NAME
sudoku - sudoku solver / generator
SYNOPSIS
sudoku [ options ] [ puzzle ... | [ < ] puzzle.file ... ]
DESCRIPTION
sudoku solves 9x9 sudoku puzzles defined by the literal puzzle operands
(containing no lower case letters) and puzzle file operands, and prints the
puzzle properties and solutions, one line per puzzle, on the standard
output. If there are no operands then puzzle descriptions are read from
the standard input. The file "-" names the standard input. A solvable
puzzle has exactly one solution.
OPTIONS
-a List all solutions. Implies -ou.
-A Force constraints to assume that constraints earlier in the default
order have already been applied.
-b Enable breadth first search which is typically slower than the default
depth first first. Breadth first search applies constraint propagation
to the lookahead candidates (guesses).
-B Batch moves within one constraint group (or constraint if -G is on)
by first identifying all moves and then committing them in one
operation. By default moves are made as they are identified.
Batching negates ordering affects between permutations of equivalent
puzzles but usually results in higher constraint counts. If -v is
also on then the move trace is prefixed by the constraint
propagation iteration count.
-cNC File puzzle input is in the Nth C separated column counting from 1.
If C is ' ' or omitted then 1 or more space characters separate the
fields. C may be followed by C separated field names. The fields may
be accessed by %(name)f, %(N)f, %(*)f (all fields, no separator), %#C(*)f
(all fields, C separator), or $N, $name in expressions.
-C[X]If X is p or omitted then list the minimum Hamming/edit distance of each
each input puzzle to the next input puzzle, the input puzzle in %v form,
and the "best" map of the input puzzle to the next, also in %v form.
Each odd-even pair of input puzzles is compared. If X contains S then
list the maximum similarity instead of Hamming distance. If X contains D
then list the default Hamming distance. If X contains f then compare
the first puzzle to each subsequent puzzle. If X contains l then compare
compare the last (previous) puzzle to the current input puzzle.
The first puzzle can be listed by specifying width 1 in the grid -f forma
t,
and the second puzzle by specifying width 2: e.g. %1v %2#.c
-d Enable depth first search which is typically faster than breadth
first. This is the default.
-DN Set debug output level to N, or increment if N omitted.
-eE Only list puzzles with constraint expression E that evaluates non-0.
-E List puzzles just after applying the last easy (non-guess) constraint.
-fF Format output according to F (default '%v # %5r %q %(CSM)#c#/q') after
each puzzle is processed. If format is "-" then per-puzzle output
is disabled. printf(3) style width, precision and fill apply.
[A][B]... before the format char treats value as an index: 0 for A, 1,
for B, etc. Information for invalid puzzles represents the solver
state when the puzzle was determined to be invalid. The formats are:
%XC Solution in canonical order. X=#a: number of non-trivial
canonical solution automorphisms.
%NF Flush the current output and: N=1: write to the standard output;
N=2: write to the standard error.
%G The number of clues (givens).
%XI Puzzle file magic string. X=#c: schema names separated by c.
X=#I: schema names with input file separator.
%XM The number of backdoors (magic cell sets).
#a: The number of cells that are part of a backdoor.
#b: The backdoor size.
#g: Estimated number of placement guesses before backdoor hit.
#u: The number of cells that are not part of a backdoor.
%XYZP Property list of Pi elements:
#c: Pi is the number of cells with i candidates.
#d: Pi is the number candidates with degree i, P1: total.
#g: Pi is the number of clue values with i-1 occurrences.
#o: Pi is the number of occurrences for clue value i.
#q: Pi is the constraint applied at constraint iteration i.
#r: Pi is the number of rookery templates for value i.
#s: Pi is the number placements at constraint iteration i.
#B: Pi is the number of clues for box i.
#C: Pi is the number of clues for col i.
#R: Pi is the number of clues for row i.
#S: #s with constraint id.
#v: Pi is the number of cells with candidate i.
Y=#S for separator S, otherwise no separator.
Z=#G for group separator G, otherwise Y separator.
-#: sorted high to low, +#: sorted low to high.
%XQ Equivalent to '%5r %-11q %#c#/q'. If #X is specified:
#m: The magic/backdoor constraints.
#q: The quick constraints.
#s: The solver (-q) constraints.
%S Default stats: equivalent to '%#nr/%02(C)x/%02(K)x/%04(I)x/%(M)x'.
%FT The current date formatted by strftime(3) using format F,
default ('%Y-%m-%d+%H:%M:%S').
%XX Miscellaneous:
#c: The number of solved cells.
#e: Elapsed time test. Format width interpreted as nsec.
#E: Simple Sudoku pencilmark exclusion list, one per line.
#f: The number of forced cells (naked singles).
#F: The full dihedral symmetry pattern id n-n-n-n-n.
#r: The current pseudo random number generator seed.
#R: The initial pseudo random number generator seed.
#t: The tuples of degree i, one per line.
#u: The solution unavoidables of size 4.
#v: The number of clue values (1-9).
%AXYc Puzzle in canonical order with solution cells listed as 0.
Leading 12345678945 omitted from row-order min-lex band values.
#A: Solution cells listed as [A-IJ-R].
#b: Band of 416 min-lex index 6-tuple, each 3-tuple sorted.
#B: #b not sorted.
#c: Grid symmetrized to minimize distance, dihedral order, and
pattern lexicographic order, ignoring the center box.
#C: #c dihedral symmetry type and distance from symmetry.
X may be:
#s: Symmetry op.
#S: Symmetry description.
#t: Symmetry type.
#d: Grid symmetrized to minimize distance, dihedral order, and
pattern lexicographic order.
#D: #d dihedral symmetry type and distance from symmetry.
X is the same as for #C above.
#e: sudz compressed minlex band/grid encoding.
#g: Gang of 44 min-lex index 6-tuple, each 3-tuple sorted. The
gang index is the largest 416 index the gang may produce.
#G: #g not sorted.
N#i: Band of 416 index N row order min-lex value.
#m: Puzzle in subgrid row order minlex canonical order with
solution cells listed as 0. The default canonical form is
based on the solution grid; subgrid canonical does not
require the grid to have a solution.
#o: The original grid dihedral symmetry id and distance from
symmetry, minimizing distance and dihedral order.
#O: The original grid dihedral symmetry type and distance from
symmetry, minimizing distance and dihedral order.
#p: Permutations mapping original to canonical.
i:identity, v:value, r:row, c:column,
a:#automorphisms.
#P: Row order pattern minlex canonical.
#s: Grid symmetrized to largest dihedral order, "assymmetric"
if no symmetry (not a fast implementation). Y=#p: if symmetric
then list the cyclic permutation that symmetrizes the input.
#t: Exemplar template with / cells in subgrid row order minlex
canonical order. Y=#a: the number of canonical form
automorphisms. Y=#p: list the cyclic permutations that
transform the exemplar to canonical order. A=(abc): exemplar
cell values weighted a=1 b=2 c=3. A=((ab)c(de)): exemplar cell
values weighted a=b=1 c=2 d=e=3. Weights are in the range
1..9.
#T: Exemplar template restricted to / cells in subgrid row order
minlex canonical order. A: same as for #t, except listed cells
are restricted to the cells specified by A.
#x: Clues treated as exemplar X in subgrid row order minlex canoni
cal
order. Y=#a: the number of canonical form automorphisms. Y=#p
: list
the cyclic permutations that transform the puzzle to canonical
order.
#X: Like #x, but list original puzzle in row order minlex canonica
l order.
#z: Like #t, but symmetrize instead of canonicalize.
#Z: Like #T, but symmetrize instead of canonicalize.
#c: Solution cells listed as 'c'.
%d The default format '%v # %5r %q %(CSM)#c#/q'.
%Xf Input file base name. X=#p: input file path name, X=(N): -c field N,
X=(@)#C: all fields separated by C, X=(@): all fields, no separator.
%XYg 9x9 text puzzle grid with no frame (default or X=#n), X=#f:
framed player's, X=#g: gordon, X=#m: metcalf, X=#p: programmer's,
forum, X=#s: simple, X=#v solver. #isupper: solution. Y=#t: template
if current input is template or exemplar, Y=#i: inverse exemplar.
Y=(:label): label appended to last grid line.
%XYh 9x9 text puzzle grid with hints (pencilmarks) and no borders,
X=#b: {hint}, X=#f: framed player's forum, X=#p: programmer's
forum, X=#s: simple, X=#v solver. Y=#0: only list strong edges,
Y=#N: only list exact N-tuples. %N..: only list candidate(s) N.
X=#e: experimental encoding.
%Xi Current identification, or %f if none.
#C: The -C id: similarity for -CS, distance otherwise.
%XYm The position list of all backdoors (magic cells) (X=#a default),
X=#A lists #a with values, X=#u lists all non-backdoor solution
cells, X=#U lists #u with values. Y=#c for 'c ' on split lines.
%Xn Various counts and ordinals:
#a: The ordinal within all listed puzzles (-f- still counts).
#A: The number of automorphisms of the input sudz grid.
#B: The band index of the input sudz grid.
#c: The -g puzzle index.
#C: The -go puzzle partition index or -C distance/similarity.
#d: The -g number of unique puzzles.
#e: The -g puzzle duplication count.
#i: The ordinal within the all input files.
#I: The window index of the input sudz grid.
#l: Input file line number of the first puzzle row.
#m: The ordinal within the current -a (multiple solution count).
#n: The number of backtrack search nodes.
#o: The ordinal within the current %i.
#p: The ordinal within the current input file.
#r: The ordinal within the current -m or -go.
#R: The current -m or -go base puzzle ordinal.
#s: The number of searched (for solution) puzzles.
#u: The number of uniqued puzzles (splay tree lookups/insertions).
#v: (default) The valid puzzle count.
#W: The number of grids in the input sudz window.
%Xq The constraints used to solve the puzzle. X may be:
#c: Constraint counts (see EXPRESSIONS below).
#d: The diamond move (hardest up to first elimination).
#D: The diamond move index (constraint ordinal).
#h: The hardest move.
#H: The hardest move index (constraint ordinal).
#i: The -q index:method map.
#I: The highest -q method index.
#m: m: minimal, M: symmetric minimal, -: otherwise.
#M: #m description.
#o: The constraints and orders used to solve the puzzle.
#p: The pearl move (hardest up to first placement).
#P: The pearl move index (constraint ordinal).
#s: Symmetry op (see -s below).
#S: Symmetry description.
#t: Symmetry type (see -s below).
Trailing 0 values are dropped and only applied constraints are
listed. The list order is CSFNBTHWXYKOZPQGMV.
%Xr The puzzle rating from 1 (very easy) to 99999 (very difficult).
Use -B or -S to normalize algorithm order bias. X=(E) uses the
rating expression E instead of the -R option value. The
90000-99999 range is exponential. X=#r for raw (no exponential
normalization), X=#s for symbolic rating, X=#. for 99999 scaled
9.9, X=#n for normalized puzzle rating from 1 (very easy) to 9
(very difficult). The usual C integer arithmetic operators, ?:,
x**y (x to the y power) and x@y (x log y) are supported.
%Xs Puzzle in original order with solution cells listed as [1-9].
X=#A: solution cells listed as [A-IJ-R].
%Xt Various times. X may be:
#e: Total elapsed time since program start.
#p: Elapsed time since the last %#pt.
#s: (default) Current puzzle solution time.
#t: Sum of all puzzle solution times.
%XYv Puzzle in original order with solution cells listed as '.'.
X and/or Y may be:
#i: inverse exemplar if current input is exemplar
#t: template if current input is template/exemplar
#N: (N non-alpha) solution cell listed as N
%Xx X=(E): the value of the expression (E).
%a Deprecated: use %#an.
%e Deprecated: use %#at.
%j Deprecated: use %#pn.
%k Deprecated: use %#in.
%l Deprecated: use %#ln.
%o Deprecated: use %#on.
%p Deprecated: use %#pt.
%t Deprecated: use %#st.
%u Deprecated: use %#tt.
%y Deprecated: use %#mn.
%z Deprecated: use %#rn.
%@ Align to <width> characters past the previous %@, e.g., %-10@.
%N/ Multi-column output with column width N. Format output after the /
will be in the next column(s).
%?V%?I If the puzzle is valid then output format V, otherwise format I.
Empty V or I, whichever is selected, means no output.
%, Space.
%: Newline.
%% Percent.
\c C style escape.
c Literal character.
-FCF Set output format default C to F. The default is empty if not specified.
C may be:
a Output after each puzzle is processed. This format is used after
all -a solutions are determined for each puzzle.
b Output before each puzzle is processed.
c -goc format (default '%#0v # %#Rn.%#cn.%#rn').
d The %d format (default '%v # %5r %q %(CSM)#c#/q').
e Value equality operator (default '').
f Output after all puzzles have been processed. Puzzle property
counters are accumulated from all puzzles.
g Value proposition (guess) operator (default '?').
h The initial -gt grid before minimization/filtering.
i Output before the first puzzle is processed.
m The %S stats format (default '%#nr/%02(C)x/%02(K)x/%04(I)x/%(M)x').
n Value inequality operator (default '^').
o The -go omitted (and possibly invalid) grid format.
p Cell position (coordinate) format (default '[%r%c]').
F is a printf-style format string with the following format
characters replacing the 'd' format:
r row offset counting from 1
R row offset counting from 0
c column offset counting from 1
C column offset counting from 0
x column offset counting from 1
X column offset counting from 0
y (10-row) offset counting from 1
Y (9-row) offset counting from 0
%Ar%ac lists the rows as ABCDEFGHI and columns as abcdefghi.
%Jr%c lists the rows as ABCDEFGHJ and columns as 123456789.
%kc%r lists the rows as 123456789 and columns as abcdefghk.
q The %Q format (default '%5r %-11q %#c#/q').
r Output as each -g solution grid is generated.
s Output after each puzzle is solved. Equivalent to -fF. Also used
for each -a solution (default '%v # %5r %q %(CSM)#c#/q').
t The default %T time format (default '%Y-%m-%d+%H:%M:%S').
v The < position operator value > format (default '%p%o%v'),
where %p is the -Fp position, %o is the -F[egn] operator, and
%v is the value.
w Default exemplar canonicalization weights (see %#tc, default '/').
x -p permutation output (default '%x').
y The exemplar characters in order (default '/X#@*$&').
Up to nine characters may be specified. The first seven have
specific meaning for the Z method.
C The default -C output format (default '%#Ci %#Cn\n%1v\n%2#.c').
-gTNA Generate puzzles of type T with optional parameters N and A.
T may be:
b Canonical solution grids for each band N class member.
If -m is also specified then only the min and max grids
for selected bands are listed. %i:band-index, %#pn:#grids,
%#in:#checked, %n:total#grids, %#on:total#checked,
%#ln:#3-band-prunes, %#mn:2-band-prunes. N,M lists
bands in the range N through M inclusive.
B Canonical band index table with leading 12345678945 omitted.
d -gC.V generates pseudo puzzles (possibly more than one solution)
with C clues (default 14) using V different values (default 3).
g Solution grids listed one per line (no other processing).
h Hint-only (pencilmark-only) puzzles with at least N (default
162) candidates. Even N => candidates evenly distributed.
H Generate using the hardest sudoku formula: at most one clue
per minirow/minicol and no repeated values in bands/chutes,
at most three clues per value, and minimal. N is the clue
fill step (default 31) -- factors of 3 are removed.
m For each puzzle list the subgrid row order minlex canonical
grid (the default canonical form is based on the solution grid,
subgrid canonical does not require the grid to have a solution).
o +/-NX...: a sequence of -N and/or +N operands with option suffixes.
-N: delete all combinations of N clues; +N: add all combinations
of N clues. Puzzles are checked for single solution after each
addition phase. X may be one or more of the following options.
Each occurrence toggles the previous value. Values are inherited
by subsequent operands in the sequence. X may be one or more of:
c use the -Fc format with no filtering, otherwise use the
default -f format, -q constraints, and -e filtering
e do not check off subpuzzles for duplicates
i add implicit (superfluous) candidates, otherwise skip them
n show off/on ops but do not execute
o list one generated puzzle per input puzzle
p the input puzzle is a pattern -- only change original clues
r -nX random {-N+M} operations per input puzzle
{...}xN repeats the bracketed group N times. {...}:N repeats the
bracketed group 0 or more times until the number of clues is <= N.
Some sequences may exhaust memory. {N}* (alternatively {N}!)
repeats {-N+N} until closure (no new puzzles generated). NOTE: a
recalcitrant bug sometimes omits a small number of puzzles in
combined off/on sequences that would otherwise be listed if the
sequence were done separately; if exact results are required (no
omissions) then use separate sequences.
p (default) Puzzle grids.
q Run the quick verification solver on all input puzzles. The output
one line containing: <#invalid> <#multi-solution> <#valid>.
s Solution grids only.
t Treat each input file as a template and generate -nN puzzles per
template, filling only the template clues. N (default 10000)
limits the number of shuffles for each solution grid projected on
the template. Template input may be a valid puzzle (but with
multiple solutions ok) or a grid with [XxOo] marking the template
clues. Some templates may not admit valid puzzles. There is no
known predictor (other than maybe 17 clues). If there are no O
cells then -v lists improvements towards 1 solution in 4 columns:
column 1 is the random solution grid count, column 2 is the solution
grid shuffle count, column 3 is the number of soultions for the grid
in column 4. At 2 solutions column 3 also contains the number of
unavoidable cells. If any template cell value is O and -m is
specified then only O template cells will be considered for
minimization. If O and clue values or candidate lists are
specified then the clue values and/or candidates are fixed in
generated solution grids, otherwise template clues are treated
as X. If there are no more than .A (default 54) candidate
then an enumerative (as opposed to random) search is used. The
enumerative search tests all candidate combinations. -gtpN.R.F.S
marks F% (default 60%) clues X cells at random, +/- S% (default
30%) and generates N random puzzles projected on the random
template. R (default 0 -- no limit) limits the number of X
assignment rounds. -gtpr ignores the enumerative checks and
forces random search. -gtpri continues the search on the modified
candidate grid. -gtm allows multiple solutions -- handy for
generating subgrids. -gtm.0 generates all subgrids. -gtpe
checks every pattern row-order minlex grid (not quite canonical
so there may be some duplication); -gtpei.N retains the first
N input grid clues; -gtpe.N.V sets clue N to V.
x Minimal superfluous (any clue unnecessary) puzzles.
P Input puzzle are treated as pattern exemplars with clue cells mapped
to X and are listed.
X Input puzzles are treated as exemplars and are listed.
-G Ignore constraint order {...} groups and treat each constraint
as if it were in its own group. Group minima/maxima and iterations
are not affected.
-h Add hints (pencilmarks) to the -P output.
-H Copy the first input file magic header, if any, to the output.
-i Force the constraint propagation iteration count to increment
with each constraint that produces a move rather than with each
constraint group. This allows the capture of in-group pencilmarks.
-jN For each -g type that operates on complete solution grids, operate
on the original and N (default 100) randomly shuffled automorphic
copies. N=0 repeatedly uses the first grid without shuffling.
-JN Solve the original puzzle and, for N (default 100) repetitions,
randomly shuffle (rows, columns, bands, stacks, values) to an
automorphic copy and re-solve. Handy for verifying order
independent metrics.
-kCR Constraint propagation iteration and/or cycle trace control. R is a
range of counts/lables on which the control is applied. C may be:
c Add XY cycles in range R to -P output with strong edges (red),
1-weak edges (green), 0-weak edges (purple), and eliminated cell
values (blue).
d Add -P 0-weak details (yellow).
i List intermediate puzzles with propagation iteration in range R.
If -v is also on then the move trace is prefixed by the constraint
propagation iteration count.
m Add -P backdoor (magic cell) values (purple).
p Add -P cycle pairs (cyan).
-KCL Constraint propagation iteration and/or cycle control. L is a list
of counts/labels on which the control is applied. C may be:
c Disable XYW cycles list L.
i Disable constraint iterations in list L.
-lX Label -P output with text X.
-LN Limit P constraint propositions to candidates with degree (number of
value/location choices) <= N. The default is 0 for no limit.
-mCN.M.X For each puzzle, derive, by dropping clues, up to N puzzles with
the minimum number of clues. If N is omitted then all minimal puzzles
are derived. If -s is also specified then symmetry will be preserved
and symmetric minimal will be considered minimal. If M is specified
only minimal puzzles with >= M clues are derived. If X is specified
only minimal puzzles with <= X clues are derived. If C is r then a
typically much quicker random search (inspired by dukosu) is used to
test N (default 10000) random minimal puzzle candidates. If C is rc
then the -Fc format is used to list the minimal puzzles; this can be
a factor of 2 faster than the default that generates puzzle stats.
-MC Restrict backdoor computations to the -q style constraints C (default
FNBTHW). Use this to avoid bad timing behavior with some constraints.
If C is - then backdoor computations are disabled.
-nN Limit -g to N puzzles (default 0 -- no limit).
-NN Set the next -f %a and %n to N.
-oF Write output to file F.
-OX Attempt to find an optimal (shortest) solution within each constraint
application. This will induce sub-optimal performance but may produce
more reasonable ratings when XY constraints are involved. Currently
limited to the XYP constraints. -O minimizes cycle size, -OA attempts to
minimize steps.
-pP If P is p or omitted then list the value (v), row (r) and column (c)
permutations that transform each input puzzle to the next input puzzle.
Each odd-even pair of input puzzles must be valid and have isomorphic
solutions. Permutations are displayed in cyclic notation, where each
(...) defines a label map, read from left to right. (137)(25) means:
1=>3, 3=>7, 7=>1, 2=>5, 5=>2; unspecified labels map to themselves.
'v', 'r' and 'c' appear to the left of the value, row, and column label
maps, respectively. 'x' denotes row-col transposition, identity v,r,c
components are omitted and 'i' denotes the identity, and 'aN' denotes
the number of automorphisms. If P is f then list permutations that
transform the first puzzle to each subsequent puzzle. If P is l then
list permutations that transform the last (previous) puzzle to the
current input puzzle. Otherwise P specifies a cyclic permutation applied
to each input puzzle as it is read in.
-P Solve the puzzle and generate postscript output. To generate images
for posting: 'convert -crop 490x490+30+20 -scale 60% t.ps t.png'.
-qC Change the current constraints using C (default FNBTHWXYKOG). If C
does not start with - or + then the current constraints are cleared.
If C is - then the input puzzles are not solved but are still listed.
Initial constraints and those following + are enabled or size specific
if a size is specified. Constraints following - are disabled.
Constraints are applied left to right as specified. {...} groups
constraints to be treated as a single constraint. : within a group
commits the moves up to that point but continues with the
remainder of the group. : outside of {...} disables G and causes
the constraints to the left to be ignored when they fail to advance
the solution. Multiple ;VARi=EXPRi may be appended; after the each -qC
is applied the VARi are evaluated and assigned. This also enables
multiple -qC options, each with its own VAR assignments. After all -qC
are applied for the current puzzle the -e expression and -f format are
are applied, where the VARi may be referenced in the expression / format.
C may also be a lower case or numeric identifier that names a common set
of options:
hardest|h Sudoku Player's Forum hardest puzzle options. Ratings are
exponentially normalized to the range 95000..99999
(fixed upper limit). Fairly consistent across
puzzle permutations, but may take few minutes
on some puzzles.
-q-G -BOX -MFN -Z'{FN}P*V(2)||{FN}BP*V(3)||{FN}B{T2H2}P*V(4)'
hardest-obsolete|o Obsolete Sudoku Player's Forum hardest options.
-q-G -B -Z'FNP(FN)V(P<=9)V(2)||-XY+P(FN)V(P<=9)V(3)||
FNP(FNB)V(P<=9)V(4)||FNP(FNBT2H2)V(5)||
FNP(FNBTHW)V(6)||FNP(FNBTHWX)V(7)||
FNP(FNBTHWXY)V(8)||FNP(FNBTHWXYO)V(9)||
FNP.3G||FNP.9G||FNBTHWXYP.9G'
inferior|i Sudoku Player's Forum inferior puzzle options.
-q{FN}-G -B -eV
1 Sudoku Player's Forum hardest puzzle Q1 rating options.
Ratings are steps+iterations plus a weight for
propositions requiring B constraints. Values fall
in the range 1 .. 99999. Much faster than -qhardest,
and may be of similar quality.
-q'{FN}-G' -Z'{FN}P*V(2)||{FN}BP*V(3)'
-B -MFN -R'steps+iterations+B*1000'
2 Sudoku Player's Forum hardest puzzle Q2 rating options.
Similar to Q1 but averages the worst case work of a
candidate degree ordered backtrack solver. Values fall
in the range 1 .. 99999.
-q'{FN}BP*-G' -B -MFN -RP?P9:I0
simple|ss A close approximation to the Simple Sudoku constraint order.
Differences will surface between SS coloring and X/Y cycles.
With guessing off (-G) some valid puzzles will have no soluti
on.
-qT1H1T2BT3T4H2W2W3XH3H4Y-G
superior|s Sudoku Player's Forum superior-plus puzzle options.
-q{[B]T1H1BT2H2W2T3H3}{[B+]W3T4H4W4}-G -eV
superior2|s2 Sudoku Player's Forum superior-2 puzzle options.
-q'{[B+]T1H1}{[B1]BT2H2W2T3H3}-G' -e'V&&S&&(W2||T3||H3)'
ulterior|u Sudoku Player's Forum ulterior puzzle options.
-q{B2:N}-G -Q!{FN} -AB -eV
Common option constraints may be modfied by appending +constraints
or -constraints to the common constraint name.
-QE Valid puzzles must additionally satisfy the constraint expression E.
E is an expression of ( ! && || == != ?: ) operators and -qC style
constraint operands. For example, -Q'!{FN}&&{N}' specifies FN-invalid
and N-valid.
-rN Set the pseudo random number generator seed to N (default time based).
The %#rX value can be used to span most pseudo-random operations across
multiple sudoku(1) invocations. Does not work for -gt.
-RX Set the %r rating expression to X. The default is
P?(G?[r]98000+G:(10000+(P5>=1000?(P5/100)*100:P5))*10+
(B0?10000000+B0*100:0)+
((T0||H0)?50000000+((T2+H2)+(T3+H3)*8+(T4+H4)*32)*32:0)):
F+N*2+B*4+(T2+T3*4+T4*16)*8+(H2+H3*8+H4*32)*64+
(W*2+W2+W3*4+W4*16)*256+(X*2+X3+(X4+X5*2+X6*4+X7*8)*4)*128+
(Y*2+Y3+(Y4+Y5*2+Y6*4+Y7*8)*16)*128+
(K*2+K3+(K4+K5*2+K6*4+K7*8)*16)*512+
(Z*2+Z2+Z3*4+Z4*8+Z5*12+Z6*14+Z7*15+Z8*16)*512+
O*128*256+(G?(G*(15**M-M1+64)+89985):0)
-sOS Generate puzzles with dihedral symmetry S (requires -g). The symmetry,
names, type, (order), and descriptions are:
f I (8) full dihedral
r II (4) full rotational (90 180 270 degrees)
hv III (4) horizontal and vertical
da IV (4) diagonal and antidiagonal
p V (2) pi rotational (180 degrees)
v VI (2) vertical
h VI (2) horizontal
d VII (2) diagonal (main)
a VII (2) antidiagonal
-sA (or -sg) cycles through all symmetries for each -g solution grid.
-sE selects a pseudo-random symmetry before for each -g solution grid.
-sF selects a pseudo-random symmetry before the first -g solution grid.
-s[248][AEF] limits the symmetries to the specified consecutive orders.
-S Restart constraint propagation after the first move in each group.
-tN Limit solution count to N.
-TM Enable test code defined by the mask M:
0x00010 List constraint orders.
0x00020 List P constraint stats.
0x00040 Don't cache P constraint propositions.
0x00080 Don't prune P constraint pairs.
0x00100 List sizeof(Grid_t) and exit.
0x00200 Disable directional y-edges.
0x00400 Enable directional y-edge ternary collapse.
0x00800 Disable x box hinges.
0x01000 Enable -v for backdoor (magic cell) search moves.
0x02000 Enable -v for -m search moves.
0x04000 List a -P empty grid and exit.
0x08000 List postscript output labels.
0x10000 Enable backdoor trace.
0x20000 Enable mutable() canonical puzzle trace.
0x40000 Disable the (not always) quick sub-solver.
-u Verify that there is exactly one puzzle solution. By default
searching stops at the first solution.
-U If -a is also set then the listed puzzle is the union of all
solutions. -f%h lists the unavoidables as pencilmark hints.
-vN Set verbose output level to N, or increment if N omitted.
-V Print the program version on the standard error and exit.
-wN Fold verbose output to be at most N characters wide (default 80).
-W Pause before exiting. Handy for viewing windows shortcut output.
-xS Puzzle data is XML tagged by S (default question).
-X Limit proposition constraint moves to eliminations (ignore backdoors).
-y Use the original -Y grids instead of implied solution grids.
-YF Generate puzzles from the solution grids for puzzles in file F rather
than random solution grids.
-ZE Puzzles not solved by the -q constraints are rechecked with the -Q
style constraint expression E. V(n) in each subexpression of a || list
can be used to label the first subexpression from the left that solves
the puzzle. The -q constraint label is 1. V(X), X not a number, within
a constraint subexpression requires the constraint subexpression to
solve the puzzle and the -e style expression X to evaluate non-zero,
otherwise the remaining subexpressions are tried.
-zX Miscellaneous options (only -I is free!). X may be:
8 Octdoku: all clues with value 9 present and not subject to minimizati
on.
b Set the process priority to "nice" ("background").
eN Exit with a diagnostic when any dictionary entry count exceeds N.
DETAILS
There are two basic solution techniques: constraint propagation ("logic")
and searching. Constraints limit candidate values for each cell. Constraint
propagation iteratively applies constraints until the puzzle is solved or
until no more progress is made. A constrained puzzle is solvable by
constraint propagation alone (pure logic / no guessing). Searching applies
candidate cell values (guesses) and checks solutions using depth first or
breadth first search. The best search strategies combine constraint
propagation with searching to prune the search tree. The amount of guessing
is determined by the strength of the constraints and the order of guesses
already applied. Some aficionados frown on solutions that require guessing,
but the lines between constraint propagation, lookahead, and guessing are
thin. This topic is very subjective, your mileage will vary, and you will
not win a "logic" vs. "trial and error" debate. I still can't figure out
out how my daughter solves so fast.
This solver uses a depth first search by default. Candidate cells with the
least amount of choices are checked first. The constraints are:
F Forced cell (naked single or T1): only one value possible.
N Only cell (hidden single or H1): only one value in row/col/box.
Bn Box claim: (2:boxline) candidates in box confined to one row/col.
(3:linebox) candidates in row/col confined to one box.
Tn (naked) tuple: order <= n (4) n exact n-tuples in row/col/box.
Hn (hidden) tuple: order <= n (4) n hidden n-tuples in row/col/box.
Wn Row/col claim: order <= n (4) pure x-wing/swordfish/jellyfish.
X Singleton cycle: strong and 1-weak edges. Requires B, included in Y.
Y Degree-2 cycle: strong, 1-weak and 0-weak (degree-2) edges. Includes X.
NOTE: y-edge (0-weak) path details are currently disabled pending
enhancement of the 2007-05-01 Y algorithm.
K Degree-2 singles knots (bivalue/location contradiction chains).
O Overlay: single digit 9-cell rookery templates (that WXY miss).
Zn Ronk's general fish finder. n is the max fish size, 2 <= n <= 8.
.m is the max number of fins, 1 <= m <= 3. Fish sizes in the range
2..n are searched. .m always denotes a fin count range from 0..m.
n.m.1 limits the size search to n.
The default is Z5.2.
Pn Constrained proposition net P(C) -- constraints C applied to candidate
propositions. n is the girth (constraint iteration) limit.
The default is 0 for no limit. .m is the candidate degree
(value/location) limit; propositions are only made on cells/values
with degree <= m, in order from least to most. The default is 0
for no limit. Degree 1 limits propositions to bivalue cells only.
The basic constraints are used if (C) is omitted. If singleton
propositions fail to advance the solution then: if P* or P..2
is specified then nested (paired) propositions are applied,
otherwise if G is enabled then normal backtrack logic is used.
-B is enabled and -S is disabled for this constraint. With -O
the minimum girth/degree that advances the solution is used,
and all propositions for that girth/degree are attempted and
counted. Without -O the minimum degree that advances the
solution is used. Related to error nets, 3D-medusa, T&E.
Qn Experimental constraints QX. Q prefix optional. Q stats account
for all Q constraints applied. X may be:
b: Red Ed's BR net which propagates 16 vertex-vertex boolean
relationships derived from strong (3) and weak (1,2) edges
by applying a triangle relationship on groups of the vertices.
Q2 is the number of sweeps, Q3 is the number of triangle
state changes. A sweep searches for triangle state changes.
Sweeps are repeated until there are no more state changes.
t(C): Michael McWilliams' red/green transport: select a degree 2
tuple and determine the common eliminations when one end and
then the other is assigned, using constraints C to propagate
(transport) the initial assignments. The quick constraints
(FN) are used if (C) is omitted. Related to the P constraint.
t1: test only bivalue cells, t2: test only bilocation cells.
Fails on puzzles matching -e 'P&&(P8!=1||P7!=2||V!=2)'.
Disables the XYKO constraints.
G Guess: backtrack search guess (T&E).
The -q option enables, disables, groups and orders constraints (G must
be explicitly disabled). The enabled constraints are called the constraint
set (default FNBTHWXYKOG). The THW constraints are further split
by size: T2H2W2T3H3W3T4H4W4.
Constraint propagation is an iterative process controlled by the -B,
-G and -S options. By default {...} constraint groups are applied as a
unit; when -G is on {...} are ignored and each constraint is applied
separately. Constraints not enclosed in {...} are considered to be a
group of size 1. Each iteration applies the constraints in order from
left to right as specified by -q. The application of each constraint is
done in an unspecified but exhaustive order over the entire puzzle. A
constraint advances the solution when it identifies a move (one or more
cell placements or candidate eliminations). An identified move may be
committed (by actually making the placement(s) or elimination(s)), or it
may be recorded for later committal. Within a single constraint
application, cells that do not advance the solution are not re-examined,
even if subsequent moves within the same constraint application would
allow them to do so. This biases solutions to use constraints earlier
in the order. A new iteration starts with the leftmost constraint.
When a constraint identifies a move:
-S The move is committed and a new iteration is started.
-B The move and all other moves in the current group are identified
and recorded. After all moves in the group have been recorded
they are committed as a unit, and a new iteration is started.
: within {...} commits any moves up to that point but continues
with the remainder of the group.
-- (default) The move is committed and all other moves in the current
group are committed as they are identified. After all moves in
the group have been committed a new iteration is started.
A step is a constraint propagation iteration that advances the
solution. The number of steps for -S and default solutions depends on
the unspecified ordering within each constraint application; this ordering
bias also allows the step counts to be different for isomorphic permutations
of a given puzzle. -B step counts, by design, are the same for a given
ordered constraint set over all isomorphic puzzles, so they are a suitable
puzzle metric.
[...] options may appear at the start of each constraint group. Options
override the -A, -B, -O and -S command line options. The options are:
A Enable -A for the group -- -A is diabled if A omitted.
B Enable -B for the group -- -B is diabled if B omitted.
O Enable -O for the group -- -O is diabled if O omitted.
S Enable -S for the group -- -S is diabled if S omitted.
* The group may supply any number of moves each application (default).
? The group must supply zero or one move each application.
+ The group must supply one or more moves each application.
n The group must supply exactly n moves each application.
n:m The group must supply between n and m moves inclusive each
application. If m is * the at least n moves must be supplied.
If F is disabled then the solution may advance to a position where only F
constraints remain; these are attributed to most recent constraint applied. -d
(default) depth first search with guesses biased to cells with the least numbe
r
of candidates after constraint propagation. X and Y are compute intensive
but produce aesthetic solutions. -qFNB typically solves the fastest (with
backtracking). If G is enabled then it is not applied until all other
other constraints fail.
Cycles are composed of edges between cells with the same candidate value
within a single row/col/box or segments between exact pairs with the
candidate value in both endpoints. The vertices of a strong edge are the
only cells with the candidate value within the row/col/box containing the
edge.
The generator generates random solution grids, and then places clues at random
from the solution grid into an empty puzzle grid, honoring -sS if specified,
and with -m generates the derived minimal puzzles for each random puzzle.
A minimal puzzle has a minimal number of clues, i.e., removing any clue
produces a puzzle with multiple solutions. There is no duplicate checking;
process generator output using -f%c as a duplicate/sort key.
A backdoor or magic cell set is the smallest set of moves that lead to a
constrained solution. A puzzle with backdoor size N (magic cell set size N)
for constraint set C is C-N-constrained. The conjecture that all puzzles
are FN-0, FN-1 or FN-2-constrained is false: a puzzle discovered by JPF has
566 FN backdoors of size 3 (3 FNBT2 backdoors of size 2). A %#nr rating in
[A-F] indicates a backdoor size of 3, 4 etc. Using -q may result in different
backdoors than the default. For this reason always qualify "backdoor"
with the constraints used.
EXPRESSIONS
Expressions are C language style signed long expressions on constraint
counters. A constraint counter is named by the constraint identifier
and one optional digit. The counters for constraint c:
c The number of constraint applications.
c0 Equivalent to c.
c1 The number of constraint placements. Grouped modifications to a cell
candidate list count as one placement.
cn Structure order 2<=n<=9 count (e.g., pair, triple, x-wing,
swordfish). c5..c9 are 0 for most constraints.
with the following (long name) exceptions and pseudo constraints:
B2 (boxline) Candidates in box confined to one row/col.
B3 (linebox) Candidates in row/col confined to one box.
C (clues) The number of clues.
C1 (minimal) 1: minimal (irreducable), 2: symmetric minimal, 0: otherwise.
C2 (inherited) The number of forced moves inherited by the last constraint.
C3 (candidates) The number of initial (pencilmark) candidates.
D (begsteps) Begin game singles steps.
D1 (begsingles) Begin game singles placements.
D2 (begbasic) Begin game basic constraint placements.
D3 (endsteps) End game singles steps.
D4 (endsingles) End game singles placements.
D5 (endbasic) End game basic constraint placements.
G (guesses) The number of backtrack guesses.
G2 (depth) Backtrack depth, 0 for no backtracking.
I (steps) The number of constraint propagation iterations.
In (group) The number of steps for constraint group 1<=n<=8.
M (backdoor) The backdoor (magic cell set) size.
M1 (magic) The number of backdoors (magic cell sets).
M2 (guesstimate) The estimated number of guesses before a backdoor hit.
P2 (propositions) The number of proposition net propositions.
P3 (luck) The number of proposition net solutions.
P4 (contradictions) The number of proposition net contradictions.
P5 (iterations) The number of proposition net iterations.
P6 (girth) Minimum maximum proposition net iteration.
P7 (degree) Minimum maximum proposition candidate degree.
P8 (nested) 2 for nested propositions, 1 for single propositions.
P9 (work) The average amount of successful proposition iterations.
R (rating) The rating from 1 (very easy) to 99999 (very difficult).
S (symmetric) The dihedral symmetry order, 0 if not symmetric.
S (symmetry) Equivalent to symmetric.
S1 (dihedral) The dihedral symmetry operation.
V (valid) 0 for invalid, the constraint index or 1 otherwise.
X2 Maximum X cycle size.
Xn X cycle sizes by range. X3:<4, X3:<8, X5:<12, ... X9:>=32.
Y2 Maximum Y cycle size.
Yn Y cycle sizes by range. Y3:<4, Y3:<8, Y5:<12, ... Y9:>=32.
bn The number of batches for hn.
hn The ordinal of the rightmost constraint applied in group n.
in The number of instances for hn.
and the following functions:
exrate exrate([SAMPLE]): suexrate style rating that is the average
number of nodes for a depth-first singles only backtrack
solver over SAMPLE (default 100) pseudo random permutations
of the current puzzle. Suexrate counts Knuth DLX nodes.
sudoku(1) is not DLX based and requires on average 9X less
nodes per solution; the sudoku(1) node counts are multiplied
by 9 to approximate suexrat(1).
determined Compute the number of candidate cell values determined by
eliminating candidates that cause any constraint to fail.
'%(determined())g' lists the grid with determined cells
included.
implicit Determine the cell values implied by the constraints in
scope. '%(implicit())g' lists the grid of implicit cells.
max max([index,]expression): return 1 if expression is greater
than the maximum value [for index 1..9], 0 otherwise.
For the final -Ff format max([index]) returns the max value.
min min([index,]expression): return 1 if expression is less
than the minimum value [for index 1..9], 0 otherwise.
For the final -Ff format min([index]) returns the min value.
mutable Determine the clues that have more than one value that results
in one solution. The format '%(mutable())h' lists the
mutable clue candidate values. The operands are:
() Number of mutable clues.
(max) Maximum number of mutable values for one clue.
(maxuni) Maximum number of unique puzzles for one clue.
(pos) Mutable clue positions.
(sig) Number of mutable candidates by clue, high to low.
(siguni) Number of unique puzzles by clue, high to low.
(tot) Total number of mutable candidate values.
(totuni) Number of unique puzzles induced by mutable clues.
required Determine the non-mutable clues and count the number of
solutions when each clue is omitted. The format
'%(required())h' lists number of solutions when each clue is
omitted, 0 for non-clue cells. The operands are:
() Number of required clues.
(max) Maximum number of solutions.
(maxpos) Cell position with the maximum solutions.
(min) Minimum number of solutions.
(minpos) Cell position with the minimum solutions.
(sig) Number of solutions, high to low.
(tot) Total number of solutions.
superfluous Determine the clues that result in one solution when omitted.
The format '%(superfluous())g' lists the grid of superfluous
clues. The operands are:
() Number of superfluous clues.
(pos) Superfluous clue positions.
symmetrize The highest dihedral symmetry order for the current puzzle.
uniq uniq() returns 1 if the current puzzle is different up to
isomorphism from the previously listed puzzles. uniq(format)
returns 1 if the -f format result is different from previous
format results. Too many different puzzles/formats may
may exhaust memory.
(%F) converts the output of the %F format string to a number and $V converts t
he
value of the environment variable to a number.
INPUT FORMAT
Puzzle input is a sequence of numbers and spaces that fill the grid
from left to right, top to bottom. Space, comma, |, and - if + or | appear,
are ignored, and all other non-digit (1 through 9) characters correspond
to an empty grid space. Most pencilmark grid forms are accepted. Multiple
descriptions may be placed in one file. A description ends after 81 cells
have been specified. If the first character in a line is a non-digit the
line is ignored. A description command line argument (file or actual
data) may be immediately followed by one or more command line argument
operations of the form [r,c]<OP><V>... that modify candidate values, where
<V> is a single digit cell value or a {...} list of candidate values, and
<OP> is = to set V, ^ to clear V. '#' starts a comment until end of line.
"IDENT" sets a puzzle identification string for %i. If there is no "IDENT"
then the last comment for the current puzzle is used for %i. A comment on the
same line following the last puzzle cell is associated with that puzzle.
Subsequent comments are associated with subsequent puzzles. For example, one
generated by -g -m -sp -q-YG -r12 -n1 -e 'V&&X' -f'"gmr11n1spq-YG"\n%#gg':
"gmr11n1spq-YG"
. 6 . 8 1 . 7 2 .
8 2 4 . . 6 . . .
. . . . . 5 3 . .
. 1 . . . . 8 . .
. 8 . . . . . 1 .
. . 9 . . . . 4 .
. . 2 1 . . . . .
. . . 6 . . 2 3 5
. 9 8 . 4 2 . 6 .
The first input line in each file may be #!sudoku followed by options that
override command line -c, -l and -x options for the duration of the file.
The previous options are restored after the file is read. For example:
#!sudoku -c5,sym,num,clues,min,puz,author,seq
The input line #!exemplar identifies the remaining input puzzles as fish
pattern exemplars. Exemplars may be listed with the %g format and subgrid
canonicalized with the %#tv, %#tc and %#Tc formats. The exemplar cell
values are:
/ empty cell: a cell that may not have a candidate
X base candidate: which may be missing
# (exo-)fin cell: a base candidate not covered by any cover sector
@ endo-fin cell: a candidate in the intersection of two base sectors
* potential exclusion if all fin cells are false
$ potential exclusion whether or not a fin cell is true
& potential exclusion of base candidate covered by two cover sectors
The octal 3 band bitmask form {O,O,O} (listed by -v2) is also recognized.
#!template may be used to mark -gt template data when -gt is not specified.
OUTPUT FORMAT
A solution is an 81 character string of [1-9A-IJ-R], [1-9] for clue cell
values, [A-I] for solution cell values, A for 1, B for 2, and so on, and
[J-R] for the backdoors, J for 1, K for 2, and so on. The solution fills
the grid from left to right, top to bottom. The default output format
for each input puzzle is '%v # %5r %q %(CSM)#c#/q' (see -f above). The
output for the example above is the single line (folded for display):
.6.81.72.824..6........53...1....8...8.....1...
9....4...21........6..235.98.42.6. # 277 FNTX C28.M/S2.p
The canonical solution is a puzzle solution transformed to have the
smallest row order lexicographic value. The canonical solution
induces an equivalence relation on the set of all solutions. The
canonical puzzle retains the labeling of the canonical solution and
induces an equivalence relation on the puzzle within the solution.
If a canonical solution has non-trivial automorphisms then the
canonical puzzle uses the canonical solution automorphism with the
smallest row order lexicographic value (with empty cells treated as
0). There may be other canonical forms. This one has an efficient
algorithm that takes advantage of the group symmetries of sudoku
solution grids.
COMPRESSED GRID FORMAT
The sudz format efficiently compresses catalogs of row order minlex
canonical solution grids. Grids are organized by the top band (top 3
rows). There are 416 essentially different minlex bands and 5472730538
essentially different grids. A byproduct of minlex ordering is that
earlier bands tend to account for more grids than later bands. For
example, band 001 contains 1007170 grids, band 006 (the largest)
contains 96229042 grids, and bands 395,397,398,400,402,403,404,406,
408,409,410,412,413,414,415 contain no grids.
The sudz format is a sequence of windows. Each window contains the number
of grids and initial band index. Each grid has the band index (if
different from the previous grid), the number of automorphisms (if > 1),
the number of cells that differ from the previous grid, and the list of
differing cell values encoded using a basic singles constraint solver.
The windows are compressed using the Burrows-Wheeler bzip library.
The entire catalog of 5472730538 essentially different grids, in minlex
order, compresses to 5.70GiB, an average 8.96 bits/grid. Uncompress rate
is ~100K grids/sec/Ghz, or ~5 hours minimum to stream through the entire
catalog on a 2.8Ghz processor.
PERFORMANCE
The solution rate for the best on average options -qFN on a collection
of posted and generated puzzles is ~1000 puzzles/second/Ghz, the -gg full
grid (81 clues) generation rate is ~5000 puzzles/second/Ghz, and the
-g -m1 -qFN generation rate is ~75 puzzles/second/Ghz. sudocoup(1),
coded for speed, solves ~7000 puzzles/second/Ghz. sudocoo(1), coded for
simplicity, is in between and is more sensitive to input grid variations.
EXAMPLES
Note: % must be entered as %% in windows .bat files and shortcut commands.
Generate 100 minimal symmetric puzzles limited to pairs and x-wings in g.dat:
sudoku -g -sg -m -q+T2H2W2-XYG -e "valid&&minimal==1" -n100 -f%v -o g.dat
Reformat and collate for player's forum inferior low/high steppers:
sudoku -f"%#tq,%(steps)x,%(clues)x,%(minimal)[-][M][SM]x,%#0v,gsf,%n" \
-q{FN}-G -B *.dat | sort -t, -k1,1 -k2,3n -k4,4r
Generate and search for player's forum ulterior low/high steppers:
sudoku -f"%#tq,%(steps)x,%(clues)x,%(minimal)[-][M][SM]x,%#0v,gsf,%n" \
-e"V&&(I<4||I>20)" -g -q{B2:N}-G -Q!{FN} -AB -m -sg
Count isomorphic puzzles and list each non-trivial representative:
sudoku -f"%C %c %4n %#0v" *.dat |
sort -k1,1 | uniq -c -s82 -w82 | grep -v "^ *1 "
Count isomorphic solutions and list each non-trivial representative:
sudoku -f"%C %c %4n %#0v" *.dat |
sort -k1,1 | uniq -c -w82 | grep -v "^ *1 "
Solve and categorize a collection of puzzles:
sudoku -F"# %t seconds" -f"%v # %7n %10t %Q" puzzles.dat > puzzles.out
Label and trace constraint interations:
sudoku -ki -v2 puzzle.dat | less
Capture the pencilmark grid after constraint iteration N:
sudoku -kiN -f%#ph puzzle.dat
Havard's "Making 17's from JPF's 19" sequence:
sudoku -go"{-1+1}x3{-2+1}{-2+2}{1+1}x9" jpf-19.dat
Generate 10 non-equivalent mimimal symmetric puzzles, each solved by a
different set of constraints, with the same solution grid as the
first puzzle in s.dat:
sudoku -gp -Ys.dat -j0 -n10 -m -sg -e 'minimal==1&&uniq()&&uniq(%q)'
Filter weak and strong pearls:
sudoku -S -qss+G -e '(%#Pq)>2'
Generate and compress the band 299 grids (133302) into 299.sudz:
sudoku -gb299 -f '%#ec' -o 299.sudz
Count the number of grid (97) with non-trivial automorphisms in 299.sudz:
sudoku -e '(%#An)>1' -f- -Ff'%#an/%n' 299.sudz
CAVEATS
This is a puzzle and algorithm analysis program.
Look elsewhere for interactive gaming and GUIs.
Its easier to solve than to generate,
and easier to generate than to rate,
and much easier to code than to document.
SEE ALSO
sudocoup(1), sudocoo(1), pseudocoup(1)
CONTRIBUTORS
Subgrid (multiple solution) row order minlex canonicalizer by Michael Deverin.
Qb BRnet constraint method by Ed Russell.
Z fish finder by Ron Kral.
IMPLEMENTATION
version sudoku (AT&T Research) 2009-05-10
author Glenn Fowler <gsf@research.att.com>
copyright Copyright (c) 2005-2009 AT&T Intellectual Property
license
http://www.opensource.org/licenses/cpl1.0.txt
by enxio27 » Fri Nov 03, 2017 3:39 am 
)
