The Science Behind SUDOKU Solving a Sudoku Puzzle Requires No Math, Not Even Arithmetic

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YOUR CHALLENGE: Insert a number from 1 to 9 in each cell without repeating any of those digits in the same row, column or subgrid (3×3 box). The solutions to the moderately difﬁ cult Sudoku in the center and to others in this article appear at www.sciam. com, along with additional puzzles.

80 SCIENTIFIC AMERICAN COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. The Science behind SUDOKU Solving a Sudoku puzzle requires no math, not even arithmetic. Even so, the game poses a number of intriguing mathematical problems BY JEAN-PAUL DELAHAYE

ne might expect a game of logic to appeal to very few diagonals add up to the same sum. Indeed, aside from the people—mathematicians, maybe, computer geeks, numbers and the grid, Sudoku has almost nothing to do with compulsive gamblers. Yet in a very short time, Sudoku the magic square—but everything to do with the Latin square has become extraordinarily popular, bringing to mind [see box on next page]. Othe Rubik’s cube craze of the early 1980s. A Latin square of order n is a matrix of n2 cells (n cells on Unlike the three-dimensional Rubik’s cube, a Sudoku puz- a side), fi lled with n symbols such that the same symbol never zle is a fl at, square grid. Typically it contains 81 cells (nine appears twice in the same row or column (each of the n sym- rows and nine columns) and is divided into nine smaller bols is thus used precisely n times). The origin of those grids squares containing nine cells each; call them subgrids. The dates back to the Middle Ages; later, mathematician Leon- game begins with numbers already printed in some cells. The hard Euler (1707–1783) named them Latin squares and stud- player must fi ll in the empty cells with the numbers 1 to 9 in ied them. such a way that no digit appears twice in the same row, col- A standard Sudoku is like an order-9 Latin square, differ- umn or subgrid. Each puzzle has one unique solution. ing only in its added requirement that each subgrid contain the Ironically, despite being a game of numbers, Sudoku numbers 1 through 9. The fi rst such puzzle appeared in the demands not an iota of mathematics of its solvers. In May 1979 edition of Dell Pencil Puzzles and Word Games fact, no operation—including addition or multipli- and, according to research done by Will Shortz, the crossword cation—helps in completing a grid, which in theo- editor of the New York Times, was apparently created by a ry could be fi lled with any set of nine different retired architect named Howard Garns. Garns died in India- symbols (letters, colors, icons and so on). Nev- napolis in 1989 (or 1981; accounts vary), too early to witness ertheless, Sudoku presents mathematicians the global success of his invention. and computer scientists with a host of chal- The game, published by Dell as “Number Place,” jumped lenging issues. to a magazine in Japan in 1984, which ultimately named it “Sudoku,” loosely translated as “single numbers.” The maga- Family Tree zine trademarked that moniker, and so copycats in Japan used on e t hi ng is not unresolved, however: the “Number Place” name. In yet another Sudoku-related the game’s roots. The ancestor of Sudoku is not, irony, then, the Japanese call the puzzle by its English name, as is commonly presumed, the magic square—an ar- and English speakers call it by its Japanese name. ray in which the integers in all the rows, columns and Sudoku owes its subsequent success to Wayne Gould, a peri-

www.sciam.com SCIENTIFIC AMERICAN 81 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. SUDOKU’S PREDECESSORS A Sudoku grid is a special kind of Latin square. Latin squares, which Cell 5 8 6 4 2 1 3 7 9 were so named by the 18th- 3 2 7 9 6 5 4 8 1 century mathematician Leonhard 1 2 3 4 9 1 4 3 7 8 6 2 5 Euler, are n×n matrices that are 2 3 4 1 ﬁ lled with n symbols in such a way 1 6 3 5 8 4 7 9 2 3 4 1 2 that the same symbol never 2 4 5 1 9 7 8 6 3 appears twice in the same row or 4 1 2 3 column. Two examples are shown. 8 7 9 6 3 2 5 1 4 Small Latin square The standard completed Sudoku (n = 4) 7 5 8 2 1 3 9 4 6 grid (also known as a solution 6 3 1 7 4 9 2 5 8 grid) is a 9×9 Latin square that Subgrid 4 9 2 8 5 6 1 3 7 meets the additional constraint of having each of its nine subgrids Latin square that is also a completed contain the digits 1 to 9. Sudoku grid (n = 9) Leonhard Euler

patetic retired judge living in Hong There are only 12 Latin squares of particular grid by elementary opera- Kong, who came across it while visiting order 3 and 576 of order 4, but 5,524,- tions. This result, from Bertram Felgen- Japan in 1997 and wrote a computer 751,496,156,892,842,531,225,600 of hauer of the Technical University of program that automatically generates order 9. Group theory, however, says Dresden in Germany and Frazer Jarvis Sudoku grids. At the end of 2004 the that a grid that can be derived from an- of the University of Sheffi eld in England, London Times accepted his proposal to other is equivalent to the original. For has now been verified several times. publish the puzzles, and in January instance, if I systematically replaced (Verifi cation matters for results that are 2005 the Daily Telegraph followed suit. each number with some other (say, 1 be- obtained this way.) Since then, several dozen daily papers in came 2, and 2 became 7, and so on), or If we count only once those grids countries all over the world have taken if I swapped two rows or columns, the that can be reduced to equivalent con- to printing the game, some even putting fi nal results would all be essentially the fi gurations, then the number shrinks to it on the cover page as a promotional same. If one counts only the reduced 5,472,730,538—slightly lower than the come-on. Specialty magazines and en- forms, then the number of Latin squares population of humans on the earth. De- tire books devoted to this diversion have of order 9 is 377,597,570,964,258,816 spite this reduction, Sudoku devotees sprung up, as have tournaments, Web (a result reported in Discrete Mathe- need not fear any shortage of puzzles. sites and blogs. matics in 1975 by Stanley E. Bammel Note that a complete Sudoku solu- and Jerome Rothstein, then at Ohio tion grid may be arrived at in more than

State University). one way from any starting, or clue, grid EMANUEL HANDMANN, 1753, ORIGINAL PASTEL AT KUNSTMUSEUM BASEL As Many Grids as Humans ) ; it did not take long for mathe- Exactly how many Sudoku grids can (that is, an incomplete grid whose solu- maticians to begin playing “how many” exist has been rather diffi cult to estab- tion is a given complete version). No- page this games with Sudoku. For instance, they lish. Today only the use of logic (to sim- body has yet succeeded in determining and soon asked how many unique fi lled-in, plify the problem) and computers (to how many different starting grids there or “solution,” grids can be constructed. examine possibilities systematically) are. Moreover, a Sudoku starting grid is Clearly, the answer has to be smaller makes it possible to estimate the number really only interesting to a mathemati- than the number of Latin squares be- of valid Sudoku solution grids: 6,670,- cian if it is minimal—that is, if removing cause of the added constraints imposed 903,752,021,072,936,960. That num- a single number will mean the solution preceding pages by the subgrids. ber includes all those derived from any is no longer unique. No one has fi gured out the number of possible minimal grids, which would amount to the ulti- Overview/Scientifi c Sudoku mate count of distinct Sudoku puzzles. ■ Sudoku is more than just an entertaining logic game for players; it also raises It is a challenge that is sure to be taken a host of deeper issues for mathematicians. up in the near future. ■ Such problems include: How many Sudoku grids can be constructed? What is Another problem of minimality also the minimal number of starting clues that will yield one unique solution? Does remains unsolved—to wit, what is the Sudoku belong to the class of hard problems known as NP-complete? smallest number of digits a puzzle mak- ■ Puzzle mavens have come up with an array of approaches to attacking er can place in a starting grid and still Sudokus and with entertaining variations on the game. guarantee a unique solution? The an-

swer seems to be 17. Gordon Royle of JEN CHRISTIANSEN; SOURCE: JEAN-PAUL DELAHAYE (

82 SCIENTIFIC AMERICAN JUNE 2006 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. the University of Western Australia has collected more than 38,000 exam- Another problem: What is the smallest ples that fi t this criterion and cannot number of digits that can be put in a starting be translated into one another by per- forming elementary operations. grid and still guarantee a unique solution? Gary McGuire of the National Uni- versity of Ireland, Maynooth, is con- The solution programs can employ counters a confl ict (which can happen ducting a search for a 16-clue puzzle several methods, but the most common very quickly), it erases the 1 just placed with a unique solution but has so far is backtracking, a systematic form of and inserts 2 or, if that is invalid, 3 or come up empty-handed. It begins to trial and error in which partial solutions the next legal number. After placing the look as if none exists. On the other are proposed and then modifi ed slightly fi rst legal number possible, it moves to hand, Royle and others working inde- as soon as they are proved wrong. the next cell and starts again with a 1. pendently have managed to fi nd one 16- The basic backtracking algorithm If the number that has to be changed clue puzzle that has just two solutions. works like this: The program places the is a 9 (which cannot be raised by one in Searchers have not yet uncovered any number 1 in the fi rst empty cell. If the a standard Sudoku grid), the program additional examples. choice is compatible with the existing backtracks and increases the number in Is anyone near to proving that no clues, it continues to the second empty the previous cell (the next-to-last num- valid Sudoku puzzle can have only 16 cell, where it places a 1. When it en- ber placed) by one. Then it moves for- clues? McGuire says no. If we could an- alyze one grid per second, looking for a HOW LOW CAN YOU GO? valid 16-clue puzzle within it, he notes, “we could search all the grids in 173 The minimum number of clues that a 9×9 Sudoku puzzle can start with and still years. Unfortunately, we cannot yet do yield a unique solution seems to be 17; an example is shown. One particular fi lled-in grid, known to Sudoku afi cionados as “Strangely Familiar,” or SF, hides 29 this, even on a fast computer.” Soon, he inequivalent 17-clue starting boards—an unusually high number. SF was once says, searching a grid might be doable in considered the grid most likely to harbor a 16-clue puzzle with a unique solution, but one minute on a powerful computer, but an exhaustive search has dashed that hope. The only known 16-clue Sudoku having at that rate the endeavor would take just two solutions appears at bottom; the fi nal grids interchange the 8’s and 9’s. 10,380 years. “Even distributed on “Strangely Familiar” Grid 10,000 computers, it would take about One 17-Clue Puzzle one year,” he adds. “We really need a 1 9 6 3 9 2 4 1 7 8 5 breakthrough in our understanding to 3 8 2 8 4 7 6 5 1 9 3 make it feasible to search all the grids. 6 5 1 7 9 8 3 6 2 4 We either need to reduce the search space or find a much better algorithm for 1 2 4 1 2 3 8 5 7 9 4 6 searching.” 7 3 7 9 6 4 3 2 8 5 1 Mathematicians do know the solu- 5 4 5 8 6 1 9 2 3 7 tion to the opposite of the minimum number of clues problem: What is the 8 6 3 4 2 1 7 8 5 6 9 maximum number of givens that do not 4 2 8 6 1 5 9 4 3 7 2 guarantee a unique solution? The an- 7 5 9 7 5 3 2 6 4 1 8 swer is 77. It is very easy to see that with 80, 79 or 78 givens, if there is a solution, 16-Clue Puzzle ...... with Two Solutions it is unique. But the same cannot be 5 2 4 5 6 2 3 89 98 4 7 1 guaranteed for 77 givens [see bottom 8 9 University of Western Australia 7 1 3 9 4 8 7 1 6 2 5 3 box on page 86]. 8 1 3 7 4 2 5 9 98 6 8 Computer Solvers 4 6 3 5 9 1 98 4 6 2 7 8 beyond the “how m a n y” ques- 7 2 98 7 4 2 6 3 1 9 5 tions, mathematicians and computer sci- 8 9 entists enjoy pondering what computers 1 2 1 6 9 5 7 3 4 8 9 8 can and cannot do when it comes to 6 2 6 8 1 5 4 2 7 3 9 8 solving and generating Sudoku puzzles. 3 1 7 2 5 6 3 9 98 1 4 For standard Sudokus (9×9), it is rela- 8 4 9 3 98 7 1 5 6 2 tively easy to write computer programs 4

JEN CHRISTIANSEN; SOURCE: GORDON ROYLEthat solve all valid starting grids.

Here are a few ways to try solving a b a Sudoku puzzle. Methods 1 and 2 are the simplest and are usually 5 1 9 6 5 1 9 6 used in tandem (a bit of one, a bit 9 5 9 5 of the other). Unfortunately, they do not always get one very far, so 5 2 7 5 2 7 “Only” a player can add in method 3 and, 4 9 1 7 cells 4 9 1 7 if that proves insuffi cient, 7 (blue) 7 method 4—which works every time but not necessarily easily. 1 3 2 1 3 2 You can also invent methods of 3 4 5 9 “Forced” 3 4 5 9 your own and try the many cells 2 8 7 1 4 (orange) 2 8 7 1 4 approaches described on the Web. 7 6 5 8 2 7 6 5 8 2 c d 5 1 9 6 9 5 5 2 7 4 9 1 7 7 1 3 2 3 4 5 9 2 8 7 1 4 7 6 5 8 2 ward until it hits a confl ict. (The pro- constraint propagation: after each new France invented Prolog in the late 1970s. gram sometimes backtracks several number is placed, the program generates For human players, the backtracking times before advancing.) In a well-writ- a table of the remaining possible num- techniques employed by computer pro- ten program, this method exhaustively bers in each empty cell and considers grams would be infeasible; they would explores all possible hypotheses and only numbers from this table. require extraordinary patience. So mor- thus fi nishes by fi nding the solution, if Backtracking techniques can be en- tals use more varied and smarter rules one exists. And if multiple solutions ex- coded by fairly short solution programs. and generally turn to trial and error only ist, as would be the case with a fl awed Indeed, concise programs have been writ- as a last resort. Some programs try to puzzle, the program fi nds all of them. ten for Sudoku in Prolog, a computer lan- mimic human methods to an extent: they Of course, refi nements are possible, guage incorporating a backtracking algo- are longer than the others but work just and they speed up the discovery of the rithm. Alain Colmerauer and Philippe as well. The programs that simulate hu- unique solution. One favorite is called Roussel of the University of Mar seilles in man reasoning are also useful for assess- ing the diffi culty of starting grids, which JEAN-PAUL DELAHAYE is professor of computer science at the University of Sciences and are ranked from “easy” (requiring simple Technologies of Lille in France and a researcher in the Computer Science Laboratory of Lille tactics) to what many call “devilish” or (LIFL) of the National Center for Scientifi c Research (CNRS) based at the university. His “diabolical” (because of the need to apply work focuses on computational game theory (such as the iterated prisoner’s dilemma) and more mind-bending rules of logic). complexity theory (such as Kolmogorov complexity) and applications of these theories to One way computer scientists think

T H E A U T H O R genetic analysis and, recently, to economics. This article expands on one Delahaye pub- about solving a Sudoku puzzle is to

lished in the December 2005 Pour la Science, the French edition of Scientiﬁ c American. equate the task to a graph-coloring prob- JEN CHRISTIANSEN; SOURCE: JEAN-PAUL DELAHAYE

METHOD 1 METHOD 2

“FORCED” CELL “ONLY” CELL This approach considers a cell to be fi xed. Here a given value is the focus—for example, the number 5. Columns one and three in By eliminating as possibilities any other grid a already have 5s, but column two so far has none. Where can that column’s 5 be? numbers in the same column, row and Not in the three fi rst cells of column two, because they sit in a subgrid that already subgrid, you see whether only one possibility has a 5. Not in the seventh cell of the column, because its subgrid, too, has a 5. remains. Such analysis of grid a will reveal Thus, the 5 of column two is either in the fourth, fi fth or sixth cell of the column. Only that the boxes containing orange numbers in the fi fth cell is free, so the number goes there. The cells marked with blue numbers grid b are “forced” cells. in grid b are “only” cells.

METHOD 3 METHOD 4

SIMPLIFYING THE RANGE OF POSSIBILITIES TRIAL AND ERROR This technique is extremely powerful but requires a pencil and eraser. In each cell, By applying methods 1 through 3, you can solve you write all possible solutions very small or use dots whose positions represent many Sudoku grids. But diabolical-level Sudokus the numbers 1 to 9. Then you apply logic to try to eliminate options. often require a phase of trial and error. When For example, grid c shows how grid a would look if it were marked up by rote, uncertainty persists, you make a random choice without methods 1 and 2 being applied fi rst. In the third column, the array of and apply all your strategies as if that were the possibilities for the second, third, fourth, fi fth and sixth cells are, respectively, correct decision. If you hit an impossibility (such {2, 3, 6, 7}, {3, 6, 9}, {2, 6}, {2, 6} and {6, 7}. The column must contain a 2 and a 6, as two identical numbers in the same column), you so these numbers must be in the two cells whose sole possibilities are 2 and 6 know you chose incorrectly. For instance, you (circled in fi rst detail). Consequently, 2 and 6 cannot be anywhere else in this might try 2 in the fourth cell of the third column in column and can be deleted from the column’s other cells (red). The range of grid c. If that fails, you begin again from the same possibilities for the column is simplifi ed to {3, 7}, {3, 9}, {2, 6}, {2, 6}, {7}. But that starting point, but this time with 6 in the box. isn’t all. Stipulating the position of 7 in turn dictates the positions of 3 and of 9 Unfortunately, sometimes you have to do (second detail). The fi nal possibilities are {3}, {9}, {2, 6}, {2, 6}, {7}. A single several rounds of trial and error, and you have to uncertainty remains: where 2 and 6 should go. be prepared to backtrack if you guess incorrectly. The general rule for simplifying possibilities is the following: if, among a set of Indeed, the idea behind the trial-and-error method possibilities (for a row, column or subgrid), you fi nd m cells that contain a subset is the same as that used by backtracking consisting only of m numbers (but not necessarily all of them in each cell), the algorithms, which computer programs can easily digits in the subset can be eliminated as possibilities from the other cells in the implement but which can sorely tax human brains. larger set. For instance, in d the arrangement {2, 3}, {1, 3}, {1, 2}, {1, 2, 4, 5}, {3, 5, 7} How remarkable it is that the method most can be simplifi ed to {2, 3}, {1, 3}, {1, 2}, {4, 5}, {5, 7}, because the cells {2, 3}, {1, 3}, effective for a machine is the least effective for {1, 2} all come from the subset {1, 2, 3} and have no other numbers. a human being.

lem in which two adjacent cells (other- lated into a coloring problem is mean- two nodes joined by an edge are assigned wise known as “two vertices joined by ingful to scientists, because that prop- the same color. In the case of Sudoku, an edge”) cannot take the same color erty links Sudoku to a class of impor- the apparently impossible challenge is and the available palette has nine colors. tant problems. In particular, Takayuki designing an efficient program that The graph contains 81 vertices, some of Yato and Takahiro Seta of the Univer- would solve Sudokus of all sizes—that is, which are colored at the outset. The col- sity of Tokyo have recently demonstrat- every grid that takes the form n2×n2, oring problem is actually quite complex ed that Sudoku belongs to the category not just the standard 32×32 (9×9) ver- because each 9×9 grid has hundreds of of NP-complete problems. Such prob- sions. No program for solving all puz- edges. Each cell is part of a row includ- lems are ones that probably cannot be zles would work efﬁ ciently because the ing eight other cells, a column including solved in a realistic time frame. Well- time required to ﬁ nd a solution increas- eight other cells, and a subgrid including known examples include the 3-color- es dramatically as n gets bigger. eight other cells (of which four have al- ability problem, which asks whether it If you have an algorithm that solves ready been counted in the column or is possible to shade each node in a graph classic Sudokus, you can use it to obtain row). So each of the 81 cells is linked to with three colors in such a way that no an algorithm that designs them. Indeed, 20 (8 + 8 + 4) other cells, which makes a grand total of 1,620 cells that share one edge with a neighbor—which, in turn, That Sudoku puzzles can be translated into means that the total number of edges is a coloring problem links the game to a 810 (1,620 divided by 2). That Sudoku puzzles can be trans- class of important mathematical problems.

www.sciam.com SCIENTIFIC AMERICAN 85 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. VARIATIONS ON A THEME a In need of something more than b standard diabolical grids? In the puzzles here, the usual rules apply, N 5 with some twists. In a, the letters in I 2 the words GRAND TIME replace 9 3 numbers, and geometric shapes T R 3 6 5 replace the square subgrids. Its R E inventor calls it a Du-Sum-Oh puzzle. 6 4 7 In b, which contains six triangular A N 1 6 subgrids, the rows and the slanted M D 8 columns may be interrupted in the 4 8 center, and when a row or column R 5 9 has only eight cells, the nearby cell A that forms a point of the “star” 3 1 2 serves as the ninth cell. In c, the G three-place numbers formed by the marked rows in the ﬁ rst two

subgrids add up to the number in the c d third subgrid. In d, greater-than and >

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less-than signs indicate where the + 9 = 6 > > > > > > > > > > > >

digits belong. In e, the dominoes at >

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6 5 8 2 > > > > > > > > > > the bottom need to be placed into > >

the empty spaces. In f, three game + 4 = 9 > >

> > >

boards overlap. Visit www.sciam. >

> > > >

6 1 8 5 7 > > > > > >

com for solutions and more games. > > >

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5 + = 8 > > > > > > > > > > > > >

8 1 4 5 6 > > > > > > > >

> > >

9 = > > > > > 5 + > > > > > > >

> > >

8 7 1 6 > > > > > > > > > > > 7 + 3 = > >

although early Sudokus were construct- ample, backtracking) is applied. If the the puzzle has various solutions, one is ed by hand, today almost all are pro- puzzle possesses a unique solution, the chosen and the algorithm then adds as ) ; b duced by computer programs based on program stops. If the partially complet- many numbers as needed to the starting ( the following approach or a similar one. ed problem has no solution, one number clues to ensure that the chosen solution Numbers are placed at random on a grid is taken away from the starting arrange- is unique. board, and a solution algorithm (for ex- ment and the program begins again. If Human Strategies TOO FEW CLUES fans who enjoy solving Sudokus manually can choose among many tac- NONZERO/OLGA LEONTIEVA

77 clues are not tics, but two basic approaches offer a © ) 1 2 THE GRAND TIME SUDOKU AND THE LAW ) ; 2 1 3 3 4 5 6 7 8 9 d necessarily enough a 4 5 6 4 5 6 7 8 1 2 3 decent starting point. First, search for to guarantee a unique 9 the most constrained empty cells: those solution. Despite 7 8 9 7 8 9 1 2 3 4 5 6 that belong to a row, column or subgrid having only four 21 12 4 4 3 9 8 5 6 7 that is already pretty well filled in. empty cells, the Sometimes eliminating impossibilities grid here has two 8 6 5 2 7 1 3 9 4 (the numbers already occupying cells in

ﬁ rst two columns ) ; 3 4 1 8 6 A K PETERS LTD (IN PRESS) ( 2 9 7 5 c you to discover the only number that the missing 1’s and 5 7 2 9 1 4 6 3 8 2’s (inset) are will work in a particular cell; in any interchangeable. 6 9 8 5 3 7 2 4 1 case, the method should greatly narrow JEN CHRISTIANSEN;OF LEFTOVERS. SOURCES: BOB HARRIS IN the options. ED PEGG, JR. (

86 SCIENTIFIC AMERICAN JUNE 2006 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. VARIATIONS ON A THEME e f 1 9 5 2 8 6 4 3 7 7 6 8 7 5 9 5 4 9 3 5 2 6 1 7 3 4 6 5 9 2 7 8 5 1 8 2 9 3 6 4 1 7 5 2 2 9 3 1 7 6 5 8 1 9 3

Second, search for where a given thus making pencil and eraser unneces- square subgrids with other structures; value might be found in a particular col- sary. Some even enable you to create still others introduce additional con- umn, row or subgrid (for example, look links between cells. Do not overlook straints. These alternatives appeal be- for the only places where a 3 might fi t in these software programs. In freeing you cause they compel you to explore new row four). Sometimes the query will from such tedious tasks as erasing, they logical strategies. Moreover, devotees have only one possible response. Other will actually spur you to greater subtle- who take only a quarter of an hour to do times just knowing that the 3 can go ty and virtuosity in this game of logic. a traditional puzzle can immerse them- only in two or three particular spots Once you have gotten bored with selves the entire day in the delights of ends up being helpful. See the box on traditional Sudokus, you can go looking combining cells and numbers in giant pages 84 and 85 for more details. Also, for the innumerable variants: some versions of Sudoku. But enough of that. visit the Web sites listed in “More to Ex- overlap multiple grids; others replace On to the next grid! plore” to fi nd a host of strategies, some of which have such creative names as MORE TO EXPLORE “swordfi sh” and “golden chain.” 1st World Sudoku Championship: www.wsc2006.com/eng/index.php A number of software programs eas- Math Games. Ed Pegg, Jr.: www.maa.org/editorial/mathgames/mathgames–09–05–05.html

) ily found on the Internet will generate The Mathematics of Su Doku. Sourendu Gupta: http://theory.tifr.res.in/ sgupta/sudoku/ f ˜ boards of speciﬁ ed difﬁ culty and help Mathematics of Sudoku. Tom Davis: www.geometer.org/mathcircles and

e you ﬁ nd solutions (without, of course, SadMan Software Sudoku techniques: www.simes.clara.co.uk/programs/sudokutechniques.htm solving the puzzle for you!). For exam- Sudoku, an overview: www.sudoku.com/howtosolve.htm ple, some allow you to put temporary Sudoku, from Wikipedia: http://en.wikipedia.org/wiki/Sudoku

ED PEGG, JR. ( marks in the cells and to erase them, A Variety of Sudoku Variants: www.sudoku.com/forums/viewtopic.php?t=995