[Paleopsych] Sigma Xi: Unwed Numbers
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American Scientist Online. The Magazine of Sigma Xi, the Scientific Research
Society
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see full issue: January-February 2006 Volume: 94 Number: 1 Page: 12
DOI: 10.1511/2006.1.12
COMPUTING SCIENCE
The mathematics of Sudoku, a puzzle that boasts "No math required!"
Brian Hayes
A few years ago, if you had noticed someone filling in a crossword puzzle with
numbers instead of letters, you might well have looked askance. Today you would
know that the puzzle is not a crossword but a Sudoku. The craze has circled the
globe. It's in the newspaper, the bookstore, the supermarket checkout line; Web
sites offer puzzles on demand; you can even play it on your cell phone.
Just in case this column might fall into the hands of the last person in North
America who hasn't seen a Sudoku, an example is given on the opposite page. The
standard puzzle grid has 81 cells, organized into nine rows and nine columns
and also marked off into nine three-by-three blocks. Some of the cells are
already filled in with numbers called givens. The aim is to complete the grid
in such a way that every row, every column and every block has exactly one
instance of each number from 1 to 9. A well-formed puzzle has one and only one
solution.
The instructions that accompany Sudoku often reassure the number-shy solver
that "No mathematics is required." What this really means is that no arithmetic
is required. You don't have to add up columns of figures; you don't even have
to count. As a matter of fact, the symbols in the grid need not be numbers at
all; letters or colors or fruits would do as well. In this sense it's true that
solving the puzzle is not a test of skill in arithmetic. On the other hand, if
we look into Sudoku a little more deeply, we may well find some mathematical
ideas lurking in the background.
A Puzzling Provenance
The name "Sudoku" is Japanese, but the game itself is almost surely an American
invention. The earliest known examples were published in 1979 in Dell Pencil
Puzzles & Word Games, where they were given the title Number Place. The
constructor of the puzzles is not identified in the magazine, but Will Shortz,
the puzzles editor of The New York Times, thinks he has identified the author
through a process of logical deduction reminiscent of what it takes to solve a
Sudoku. Shortz examined the list of contributors in several Dell magazines; he
found a single name that was always present if an issue included a Number Place
puzzle, and never present otherwise. The putative inventor identified in this
way was Howard Garns, an architect from Indianapolis who died in 1989. Mark
Lagasse, senior executive editor of Dell Puzzle Magazines, concurs with
Shortz's conclusion, although he says Dell has no records attesting to Garns's
authorship, and none of the editors now on the staff were there in 1979.
The later history is easier to trace. Dell continued publishing the puzzles,
and in 1984 the Japanese firm Nikoli began including puzzles of the same design
in one of its magazines. (Puzzle publishers, it seems, are adept at the
sincerest form of flattery.) Nikoli named the puzzle "suji wa dokushin ni
kagiru," which I am told means "the numbers must be single"--single in the
sense of unmarried. The name was soon shortened to Sudoku, which is usually
translated as "single numbers." Nikoli secured a trademark on this term in
Japan, and so later Japanese practitioners of sincere flattery have had to
adopt other names. Ed Pegg, writing in the Mathematical Association of
America's MAA Online, points out an ironic consequence: Many Japanese know the
puzzle by its English name Number Place, whereas the English-speaking world
prefers the Japanese term Sudoku.
The next stage in the puzzle's east-to-west circumnavigation was a brief detour
to the south. Wayne Gould, a New Zealander who was a judge in Hong Kong before
the British lease expired in 1997, discovered Sudoku on a trip to Japan and
wrote a computer program to generate the puzzles. Eventually he persuaded The
Times of London to print them; the first appeared in November 2004. The
subsequent fad in the U.K. was swift and intense. Other newspapers joined in,
with The Daily Telegraph running the puzzle on its front page. There was
boasting about who had the most and the best Sudoku, and bickering over the
supposed virtues of handmade versus computer-generated puzzles. In July 2005 a
Sudoku tournament was televised in Britain; the event was promoted by carving a
275-foot grid into a grassy hillside near Bristol. (It soon emerged that this
"world's largest Sudoku" was defective.)
Sudoku came back to the U.S. in the spring of 2005. Here too the puzzle has
become a popular pastime, although perhaps not quite the all-consuming
obsession it was in the U.K. I don't believe anyone will notice a dip in the
U.S. gross domestic product as a result of this mass distraction. On the other
hand, I must report that my own motive for writing on the subject is partly to
justify the appalling number of hours I have squandered solving Sudoku.
Hints and Heuristics
If you take a pencil to a few Sudoku problems, you'll quickly discover various
useful rules and tricks. The most elementary strategy for solving the puzzle is
to examine each cell and list all its possible occupants--that is, all the
numbers not ruled out by a conflict with another cell. If you find a cell that
has only one allowed value, then obviously you can write that value in. The
complementary approach is to note all the cells within a row, a column or a
block where some particular number can appear; again, if there is a number that
can be put in only one position, then you should put it there. In either case,
you can eliminate the selected number as a candidate in all other cells in the
same neighborhood.
Some Sudoku can be solved by nothing more than repeated application of these
two rules--but if all the puzzles were so straightforward, the fad would not
have lasted long. Barry Cipra, a mathematician and writer in Northfield,
Minnesota, describes a hierarchy of rules of increasing complexity. The rules
mentioned above constitute level 1: They restrict a cell to a single value or
restrict a value to a single cell. At level 2 are rules that apply to pairs of
cells within a row, column or block; when two such cells have only two possible
values, those values are excluded elsewhere in the neighborhood. Level-3 rules
work with triples of cells and values in the same way. In principle, the tower
of rules might rise all the way to level 9.
This sequence of rules suggests a simple scheme for rating the difficulty of
puzzles. Unfortunately, however, not all Sudoku can be solved by these rules
alone; some of the puzzles seem to demand analytic methods that don't have a
clear place in the hierarchy. A few of these tactics have even acquired names,
such as "swordfish" and "x-wing." The subtlest of them are nonlocal rules that
bring together information from across a wide swath of the matrix.
When you are solving a specific puzzle, the search for patterns that trigger
the various rules is where the fun is (assuming you go in for that sort of
thing). But if you are trying to gain a higher-level understanding of Sudoku,
compiling a catalog of such techniques doesn't seem very promising. The rules
are too many, too various and too specialized.
Rather than discuss methods for solving specific puzzles, I want to ask some
more-general questions about Sudoku, and look at it as a computational problem
rather than a logic puzzle. How hard a problem is it? Pencil-and-paper
experience suggests that some instances are much tougher than others, but are
there any clear-cut criteria for ranking or classifying the puzzles?
Counting Solutions
In the search for general principles, a first step is to generalize the puzzle
itself. The standard 81-cell Sudoku grid is not the only possibility. For any
positive integer n, we can draw an order-n Sudoku grid with n ^2 rows, n ^2
columns and n ^2 blocks; the grid has a total of n ^4 cells, which are to be
filled with numbers in the range from 1 to n ^2. The standard grid with 81
cells is of order 3. Some publishers produce puzzles of order 4 (256 cells) and
order 5 (625 cells). On the smaller side, there's not much to say about the
order-1 puzzle. The order-2 Sudoku (with 4 rows, columns and blocks, and 16
cells in all) is no challenge as a puzzle, but it does serve as a useful test
case for studying concepts and algorithms.
click for full image and caption Order-2 Sudoku...
How many Sudoku solutions exist for each n? To put the question another way:
Starting from a blank grid--with no givens at all--how many ways can the
pattern be completed while obeying the Sudoku constraints? As a first
approximation, we can simplify the problem by ignoring the blocks in the Sudoku
grid, allowing any solution in which each column and each row has exactly one
instance of each number. A pattern of this kind is known as a Latin square, and
it was already familiar to Leonhard Euler more than 200 years ago.
Consider the 4 x 4 Latin square (which corresponds to the order-2 Sudoku).
Euler counted them: There are exactly 576 ways of arranging the numbers 1, 2, 3
and 4 in a square array with no duplications in any row or column. It follows
that 576 is an upper limit on the number of order-2 Sudoku. (Every Sudoku
solution is necessarily a Latin square, but not every Latin square is a valid
Sudoku.) In a series of postings on the Sudoku Programmers Forum, Frazer Jarvis
of the University of Sheffield showed that exactly half the 4 x 4 Latin squares
are Sudoku solutions; that is, there are 288 valid arrangements. (The method of
counting is summarized in the illustration on the next page.)
Moving to higher-order Sudoku and larger Latin squares, the counting gets
harder in a hurry. Euler got only as far as the 5 x 5 case, and the 9 x 9 Latin
squares were not enumerated until 1975; the tally is
5,524,751,496,156,892,842,531,225,600, or about 6 x 10^27. The order-3 Sudoku
must be a subset of these squares. They were counted in June 2005 by Bertram
Felgenhauer of the Technical University of Dresden in collaboration with
Jarvis. The total they computed is 6,670,903,752,021,072,936,960, or 7 x 10^21.
Thus, among all the 9 x 9 Latin squares, a little more than one in a million
are also Sudoku grids.
It's a matter of definition, however, whether all those patterns are really
different. The Sudoku grid has many symmetries. If you take any solution and
rotate it by a multiple of 90 degrees, you get another valid grid; in the
tabulations above, these variants are counted as separate entries. Beyond the
obvious rotations and reflections, you can permute the rows within a horizontal
band of blocks or the columns within a vertical stack of blocks, and you can
also freely shuffle the bands and stacks themselves. Furthermore, the numerals
in the cells are arbitrary markers, which can also be permuted; for example, if
you switch all the 5s and 6s in a puzzle, you get another valid puzzle.
When all these symmetries are taken into account, the number of essentially
different Sudoku patterns is reduced substantially. In the case of the order-2
Sudoku, it turns out there are actually only two distinct grids! All the rest
of the 288 patterns can all be generated from these two by applying various
symmetry operations. In the order-3 case, the reduction is also dramatic,
although it still leaves an impressive number of genuinely different solutions:
3,546,146,300,288, or 4 x 10^12.
Does the large number of order-3 Sudoku grids tell us anything about the
difficulty of solving the puzzle? Maybe. If we set out to solve it by some kind
of search algorithm, then the number of patterns to be considered is a relevant
factor. But any strategy that involves generating all
6,670,903,752,021,072,936,960 grids is probably not the best way to go about
solving the puzzle.
NP or Not NP, That Is the Question
Computer science has an elaborate hierarchy for classifying problems according
to difficulty, and the question of where Sudoku fits into this scheme has
elicited some controversy and confusion. It is widely reported that Sudoku
belongs in the class NP, a set of notoriously difficult problems; meanwhile,
however, many computer programs effortlessly solve any order-3 Sudoku puzzle.
There is actually no contradiction in these facts, but there is also not much
help in dispelling the confusion.
Complexity classes such as NP do not measure the difficulty of any specific
problem instance but rather describe the rate at which difficulty grows as a
function of problem size. If we can solve an order-n Sudoku, how much harder
will we have to work to solve a puzzle of order n + 1? For problems in NP, the
effort needed grows exponentially.
Most discussions of the complexity of Sudoku refer to the work of Takayuki Yato
and Takahiro Seta of the University of Tokyo, whose analysis relates the task
of solving Sudoku to the similar problem of completing a partially specified
Latin square. The latter problem in turn has been connected with others that
are already known to be in NP. This process of "reduction" from one problem to
another is the standard way of establishing the complexity classes of
computational problems. Yato and Seta employ an unusual form of reduction that
addresses the difficulty of finding an additional solution after a first
solution is already known. In Sudoku, of course, well-formed puzzles are
expected to have only one solution. Yato and Seta say their result applies
nonetheless. I don't quite follow their reasoning on this point, but the
literature of complexity theory is vast and technical, and the fault is likely
my own.
When you lay down your pencil on a completed Sudoku, the thought that you've
just dispatched a problem in the class NP may boost your psychological
wellbeing, but the NP label doesn't say anything about the relative difficulty
of individual Sudoku puzzles. For that, a different kind of hierarchy is
needed.
Many publishers rank their Sudoku on a scale from easy to hard (or from gentle
to diabolical). The criteria for these ratings are not stated, and it's a
common experience to breeze through a "very hard" puzzle and then get stuck on
a "medium."
One easily measured factor that might be expected to influence difficulty is
the number of givens. In general, having fewer cells specified at the outset
ought to make for a harder puzzle. At the extremes of the range, it's clear
that having all the cells filled in makes a puzzle very easy indeed, and having
none filled in leaves the problem under-specified. What is the minimum number
of givens that can ensure a unique solution? For an order-n grid, there is a
lower bound of n ^2 - 1. For example, on an order-3 grid with fewer than eight
givens, there must be at least two numbers that appear nowhere among the
givens. With no constraints on those symbols, there are at least two solutions
in which their roles are interchanged.
Can the n ^2 - 1 bound be achieved in practice? For n = 1 the answer is yes. On
the order-2 grid there are uniquely solvable puzzles with four givens but not,
I think, with three. (Finding the arrangements with just four givens is itself
a pleasant puzzle.) For order 3, the minimum number of givens is unknown.
Gordon Royle of the University of Western Australia has collected more than
24,000 examples of uniquely solvable grids with 17 givens, and he has found
none with fewer than 17, but a proof is lacking.
Published puzzles generally have between 25 and 30 givens. Within this range,
the correlation between number of givens and difficulty rating is weak. In one
book, I found that the "gentle" puzzles averaged 28.3 givens and the
"diabolical" ones 28.0.
Logic Rules
Many puzzle constructors distinguish between puzzles that can be solved "by
logic alone" and those that require "trial and error." If you solve by logic,
you never write a number into a cell until you can prove that only that number
can appear in that position. Trial and error allows for guessing: You fill in a
number tentatively, explore the consequences, and if necessary backtrack,
removing your choice and trying another. A logic solver can work with a pen; a
backtracker needs a pencil and eraser.
For the logic-only strategy to work, a puzzle must have a quality of
progressivism: At every stage in the solution, there must be at least one cell
whose value can be determined unambiguously. Filling in that value must then
uncover at least one other fully determined value, and so on. The backtracking
protocol dispenses with progressivism: When you reach a state where no choice
is forced upon you--where every vacant cell has at least two candidates--you
choose a path arbitrarily.
The distinction between logic and backtracking seems like a promising criterion
for rating the difficulty of puzzles, but on a closer look, it's not clear the
distinction even exists. Is there a subset of Sudoku puzzles that can be solved
by backtracking but not by "logic"? Here's another way of asking the question:
Are there puzzles that have a unique solution, and yet at some intermediate
stage reach an impasse, where no cell has a value that can be deduced
unambiguously? Not, I think, unless we impose artificial restrictions on the
rules allowed in making logical deductions.
Backtracking itself can be viewed as a logical operation; it supplies a proof
by contradiction. If you make a speculative entry in one cell and, as a
consequence, eventually find that some other cell has no legal entry, then you
have discovered a logical relation between the cells. The chain of implication
could be very intricate, but the logical relation is no different in kind from
the simple rule that says two cells in the same row can't have the same value.
(David Eppstein of the University of California at Irvine has formulated some
extremely subtle Sudoku rules, which capture the kind of information gleaned
from a backtracking analysis, yet work in a forward-looking, nonspeculative
mode.)
A Satisfied Mind
From a computational point of view, Sudoku is a constraint-satisfaction
problem. The constraints are the rules forbidding two cells in the same
neighborhood to have the same value; a solution is an assignment of values to
cells that satisfies all the constraints simultaneously. In one obvious
encoding, there are 810 constraints in an order-3 grid.
It's interesting to observe how differently one approaches such a problem when
solving it by computer rather than by hand. A human solver may well decide that
logic is all you need, but backtracking is the more appealing option for a
program. For one thing, backtracking will always find the answer, if there is
one. It can even do the right thing if there are multiple solutions or no
solution. To make similar claims for a logic-only program, you would have to
prove you had included every rule of inference that might possibly be needed.
Backtracking is also the simpler approach, in the sense that it relies on one
big rule rather than many little ones. At each stage you choose a value for
some cell and check to see if this new entry is consistent with the rest of the
grid. If you detect a conflict, you have to undo the choice and try another. If
you have exhausted all the candidates for a given cell, then you must have
taken a wrong turn earlier, and you need to backtrack further. This is not a
clever algorithm; it amounts to a depth-first search of the tree of all
possible solutions--a tree that could have 9^81 leaves. There is no question
that we are deep in the exponential territory of NP problems here. And yet, in
practice, solving Sudoku by backtracking is embarrassingly easy.
There are many strategies for speeding up the search, mostly focused on making
a shrewd choice of which branch of the tree to try next. But such optimizations
are hardly needed. On an order-3 Sudoku grid, even a rudimentary backtracking
search converges on the solution in a few dozen steps. Evidently, competing
against a computer in Sudoku is never going to be much fun.
Does that ruin the puzzle for the rest of us? In moments of frustration, when
I'm struggling with a recalcitrant diabolical, the thought that the machine
across the room could instantly sweep away all my cobwebs of logic is indeed
dispiriting. I begin to wonder whether this cross-correlation of columns, rows
and blocks is a fit task for the human mind. But when I do make a breakthrough,
I take more pleasure in my success than the computer would. © Brian Hayes
Bibliography
* Bammel, Stanley E., and Jerome Rothstein. 1975. The number of 9 x 9 Latin
squares. Discrete Mathematics 11:93-95.
* Eppstein, David. Preprint. Nonrepetitive paths and cycles in graphs with
application to sudoku. http://arxiv.org/abs/cs.DS/0507053
* Felgenhauer, Bertram, and Frazer Jarvis. Preprint. Enumerating possible
Sudoku grids. http://www.shef.ac.uk/~pm1afj/sudoku/sudoku.pdf
* Pegg, Ed. 2005. Math Games: Sudoku variations. MAA Online.
http://www.maa.org/editorial/mathgames/mathgames_09_05_05.html
* Royle, Gordon F. 2005. Minimum sudoku.
http://www.csse.uwa.edu.au/~gordon/sudokumin.php
* Simonis, Helmut. 2005. Sudoku as a constraint problem. In Proceedings of
the Fourth International Workshop on Modelling and Reformulating Constraint
Satisfaction Problems, pp. 13-27.
* Sudoku Programmers Forum. 2005. Discussion thread, May 5, 2005, through
June 11, 2005. http://www.setbb.com/phpbb/viewtopic.php?t=27&mforum=sudoku
* Wikipedia. 2005. Sudoku. http://en.wikipedia.org/wiki/Sudoku
* Yato, Takayuki, and Takahiro Seta. 2002. Complexity and completeness of
finding another solution and its application to puzzles.
http://www-imai.is.s.u-tokyo.ac.jp/~yato/data2/SIGAL87-2.pdf
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