The mathematics of Sudoku, a puzzle that boasts "No math required!"
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?
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.
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 1027.
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 1021.
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 1012.
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.
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 981
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