Sudoku Puzzles¶
This module provides algorithms to solve Sudoku puzzles, plus tools
for inputting, converting and displaying various ways of writing a
puzzle or its solution(s). Primarily this is accomplished with the
sage.games.sudoku.Sudoku
class, though the legacy top-level
sage.games.sudoku.sudoku()
function is also available.
AUTHORS:
Tom Boothby (2008/05/02): Exact Cover, Dancing Links algorithm
Robert Beezer (2009/05/29): Backtracking algorithm, Sudoku class
- class sage.games.sudoku.Sudoku(puzzle, verify_input=True)¶
Bases:
sage.structure.sage_object.SageObject
An object representing a Sudoku puzzle. Primarily the purpose is to solve the puzzle, but conversions between formats are also provided.
INPUT:
- puzzle – the first argument can take one of three forms
list - a Python list with elements of the puzzle in row-major order, where a blank entry is a zero
matrix - a square Sage matrix over \(\ZZ\)
string - a string where each character is an entry of the puzzle. For two-digit entries, a = 10, b = 11, etc.
verify_input – default =
True
, useFalse
if you know the input is valid
EXAMPLES:
sage: a = Sudoku('5...8..49...5...3..673....115..........2.8..........187....415..3...2...49..5...3') sage: print(a) +-----+-----+-----+ |5 | 8 | 4 9| | |5 | 3 | | 6 7|3 | 1| +-----+-----+-----+ |1 5 | | | | |2 8| | | | | 1 8| +-----+-----+-----+ |7 | 4|1 5 | | 3 | 2| | |4 9 | 5 | 3| +-----+-----+-----+ sage: print(next(a.solve())) +-----+-----+-----+ |5 1 3|6 8 7|2 4 9| |8 4 9|5 2 1|6 3 7| |2 6 7|3 4 9|5 8 1| +-----+-----+-----+ |1 5 8|4 6 3|9 7 2| |9 7 4|2 1 8|3 6 5| |3 2 6|7 9 5|4 1 8| +-----+-----+-----+ |7 8 2|9 3 4|1 5 6| |6 3 5|1 7 2|8 9 4| |4 9 1|8 5 6|7 2 3| +-----+-----+-----+
- backtrack()¶
Return a generator which iterates through all solutions of a Sudoku puzzle.
This function is intended to be called from the
solve()
method when thealgorithm='backtrack'
option is specified. However it may be called directly as a method of an instance of a Sudoku puzzle.At this point, this method calls
backtrack_all()
which constructs all of the solutions as a list. Then the present method just returns the items of the list one at a time. Once Cython supports closures and a yield statement is supported, then the contents ofbacktrack_all()
may be subsumed into this method and thesage.games.sudoku_backtrack
module can be removed.This routine can have wildly variable performance, with a factor of 4000 observed between the fastest and slowest \(9\times 9\) examples tested. Examples designed to perform poorly for naive backtracking, will do poorly (such as
d
below). However, examples meant to be difficult for humans often do very well, with a factor of 5 improvement over the \(DLX\) algorithm.Without dynamically allocating arrays in the Cython version, we have limited this function to \(16\times 16\) puzzles. Algorithmic details are in the
sage.games.sudoku_backtrack
module.EXAMPLES:
This example was reported to be very difficult for human solvers. This algorithm works very fast on it, at about half the time of the DLX solver. [sudoku:escargot]
sage: g = Sudoku('1....7.9..3..2...8..96..5....53..9...1..8...26....4...3......1..4......7..7...3..') sage: print(g) +-----+-----+-----+ |1 | 7| 9 | | 3 | 2 | 8| | 9|6 |5 | +-----+-----+-----+ | 5|3 |9 | | 1 | 8 | 2| |6 | 4| | +-----+-----+-----+ |3 | | 1 | | 4 | | 7| | 7| |3 | +-----+-----+-----+ sage: print(next(g.solve(algorithm='backtrack'))) +-----+-----+-----+ |1 6 2|8 5 7|4 9 3| |5 3 4|1 2 9|6 7 8| |7 8 9|6 4 3|5 2 1| +-----+-----+-----+ |4 7 5|3 1 2|9 8 6| |9 1 3|5 8 6|7 4 2| |6 2 8|7 9 4|1 3 5| +-----+-----+-----+ |3 5 6|4 7 8|2 1 9| |2 4 1|9 3 5|8 6 7| |8 9 7|2 6 1|3 5 4| +-----+-----+-----+
This example has no entries in the top row and a half, and the top row of the solution is
987654321
and therefore a backtracking approach is slow, taking about 750 times as long as the DLX solver. [sudoku:wikipedia]sage: c = Sudoku('..............3.85..1.2.......5.7.....4...1...9.......5......73..2.1........4...9') sage: print(c) +-----+-----+-----+ | | | | | | 3| 8 5| | 1| 2 | | +-----+-----+-----+ | |5 7| | | 4| |1 | | 9 | | | +-----+-----+-----+ |5 | | 7 3| | 2| 1 | | | | 4 | 9| +-----+-----+-----+ sage: print(next(c.solve(algorithm='backtrack'))) +-----+-----+-----+ |9 8 7|6 5 4|3 2 1| |2 4 6|1 7 3|9 8 5| |3 5 1|9 2 8|7 4 6| +-----+-----+-----+ |1 2 8|5 3 7|6 9 4| |6 3 4|8 9 2|1 5 7| |7 9 5|4 6 1|8 3 2| +-----+-----+-----+ |5 1 9|2 8 6|4 7 3| |4 7 2|3 1 9|5 6 8| |8 6 3|7 4 5|2 1 9| +-----+-----+-----+
- dlx(count_only=False)¶
Return a generator that iterates through all solutions of a Sudoku puzzle.
INPUT:
count_only – boolean, default = False. If set to
True
the generator returned as output will simply generateNone
for each solution, so the calling routine can count these.
OUTPUT:
A generator that iterates over all the solutions.
This function is intended to be called from the
solve()
method with thealgorithm='dlx'
option. However it may be called directly as a method of an instance of a Sudoku puzzle if speed is important and you do not need automatic conversions on the output (or even just want to count solutions without looking at them). In this case, inputting a puzzle as a list, withverify_input=False
is the fastest way to create a puzzle.Or if only one solution is needed it can be obtained with one call to
next()
, while the existence of a solution can be tested by catching theStopIteration
exception with atry
. Calling this particular method returns solutions as lists, in row-major order. It is up to you to work with this list for your own purposes. If you want fancier formatting tools, use thesolve()
method, which returns a generator that createssage.games.sudoku.Sudoku
objects.EXAMPLES:
A \(9\times 9\) known to have one solution. We get the one solution and then check to see if there are more or not.
sage: e = Sudoku('4.....8.5.3..........7......2.....6.....8.4......1.......6.3.7.5..2.....1.4......') sage: print(next(e.dlx())) [4, 1, 7, 3, 6, 9, 8, 2, 5, 6, 3, 2, 1, 5, 8, 9, 4, 7, 9, 5, 8, 7, 2, 4, 3, 1, 6, 8, 2, 5, 4, 3, 7, 1, 6, 9, 7, 9, 1, 5, 8, 6, 4, 3, 2, 3, 4, 6, 9, 1, 2, 7, 5, 8, 2, 8, 9, 6, 4, 3, 5, 7, 1, 5, 7, 3, 2, 9, 1, 6, 8, 4, 1, 6, 4, 8, 7, 5, 2, 9, 3] sage: len(list(e.dlx())) 1
A \(9\times 9\) puzzle with multiple solutions. Once with actual solutions, once just to count.
sage: h = Sudoku('8..6..9.5.............2.31...7318.6.24.....73...........279.1..5...8..36..3......') sage: len(list(h.dlx())) 5 sage: len(list(h.dlx(count_only=True))) 5
A larger puzzle, with multiple solutions, but we just get one.
sage: j = Sudoku('....a..69.3....1d.2...8....e.4....b....5..c.......7.......g...f....1.e..2.b.8..3.......4.d.....6.........f..7.g..9.a..c...5.....8..f.....1..e.79.c....b.....2...6.....g.7......84....3.d..a.5....5...7..e...ca.....3.1.......b......f....4...d..e..g.92.6..8....') sage: print(next(j.dlx())) [5, 15, 16, 14, 10, 13, 7, 6, 9, 2, 3, 4, 11, 8, 12, 1, 13, 3, 2, 12, 11, 16, 8, 15, 1, 6, 7, 14, 10, 4, 9, 5, 1, 10, 11, 6, 9, 4, 3, 5, 15, 8, 12, 13, 16, 7, 14, 2, 9, 8, 7, 4, 12, 2, 1, 14, 10, 5, 16, 11, 6, 3, 15, 13, 12, 16, 4, 1, 13, 14, 9, 10, 2, 7, 11, 6, 8, 15, 5, 3, 3, 14, 5, 7, 16, 11, 15, 4, 12, 13, 8, 9, 1, 2, 10, 6, 2, 6, 13, 11, 1, 8, 5, 3, 4, 15, 14, 10, 7, 9, 16, 12, 15, 9, 8, 10, 2, 6, 12, 7, 3, 16, 5, 1, 4, 14, 13, 11, 8, 11, 3, 15, 5, 10, 4, 2, 13, 1, 6, 12, 14, 16, 7, 9, 16, 12, 14, 13, 7, 15, 11, 1, 8, 9, 4, 5, 2, 6, 3, 10, 6, 2, 10, 5, 14, 12, 16, 9, 7, 11, 15, 3, 13, 1, 4, 8, 4, 7, 1, 9, 8, 3, 6, 13, 16, 14, 10, 2, 5, 12, 11, 15, 11, 5, 9, 8, 6, 7, 13, 16, 14, 3, 1, 15, 12, 10, 2, 4, 7, 13, 15, 3, 4, 1, 10, 8, 5, 12, 2, 16, 9, 11, 6, 14, 10, 1, 6, 2, 15, 5, 14, 12, 11, 4, 9, 7, 3, 13, 8, 16, 14, 4, 12, 16, 3, 9, 2, 11, 6, 10, 13, 8, 15, 5, 1, 7]
The puzzle
h
from above, but purposely made unsolvable with addition in second entry.sage: hbad = Sudoku('82.6..9.5.............2.31...7318.6.24.....73...........279.1..5...8..36..3......') sage: len(list(hbad.dlx())) 0 sage: next(hbad.dlx()) Traceback (most recent call last): ... StopIteration
A stupidly small puzzle to test the lower limits of arbitrary sized input.
sage: s = Sudoku('.') sage: print(next(s.solve(algorithm='dlx'))) +-+ |1| +-+
ALGORITHM:
The
DLXCPP
solver finds solutions to the exact-cover problem with a “Dancing Links” backtracking algorithm. Given a \(0-1\) matrix, the solver finds a subset of the rows that sums to the all \(1\)’s vector. The columns correspond to conditions, or constraints, that must be met by a solution, while the rows correspond to some collection of choices, or decisions. A \(1\) in a row and column indicates that the choice corresponding to the row meets the condition corresponding to the column.So here, we code the notion of a Sudoku puzzle, and the hints already present, into such a \(0-1\) matrix. Then the
sage.combinat.matrices.dlxcpp.DLXCPP
solver makes the choices for the blank entries.
- solve(algorithm='dlx')¶
Return a generator object for the solutions of a Sudoku puzzle.
INPUT:
algorithm – default =
'dlx'
, specify choice of solution algorithm. The two possible algorithms are'dlx'
and'backtrack'
.
OUTPUT:
A generator that provides all solutions, as objects of the
Sudoku
class.Calling
next()
on the returned generator just once will find a solution, presuming it exists, otherwise it will return aStopIteration
exception. The generator may be used for iteration or wrapping the generator withlist()
will return all of the solutions as a list. Solutions are returned as new objects of theSudoku
class, so may be printed or converted using other methods in this class.Generally, the DLX algorithm is very fast and very consistent. The backtrack algorithm is very variable in its performance, on some occasions markedly faster than DLX but usually slower by a similar factor, with the potential to be orders of magnitude slower. See the docstrings for the
dlx()
andbacktrack_all()
methods for further discussions and examples of performance. Note that the backtrack algorithm is limited to puzzles of size \(16\times 16\) or smaller.EXAMPLES:
This puzzle has 5 solutions, but the first one returned by each algorithm are identical.
sage: h = Sudoku('8..6..9.5.............2.31...7318.6.24.....73...........279.1..5...8..36..3......') sage: h +-----+-----+-----+ |8 |6 |9 5| | | | | | | 2 |3 1 | +-----+-----+-----+ | 7|3 1 8| 6 | |2 4 | | 7 3| | | | | +-----+-----+-----+ | 2|7 9 |1 | |5 | 8 | 3 6| | 3| | | +-----+-----+-----+ sage: next(h.solve(algorithm='backtrack')) +-----+-----+-----+ |8 1 4|6 3 7|9 2 5| |3 2 5|1 4 9|6 8 7| |7 9 6|8 2 5|3 1 4| +-----+-----+-----+ |9 5 7|3 1 8|4 6 2| |2 4 1|9 5 6|8 7 3| |6 3 8|2 7 4|5 9 1| +-----+-----+-----+ |4 6 2|7 9 3|1 5 8| |5 7 9|4 8 1|2 3 6| |1 8 3|5 6 2|7 4 9| +-----+-----+-----+ sage: next(h.solve(algorithm='dlx')) +-----+-----+-----+ |8 1 4|6 3 7|9 2 5| |3 2 5|1 4 9|6 8 7| |7 9 6|8 2 5|3 1 4| +-----+-----+-----+ |9 5 7|3 1 8|4 6 2| |2 4 1|9 5 6|8 7 3| |6 3 8|2 7 4|5 9 1| +-----+-----+-----+ |4 6 2|7 9 3|1 5 8| |5 7 9|4 8 1|2 3 6| |1 8 3|5 6 2|7 4 9| +-----+-----+-----+
Gordon Royle maintains a list of 48072 Sudoku puzzles that each has a unique solution and exactly 17 “hints” (initially filled boxes). At this writing (May 2009) there is no known 16-hint puzzle with exactly one solution. [sudoku:royle] This puzzle is number 3000 in his database. We solve it twice.
sage: b = Sudoku('8..6..9.5.............2.31...7318.6.24.....73...........279.1..5...8..36..3......') sage: next(b.solve(algorithm='dlx')) == next(b.solve(algorithm='backtrack')) True
These are the first 10 puzzles in a list of “Top 95” puzzles, [sudoku:top95] which we use to show that the two available algorithms obtain the same solution for each.
sage: top =['4.....8.5.3..........7......2.....6.....8.4......1.......6.3.7.5..2.....1.4......', ....: '52...6.........7.13...........4..8..6......5...........418.........3..2...87.....', ....: '6.....8.3.4.7.................5.4.7.3..2.....1.6.......2.....5.....8.6......1....', ....: '48.3............71.2.......7.5....6....2..8.............1.76...3.....4......5....', ....: '....14....3....2...7..........9...3.6.1.............8.2.....1.4....5.6.....7.8...', ....: '......52..8.4......3...9...5.1...6..2..7........3.....6...1..........7.4.......3.', ....: '6.2.5.........3.4..........43...8....1....2........7..5..27...........81...6.....', ....: '.524.........7.1..............8.2...3.....6...9.5.....1.6.3...........897........', ....: '6.2.5.........4.3..........43...8....1....2........7..5..27...........81...6.....', ....: '.923.........8.1...........1.7.4...........658.........6.5.2...4.....7.....9.....'] sage: p = [Sudoku(top[i]) for i in range(10)] sage: verify = [next(p[i].solve(algorithm='dlx')) == next(p[i].solve(algorithm='backtrack')) for i in range(10)] sage: verify == [True]*10 True
- to_ascii()¶
Construct an ASCII-art version of a Sudoku puzzle. This is a modified version of the ASCII version of a subdivided matrix.
EXAMPLES:
sage: s = Sudoku('.4..32....14..3.') sage: print(s.to_ascii()) +---+---+ | 4| | |3 2| | +---+---+ | |1 4| | |3 | +---+---+ sage: s.to_ascii() '+---+---+\n| 4| |\n|3 2| |\n+---+---+\n| |1 4|\n| |3 |\n+---+---+'
- to_latex()¶
Create a string of \(\LaTeX\) code representing a Sudoku puzzle or solution.
EXAMPLES:
sage: s = Sudoku('.4..32....14..3.') sage: print(s.to_latex()) \begin{array}{|*{2}{*{2}{r}|}}\hline &4& & \\ 3&2& & \\\hline & &1&4\\ & &3& \\\hline \end{array}
- to_list()¶
Construct a list representing a Sudoku puzzle, in row-major order.
EXAMPLES:
sage: s = Sudoku('1.......2.9.4...5...6...7...5.9.3.......7.......85..4.7.....6...3...9.8...2.....1') sage: s.to_list() [1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 9, 0, 4, 0, 0, 0, 5, 0, 0, 0, 6, 0, 0, 0, 7, 0, 0, 0, 5, 0, 9, 0, 3, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 8, 5, 0, 0, 4, 0, 7, 0, 0, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 0, 9, 0, 8, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1]
- to_matrix()¶
Construct a Sage matrix over \(\ZZ\) representing a Sudoku puzzle.
EXAMPLES:
sage: s = Sudoku('.4..32....14..3.') sage: s.to_matrix() [0 4 0 0] [3 2 0 0] [0 0 1 4] [0 0 3 0]
- to_string()¶
Construct a string representing a Sudoku puzzle.
Blank entries are represented as periods, single digits are not converted and two digit entries are converted to lower-case letters where
10 = a
,11 = b
, etc. This scheme limits puzzles to at most 36 symbols.EXAMPLES:
sage: b = matrix(ZZ, 9, 9, [ [0,0,0,0,1,0,9,0,0], [8,0,0,4,0,0,0,0,0], [2,0,0,0,0,0,0,0,0], [0,7,0,0,3,0,0,0,0], [0,0,0,0,0,0,2,0,4], [0,0,0,0,0,0,0,5,8], [0,6,0,0,0,0,1,3,0], [7,0,0,2,0,0,0,0,0], [0,0,0,8,0,0,0,0,0] ]) sage: Sudoku(b).to_string() '....1.9..8..4.....2.........7..3..........2.4.......58.6....13.7..2........8.....'
- sage.games.sudoku.sudoku(m)¶
Solves Sudoku puzzles described by matrices.
INPUT:
m
- a square Sage matrix over \(\ZZ\), where zeros are blank entries
OUTPUT:
A Sage matrix over \(\ZZ\) containing the first solution found, otherwise
None
.This function matches the behavior of the prior Sudoku solver and is included only to replicate that behavior. It could be safely deprecated, since all of its functionality is included in the
Sudoku
class.EXAMPLES:
An example that was used in previous doctests.
sage: A = matrix(ZZ,9,[5,0,0, 0,8,0, 0,4,9, 0,0,0, 5,0,0, 0,3,0, 0,6,7, 3,0,0, 0,0,1, 1,5,0, 0,0,0, 0,0,0, 0,0,0, 2,0,8, 0,0,0, 0,0,0, 0,0,0, 0,1,8, 7,0,0, 0,0,4, 1,5,0, 0,3,0, 0,0,2, 0,0,0, 4,9,0, 0,5,0, 0,0,3]) sage: A [5 0 0 0 8 0 0 4 9] [0 0 0 5 0 0 0 3 0] [0 6 7 3 0 0 0 0 1] [1 5 0 0 0 0 0 0 0] [0 0 0 2 0 8 0 0 0] [0 0 0 0 0 0 0 1 8] [7 0 0 0 0 4 1 5 0] [0 3 0 0 0 2 0 0 0] [4 9 0 0 5 0 0 0 3] sage: sudoku(A) [5 1 3 6 8 7 2 4 9] [8 4 9 5 2 1 6 3 7] [2 6 7 3 4 9 5 8 1] [1 5 8 4 6 3 9 7 2] [9 7 4 2 1 8 3 6 5] [3 2 6 7 9 5 4 1 8] [7 8 2 9 3 4 1 5 6] [6 3 5 1 7 2 8 9 4] [4 9 1 8 5 6 7 2 3]
Using inputs that are possible with the
Sudoku
class, other than a matrix, will cause an error.sage: sudoku('.4..32....14..3.') Traceback (most recent call last): ... ValueError: sudoku function expects puzzle to be a matrix, perhaps use the Sudoku class