others which contain a higher number of revealed than the average can be very difficult to solve.

Knowing the complexity of the rules, the software of automatic resolution can estimate the difficulty for human of finding a solution. This estimate is in general sufficiently precise to make it possible to the editors to provide it. Some editors on line also provide this estimate.

Several factors influence the difficulty of these problems. The basic equation holds account modulo a certain weighting:

of the number of cells to be filled;
of the number of cells filled by elimination;
of the number of groups independent of numerically dependant, manageable multiples according to only one dimension; area, line or column;
of the number of groups independent of numerically dependant, manageable multiples according to two dimensions at the same time; area X line, area X column or column X line;
of the number of groups independent of numerically dependant, manageable multiples according to three dimensions at the same time; area X line X column;
of the number of assumptions to be made in the event of temporary blocking;
of the iteration count of the heuristics of resolution;
of the number of research to make to supplement the grid.

The question of the difficulty is very difficult and is the subject of many debates in the forums on Sudoku, because it is related to the concepts and visual representations that each one is ready to adopt. But it can be completely elucidated by the adoption of a hierarchy (of simple with the complex) of the techniques and processes which one can use to make a success of a grid, and by our manner of playing by observing certain rules of handicap, such as for example the integral resolution by mental reasoning only, or absolute prohibition to reproduce the grid-problem in several grids by making assumptions, etc

Moreover, " should not be confused; the level of the joueur" with " the degree of difficulty of a grill". Certain players are able to make a success of a grid while reasoning mentally, without writing multiples in the boxes which do not receive thereafter, each one, which the good figure, whereas others pain with boxes introducing several candidates, or with several grids coming from the assumptions put forth free, or worked out according to the categories (lines, columns and areas) of which thejoint one for example, which includes in fact a certain wide table of resolution. This is why one prefers to classify the grid-problems in five types, in which, one finds various levels of difficulty (see the typology of the grid-problems of Known-Doku worked out by Farid MITA) ":

Type 1:

Use of the simple techniques of which the “research of the good box for a figure and an area given” by reduction by cross, and “seeks it, of the good figure for a given cell”, by calculation, although the latter is a little more tiresome than the first. In theory, for this type of grids, the reasoning is done mentally, without one being obliged to register the possible candidates in a given cell, and the filling of the grid is done gradually while following one of the innumerable tracks or sequences which arise. It is this type of grids which you frequently find in the sites, newspapers and magazines or generated by software, classifying wrongly some of them, in the category of the " difficiles" or even " diaboliques" ! The reason is that there exists a class of grids of the type 1, really difficult to succeed by mental calculation. And thus, do not underestimate the grids of the type 1: there are " of them; faciles" , " moyennes" and even " difficiles".

Type 2:

Use of the techniques allowing the treatment of the cell-with-candidate-multiples according to only one dimension; line, column or area, of which “elimination because of the naked pairs”, “the degreasing of the hidden candidates” and “the degreasing of the camouflaged pairs”. Certain grids of the type 2 can be succeeded, as for type 1, mentally. Others, of a higher level, require that one registers, progressively, the candidates in the cells of an area, a line or a column, without however doing it for all the blank cells, and seeing whether one can simplify the multiples by one of the three preceding techniques. Most difficult of the grids of this type 2 do not lend themselves to the resolution that once all the cells contain their probable candidates. In this case, it is necessary to try to arrive at the optimal situation of the grid: in each category (line, column and area), the groups of the “numerically dependant multiples” must be “independent”. Other simple techniques of treatment according to only one dimension can be used, of which “elimination because of the naked triplets” and “the degreasing of the camouflaged triplets”. The latter is more painful to make! One will be able to also eliminate certain figures by a simple technique from treatment, this time, with two dimensions line X area or column X area: “the distribution of blocks in four complementary or alternate fields”. Therefore, if you choose a mental exercise, this type of grids proposes good of it to you difficult. And there if you allow yourselves to register the multiples in the cells, you have very beautiful exercises of drive on the strategy of treatment of the “groups independent of numerically dependant multiples”.

Type 3:

Use of the techniques allowing the simplification of the cell-with-candidate-multiples, initially as for type 2, according to only one dimension; line, column or area, but with a size greater of which “elimination because of the quadruplets and quintuples naked” and “the degreasing of the quadruplets and quintuples hidden”. To proceed by this last technique, which is more tiresome besides to carry out, it is in makes use “elimination because of one or two naked groups” but of lower size!

Certain grids of the type 3 require a treatment according to two dimensions (lines X columns, lines X areas and/or columns X areas) by using processes much more astute, but justifiable of which X-Wing, Swordfish, Jellyfish, Squirmbag or the TPU, technique rising from the “principle of the unicity of the solution”. Thus for this type of grids, one should not hope to lead to the solution only while reasoning mentally, without to have henceforth put all the possible candidates in all the cells. Three degrees of difficulty are possible, according to the size of the naked or camouflaged groups, but also according to their number.

Type 4:

The strategy adopted for the grids of this type, presenting more complex cases of figures, consists in taking into account simultaneously three dimensions (lines X columns X areas). " thus should be able; sauter" of an area with another, through the boxes, by using " passerelles" materialized either by a line, a column or an area. In short, " should be created; chemins" between the various boxes. Thus, one will recognize processes similar to those already used by two dimension treatment of which X-Wing for example (the tops are not any more those of a rectangle, but among those of a polygon). Let us specify that this strategy is based on bivalent logic (for a figure NR fixed and a given box of multiples, p: " NR is the valeur" or not (p): " NR is not the valeur").

Seen of a certain angle, it is a question of superimposing two or several grids on the same grid-problem initial, of making a logical conjugation of the various proposals (concretized by ways) and of determining those of the grids which end in a contradiction with one of the rules which govern the play sudoku, to discover the good solution. It is thus as if one proceeds by formulation by assumption, but in a diverted way! It is necessary to acknowledge that this manner of making gets more pleasure to play and apply processes to put forth assumptions to obtain grids " pauvres" on the intellectual level! Use colored pencils. Those which are already initiated with this technique will recognize easy, average and even difficult grids.

Type 5:

Certain newspapers, magazines, sites and software deliver grids to us say “diabolic”. Generally, it of it is not nothing like it! These grids can be solved by the techniques developped at the point so far. A great majority can be filled “mentally” even!

From now on, it is the only means to lead to the solution, while waiting for the development of new “manual” processes.

Let us announce finally, that on the level of the construction of the grid-problems, it is frequently easier to obtain a grid of the type 1, and almost rare to fall on a grid of the type 4 or 5. The software worked out so far leaves of course the various processes of resolution, to manufacture a problem, but the desired level drops, alas, generally one or even of two degrees! Statistically, one notes that the frequency distribution by type turns around 46%,32%,11%,8% and 3%, of the first type to the fifth.

Construction

It would seem that the grids of Dell Magazines , the pioneer in the field of the publication, are generated by computer. They are usually made up of 30 revealed digits distributed with the Hasard. The author of the grids is unknown. During the Winter 2000, Wei-Hwa Huang affirmed that he was the author of the program which generates these grids; according to him, the former grids were built with the hand. The generator would be written in C++ and, although it offers certain options to satisfy the Japanese market (Symétrie and less figures), Dell prefers not to use them. Some speculate that Dell continuous to use this program, but no proof supports their assertion.

The sudoku of Nikoli, important creator of sudoku in Japan, are built with the hand, the name of the author appearing with each grid published; revealed are always presented in a symmetrical way. This exploit is possible by knowing in advance the place where will be revealed and by assigning thereafter a figure to the cells thus chosen. The Number Place Challenger of Dell posts also the name of the author. The grids published in the majority of the British newspapers would be generated automatically, but call upon symmetry, which would let imply that human creates them. The Guardian affirms that its grids are created with the hand by Japanese, but no mention of the author is made. They would be built by people of Nikoli. The Guardian affirmed that since they are built with the hand, they contain “subtle highly improbable allusions” in the grids built by computer.

It is possible to build grids with multiple solutions and unresolved, but those are not regarded as authentic sudoku. As for the other logical plays, a single solution is necessary. An great attention is thus necessary during the construction of a grid, since only one figure badly placed risk to make the resolution of this one impossible.

Let us point out how the basic principle of Known-Doku resides in the fact that only are allowed like problems to be solved, the grids which lead to one and only one solution! However, certain sites and specialized magazines publish grid-problems proposing less data at the beginning and having even symmetries being able to be more attractive, sometimes whimsical, but admitting more than one solution. But, there is not that the problem of the unicity of the solution, certain experienced players noticed that, for certain grids, one or more figures are revealed in way " gratuite" , because they can be logically deduced by considering the remainder from the figures from the grid. What wants to say that one could propose the grid with less figures while guaranteeing the result with same and single solution. It is a question of optimization of the grid-problems: less figures from which none can be deduced from the others.

See too

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Mathematical of Sudoku

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Generalized symmetries and wide table of resolution

Grid-joint; to change situation for more thorough resolution

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Strategy of last recourse: clear and clear formulation of the assumptions

Software solutions

Degrees of difficulty

Type 1:

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