''When you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth.'' -- Sherlock Holmes
When it comes to sudoku, there is no escape. The grids of these puzzles seem to shut down the mental apparatus, enclosing one's faculties in a tightly constrained universe -- a 9 by 9 array that must be carefully filled up with the numbers 1 to 9, following certain rules.
That enclosure is hypnotic. Publisher's Weekly recently counted 23 sudoku books in print with total sales of 5.7 million copies. Newspapers wage circulation wars by running sudoku in their pages. And sudoku Web sites and forums proliferate internationally (for example, see sudoku.jouwpagina.nl). The teams in the first World Sudoku Championship -- held in Lucca, Italy, in March -- came from 22 countries, including the Philippines, India, Venezuela, and Croatia. The winner was a 31-year-old woman, a Czech accountant. The Independent of London recently reported that a 700 percent increase in the sale of pencils has been attributed to the sudoku craze. Last November, British Airways sent a memo to all its cabin crews, forbidding them to work on sudoku puzzles during takeoffs and landings.
The international appeal, of course, may have something to do with the fact that no language is needed to solve sudoku puzzles; neither, for that matter, is any mathematics. The puzzler is given an 81-square grid, with about 20 squares filled in (the ''givens''). That large grid is itself divided into nine 3 x 3 grids. The challenge is to fill in the blanks so that each nine-cell row, each nine-cell column and each nine-cell mini-grid contains all the numbers from 1 to 9, with no repetitions or omissions.
This is not a novel challenge. Magic squares of various kinds were part of many ancient cultures. Benjamin Franklin published a paper about magic squares, and he obsessively fiddled with them during the same years he was helping to form a more perfect union. The 18th-century Swiss mathematician Leonhard Euler studied the properties of Latin Squares in which each row and column would contain a complete list of the elements of a set of numbers or letters.
Sudoku-style puzzles -- which add the twist of the mini-grids within a larger array -- were titled Number Place when they began appearing anonymously in 1979 in the periodical Dell Pencil Puzzles and Word Games. Will Shortz, the crossword puzzle editor of The New York Times, deduced the author's identity with sudoku-style argument: anytime the Dell publication contained one of these puzzles -- and never otherwise -- the list of contributors included Howard Garns, an architect from Indianapolis; Mr. Garns died in 1989.
The American-born puzzles made their way to Japan in 1984, where the publisher Nikoli ended up calling them sudoku -- meaning single numbers. And in 1997, Wayne Gould, a New Zealander who had served as a judge in Hong Kong, came across them while vacationing in Tokyo. In 2004 he successfully lobbied The Times of London to introduce the puzzles to Britain, beginning the craze in the West. Strangely, in Japan, where Nikoli has trademark control over the name sudoku, the puzzles are still familiarly known by the English title Number Place, while in the English-speaking work, their Japanese pedigree is widely assumed.
But what is their lure? A mathematician I spoke with dismissed the puzzles as mere ''bookkeeping'' -- keeping track of where things go. And there surely is some of that, since one technique for solving them involves tentatively writing miniature numbers in each little square to figure out the various possibilities. The grid for a difficult puzzle can begin to look like the first draft of a major corporation's balance sheet. This is hardly higher mathematics. In fact, numbers are hardly necessary: the same puzzle can be posed using nine colors or nine national flags.
Yet mathematicians have been taking more of an interest in sudoku -- not necessarily in solving the puzzles, but in understanding more about their character. In a recent essay in The American Scientist, Brian Hayes described the difficulty of determining the difficulty of these puzzles: it bears little connection with how many numbers are given at the start. And while numbers are really not that important to sudoku itself, they certainly proliferate in discussions about it. In MathWorld, an online mathematical journal (mathworld.wolfram.com), Eric W. Weisstein cited research showing that 6,670,903,752,021,072,936,960 completed grids are possible for sudoku puzzles, though only 5,472,730,538 unique grids remain once equivalent solutions are eliminated.
In a September 2005 column by Ed Pegg Jr. on the site of the Mathematical Association of America (maa.org/news/mathgames.html) also pointed out that the graph theorist Gordon Royle had collected more than 10,000 sudoku puzzles, each with 17 givens. That may be the smallest number of givens that will yield unique solutions (any fewer givens will allow multiple answers), though apparently that hypothesis has not yet been proven. The exhaustive Wikipedia article on sudoku goes into even greater detail (en.wikipedia.org/wiki/Sudoku).
What many of these studies focus on, though, is how many sudoku possibilities there are. Each puzzle, using only the simplest of elements, combined according to the simplest of rules, pulls a single solution out of a mind-boggling number of possibilities. The puzzle is an act of reduction and elimination.
Often, in solving a puzzle, we work toward an answer, accumulating information. Something must be calculated or produced. In a crossword puzzle, the words have to be pulled out of one's experience. Here, though, each square has only nine possibilities and the work mainly involves not finding the possible but eliminating the impossible, filtering away everything that does not fit -- ruling out the number 4 for a particular square, for example, if a 4 appears in the same row or column or mini-grid. Sudoku does not open up into the world; it reduces the world to its boundaries, forcing everything extraneous to be discarded. There is something more technological about it than mathematical: we know what we must produce; the problem is in getting rid of everything that doesn't fit.
This must help account for sudoku's tremendous appeal: it seems to distill complication into elemental clarity, even when that task becomes difficult. On the blog onigame.livejournal.com, the puzzle master Wei-Hwa Huang (who came in third at the world competition in March) devised a unusual system, using colored loops, for solving sudoku that is more knotty than seems possible given the puzzle itself. But the solution is still something easily understood once complete. And it is always immensely satisfying, because finally, all impossibilities have been eliminated, leaving behind a neat array of 81 numbers, that however improbably reveal the trivial truth.