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I bet Sudoku is behind this thread. I wrote a bruteforce solver last week, takes about 10 seconds for the "hard" ones on my three years old PC. I'll post it once I get to the office. I do have some more ideas, a wee bit less bruteforcish, though not at all similar to this.

Update: Here is the blindly bruteforcish implementation. Needs about 11s to solve the "hard" sudokus on my Pentium 4. Making something like 400000 "what numbers are allowed on this place so far" tests.

```use integer;
use strict;
use Data::Dumper;

my \$str = <<'*END*';
5 - - - 7 - 6 8 2
- - - 5 9 6 - - -
- - - - - - - - -
- - - - 8 - - 4 9
- 3 6 - - - - - -
- - - - - - - - -
- - 8 - - 7 - - 1
- - 3 - - 4 - - 7
6 4 - 3 - - - 2 -
*END*

sub parse_data {
my \$data = shift;

\$data =~ s/\r\n/\n/g;
\$data =~ /(([0-9\-] ){8}[0-9\-]\n){9}/s or die "Not well formatted
+";

\$data =~ s/\n/ /g;
my @data = map {\$_ eq '-' ? undef : \$_+0} split / /, \$data;
return \@data;
}

my (@id2row,@id2col,@id2square);
for my \$id (0..9*9-1) {
\$id2row[\$id] = \$id / 9;
\$id2col[\$id] = \$id % 9;
\$id2square[\$id] = int(\$id2col[\$id] / 3) + 3*int(\$id2row[\$id] / 3);
}

my (@num2bit, %bit2num);
for my \$num (1 .. 9) {
my \$bit = 1 << (\$num - 1);
\$num2bit[\$num] = \$bit;
\$bit2num{\$bit} = \$num;
}

sub prepare_struct {
my \$data = shift;

my \$struct = {
data => \$data,
rows => [(0) x 9],
cols => [(0) x 9],
squares => [(0) x 9],
fixed => {},
};
foreach my \$id (0 .. (9*9-1)) {
next unless \$data->[\$id];
my \$bit = \$num2bit[\$data->[\$id]];
\$data->[\$id] = \$bit;
\$struct->{rows}[\$id2row[\$id]] |= \$bit;
\$struct->{cols}[\$id2col[\$id]] |= \$bit;
\$struct->{squares}[\$id2square[\$id]] |= \$bit;
\$struct->{fixed}{\$id} = 1;
}
return \$struct;
}

my \$data = prepare_struct(parse_data(\$str));

print_solution(\$data);

sub ALL_TAKEN () {0b111111111} # all 9 numbers used

sub get_allowed {
my (\$data, \$id) = @_;

return \$data->{data}[\$id] if defined(\$data->{data}[\$id]); # preset

my \$map = ALL_TAKEN & ~(\$data->{rows}[\$id2row[\$id]] | \$data->{cols
+}[\$id2col[\$id]] | \$data->{squares}[\$id2square[\$id]]);

return if \$map == 0;
return \$map if exists \$bit2num{\$map};

return [grep( (\$_ && (\$_&\$map)), @num2bit)];
}

sub set_number {
my (\$data, \$id, \$bit) = @_;

return if \$data->{data}[\$id] == \$bit;

die "Huuumpf, trying to set id \$id (row \$id2row[\$id], col \$id2col[\$id]
+) from \$data->{data}[\$id] to \$bit!!!\n"
if \$data->{data}[\$id];

\$data->{data}[\$id] = \$bit;
\$data->{rows}[\$id2row[\$id]] |= \$bit;
\$data->{cols}[\$id2col[\$id]] |= \$bit;
\$data->{squares}[\$id2square[\$id]] |= \$bit;
}
sub unset_number {
my (\$data, \$id, \$bit) = @_;
return if \$data->{fixed}{\$id};
\$data->{data}[\$id] = undef;
\$bit = ~\$bit;
\$data->{rows}[\$id2row[\$id]] &= \$bit;
\$data->{cols}[\$id2col[\$id]] &= \$bit;
\$data->{squares}[\$id2square[\$id]] &= \$bit;
}

my @backtrack = (undef) x (9*9);

my \$pos = 0;
\$backtrack[\$pos] = get_allowed( \$data, \$pos);
if (ref \$backtrack[\$pos]) {
set_number( \$data, \$pos, \$backtrack[\$pos][0]);
} else {
set_number( \$data, \$pos, \$backtrack[\$pos]);
}
\$pos++;

while (\$backtrack[0] and \$pos < (9*9)) {

my \$allowed = get_allowed( \$data, \$pos);

if (\$allowed) {
if (ref \$allowed) {
#print " " x \$pos, '>', \$allowed->[0], "\n";
set_number( \$data, \$pos, \$allowed->[0]);
} else {
#print " " x \$pos, '>', \$allowed, "\n";
set_number( \$data, \$pos, \$allowed);
}
\$backtrack[\$pos] = \$allowed;
\$pos++;
} else {
\$pos--;
while (1) {

if (ref \$backtrack[\$pos]) {
#print " " x \$pos, '<', \$backtrack[\$pos][0], "\n";
unset_number( \$data, \$pos, \$backtrack[\$pos][0]);

if (@{\$backtrack[\$pos]} > 1) {
shift(@{\$backtrack[\$pos]});
#print " " x \$pos, '>', \$backtrack[\$pos][0], "\n";
set_number( \$data, \$pos, \$backtrack[\$pos][0]);
\$pos++;
last;
} else {
\$pos--;
}
} else {
#print " " x \$pos, '<', \$backtrack[\$pos], "\n";
unset_number( \$data, \$pos, \$backtrack[\$pos]);
\$pos--;
}
}
}

}

print_solution(\$data);

sub print_solution {
my \$data = shift;

foreach my \$r (0 .. 8) {
foreach my \$c (0 .. 8) {
print( (\$bit2num{\$data->{data}[\$r*9 + \$c]} || ' ')." ");
}
print "\n"
}
print "\n";
}

print join(',', times);

Update 2: Here is the promised improved version. It's quite similar except that there is an additional step. Whenever I make a guess I scan the whole plan for fields that are clear according to the fields already set and continue scanning and setting fields until there are no more clear ones and I have to find the next unset field in which I have to guess. This has brought the time from 11s down to 0.3s.

```use integer;
use strict;
use Data::Dumper;

=rem
my \$str = <<'*END*';
9 - - 2 - - - - 4
- 6 - - - 7 9 - -
- 4 - 6 - - - - -
- - 5 - - - 3 - -
- - - - - - 2 5 -
- 2 - - 4 - - - 8
- - 4 - - - - 2 3
- 1 7 8 - - 6 - -
- 5 - - 3 - - - 1
*END*
=cut

=rem
my \$str = <<'*END*';
- - - - - - - - -
- - - - - - - - -
- - - - - - - - -
- - - - - - - - -
- - - - - - - - -
- - - - - - - - -
- - - - - - - - -
- - - - - - - - -
- - - - - - - - -
*END*
=cut

my \$str = <<'*END*';
5 - - - 7 - 6 8 2
- - - 5 9 6 - - -
- - - - - - - - -
- - - - 8 - - 4 9
- 3 6 - - - - - -
- - - - - - - - -
- - 8 - - 7 - - 1
- - 3 - - 4 - - 7
6 4 - 3 - - - 2 -
*END*

sub parse_data {
my \$data = shift;

\$data =~ s/\r\n/\n/g;
\$data =~ /(([0-9\-] ){8}[0-9\-]\n){9}/s or die "Not well formatted
+";

\$data =~ s/\n/ /g;
my @data = map {\$_ eq '-' ? undef : \$_+0} split / /, \$data;
return \@data;
}

my (@id2row,@id2col,@id2square);
for my \$id (0..9*9-1) {
\$id2row[\$id] = \$id / 9;
\$id2col[\$id] = \$id % 9;
\$id2square[\$id] = int(\$id2col[\$id] / 3) + 3*int(\$id2row[\$id] / 3);
}

my (@num2bit, %bit2num);
for my \$num (1 .. 9) {
my \$bit = 1 << (\$num - 1);
\$num2bit[\$num] = \$bit;
\$bit2num{\$bit} = \$num;
}

sub prepare_struct {
my \$data = shift;

my \$struct = {
data => \$data,
rows => [(0) x 9],
cols => [(0) x 9],
squares => [(0) x 9],
fixed => {},
};
foreach my \$id (0 .. (9*9-1)) {
next unless \$data->[\$id];
my \$bit = \$num2bit[\$data->[\$id]];
\$data->[\$id] = \$bit;
\$struct->{rows}[\$id2row[\$id]] |= \$bit;
\$struct->{cols}[\$id2col[\$id]] |= \$bit;
\$struct->{squares}[\$id2square[\$id]] |= \$bit;
\$struct->{fixed}{\$id} = 1;
}
return \$struct;
}

my \$data = prepare_struct(parse_data(\$str));

print_solution(\$data);

sub ALL_TAKEN () {0b111111111} # all 9 numbers used

sub get_allowed {
my (\$data, \$id) = @_;

return [\$data->{data}[\$id]] if defined(\$data->{data}[\$id]); # pres
+et

my \$map = ALL_TAKEN & ~(\$data->{rows}[\$id2row[\$id]] | \$data->{cols
+}[\$id2col[\$id]] | \$data->{squares}[\$id2square[\$id]]);

return if \$map == 0;
return \$map if exists \$bit2num{\$map};

return [grep( (\$_ && (\$_&\$map)), @num2bit)];
}

my (\$data) = @_;

my \$cnt = 0;
eval {
do {
\$cnt = 0;
foreach my \$id (0 .. (9*9-1)) {
next if defined(\$data->{data}[\$id]); # preset

my \$map = ALL_TAKEN & ~(\$data->{rows}[\$id2row[\$id]] |
+\$data->{cols}[\$id2col[\$id]] | \$data->{squares}[\$id2square[\$id]]);

die "failed" if \$map == 0;
if (exists \$bit2num{\$map}) { # only a single bit is se
+t
set_number( \$data, \$id, \$map);
\$cnt++;
}
};
} while (\$cnt > 0);
};
if (\$@ =~ /^failed/) {
unset_number( \$data, \$_);
};
die "failed";
}
}

sub set_number {
my (\$data, \$id, \$bit) = @_;

return if \$data->{data}[\$id] == \$bit;

die "Huuumpf, trying to set id \$id (row \$id2row[\$id], col \$id2col[
+\$id]) from \$data->{data}[\$id] to \$bit!!!\n"
if \$data->{data}[\$id];

\$data->{data}[\$id] = \$bit;
\$data->{rows}[\$id2row[\$id]] |= \$bit;
\$data->{cols}[\$id2col[\$id]] |= \$bit;
\$data->{squares}[\$id2square[\$id]] |= \$bit;
}
sub unset_number {
my (\$data, \$id) = @_;
return if \$data->{fixed}{\$id} or !\$data->{data}[\$id];

my \$bit = \$data->{data}[\$id];

\$data->{data}[\$id] = undef;
\$bit = ~\$bit;
\$data->{rows}[\$id2row[\$id]] &= \$bit;
\$data->{cols}[\$id2col[\$id]] &= \$bit;
\$data->{squares}[\$id2square[\$id]] &= \$bit;
}

#script

{
my \$clear = add_single_allowed(\$data); # find the items that only
+have one option possible
foreach (@\$clear) {
\$data->{fixed}{\$_} = 1;
}
}
#print_solution(\$data);

my @backtrack = (undef) x (9*9);

my \$pos = 0;

FIELD: while (\$pos >= 0 and \$pos < (9*9)) {
if (\$data->{data}[\$pos]) { # already known
\$backtrack[\$pos] = undef;
\$pos++;
next;
}

my \$allowed = get_allowed( \$data, \$pos);

if (\$allowed) {
ALLOWED: while (@\$allowed) {
set_number( \$data, \$pos, \$allowed->[0]);
if (eval {
\$backtrack[\$pos] = [\$allowed, \$clear];
}) {
\$pos++;
next FIELD;
} else {
unset_number( \$data, \$pos);
shift(@\$allowed);
next ALLOWED;
}
}
# we ran out of allowed values for the current field
}

\$pos--;
while (\$pos >= 0) {
if (\$backtrack[\$pos]) {
if (\$backtrack[\$pos][1]) {
foreach (@{\$backtrack[\$pos][1]}) { # unset those that
+were "clear"
unset_number( \$data, \$_);
}
}
unset_number( \$data, \$pos);

if (ref(\$backtrack[\$pos]) and ref(\$backtrack[\$pos][0]) and
+ @{\$backtrack[\$pos][0]} > 1) {
shift(@{\$backtrack[\$pos][0]});
\$allowed = \$backtrack[\$pos][0];
\$backtrack[\$pos] = undef;
goto ALLOWED;
} else {
\$pos--;
}
} else {
\$pos--
}
}

}

print_solution(\$data);

sub print_solution {
my \$data = shift;

print "-" x 18, "\n";
foreach my \$r (0 .. 8) {
foreach my \$c (0 .. 8) {
print( (\$bit2num{\$data->{data}[\$r*9 + \$c]} || ' ')." ");
}
print "\n"
}
print "-" x 18, "\n";
print "\n";
}

print join(',', times);

In reply to Re^2: decomposing binary matrices by Jenda
in thread decomposing binary matrices by hv

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