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This code "solves" artist's puzzle posted in Matrix Formation. The problem presented was to find the largest divisor for an N x N matrix of digits such that the numbers formed by joining the digits in each row and numbers formed by joining the digits in each column all have a common divisor. In addition, the following restrictions were added:
2. No two formed-numbers are allowed to be the same.
For example, a 2 x 2 matrix is solved as:

2 1

8 4

The numbers formed are: 21, 84, 28, and 14 which all have a divisor of 7.

The code below has come up with the follwing solutions:

• 2x2: Divisor 7, Rows 21, 84
• 3x3: Divisor 44, Rows 132, 792, 660
• 4x4: Divisor 416, Rows 2496, 9152, 1664, 2080
```#!/usr/bin/perl
#
# NOTES: This code will not solve a 1 x 1 matrix as all numbers are "r
+eused"
#
use strict;
use warnings;

use constant TRUE       => 1;
use constant FALSE      => 0;

use constant MATRIX_DIM => 3; # Set your matrix size here
use constant MAX_NUM    => (10 ** MATRIX_DIM) - 1;

use constant DEBUG      => 0; # 0=Nothing, 1=Summary, 2=Detail

use constant SEPARATOR  => '-';

# Since you need to find 2 times the matrix width distinct numbers, yo
+u cannot
# have a divisor greater than the maximum number divided by that produ
+ct.
#
# For example, a 3 x 3 matrix has a maximum number of 999. You need 6
+distinct
# numbers to "solve" the matrix, so the maximum possible divisor would
+ be 166.
# 166 would result in 166, 332, 498, 664, 830, and 996.
#
my \$Divisor = int(MAX_NUM / (MATRIX_DIM * 2));

my @Num;
my @PermPtrs; # Saves the position of the pointers between GetNextPerm
+() calls
my @Sol=();

# GetNextPerm generates purmutations from the passed array (one at a t
+ime)
# and uses a global array of pointers (@PermPtrs). This method seems t
+o be
# fast and uses less memory than computing all the permutations at onc
+e.
#
# NOTE: The first time this sub is called, @PermPtrs should be an empt
+y array.
#
sub GetNextPerm
{
my \$pNum=shift;

my \$Perm='';
my @PtrStack;

if (\$#PermPtrs > -1)
{
my \$Ptr=MATRIX_DIM-1;
while (TRUE)
{
\$PermPtrs[\$Ptr]++;
if (\$PermPtrs[\$Ptr] > \$#\$pNum)
{
push @PtrStack,\$Ptr;
\$Ptr--;
return undef if (\$Ptr == -1);
next;
}
else
{
my \$OkInc=TRUE;
foreach (0..MATRIX_DIM-1)
{
next if (\$_ == \$Ptr);
\$OkInc=FALSE if (\$PermPtrs[\$_] == \$PermPtrs[\$Ptr]);
}
next unless(\$OkInc)
}
last unless (\$#PtrStack > -1);
\$Ptr=pop(@PtrStack);
\$PermPtrs[\$Ptr]=-1;
}
}
else
{
@PermPtrs=(0..MATRIX_DIM-1);
}

foreach (@PermPtrs)
{
\$Perm.=SEPARATOR if (length(\$Perm));
\$Perm.=\$\$pNum[\$_];
}

return \$Perm;
}

sub CheckSolution
{
my \$pNum=shift;

my \$CheckNum;
my \$PossSol;
my \$Ptr;
my @Sol=();
my %Check;
my %Num;

#
# This code checks every permutation of possible numbers (from the p
+assed
# numberarray) as rows and checks if there are MATRIX_DIM other numb
+ers
# (also in the passed array) which can act as columns.
#
# For example, with a 3 x 3 matrix, this code should go through each
+
# permutation of three number (for rows) from the passed array and s
+ee if
# there are three other numbers which would work with the selected r
+ows as
# their columns.
#
# WISH LIST: Verify members of each permutation for "fitness" in the
+ir
#            assigned spot. "Fitness" means you do not put a number
+in the
#            top row whose first digit is not the first digit of at
+least
#            one other number... whose second digit is not the first
+ digit
#            of at least one other number... etc... This may speed u
+p this
#            section of the code.
#
#            Benchmark the code which checks the columns (foreach \$P
+tr block)
#            That code gets executed a lot, and *may* benefit from r
+ecoding.
#

@PermPtrs=();
while(\$PossSol=GetNextPerm(\$pNum))
{
print "\t",'Checking ',\$PossSol,'... ' if (DEBUG >= 2);

%Num=map { \$_ => TRUE } @\$pNum;
@Sol=split(SEPARATOR,\$PossSol);
delete \$Num{\$_} foreach (@Sol);
foreach \$Ptr (0..length(\$Sol[0])-1)
{
\$CheckNum='';
\$CheckNum.=substr(\$_,\$Ptr,1) foreach (@Sol);
if (defined(\$Num{\$CheckNum}))
{
delete \$Num{\$CheckNum};
}
else
{
print 'nope',"\n" if (DEBUG >= 2);
@Sol=();
last;
}
}
last if (\$#Sol > -1);
}

return @Sol;
}

while (\$Divisor >= 1)
{
@Num=();
foreach (1..int(MAX_NUM / \$Divisor))
{
next if ((\$Divisor * \$_) < (MAX_NUM / 10)); # Numbers which begin
+with zero
push @Num, (\$Divisor * \$_);
}
print 'Checking solution for divisor ',\$Divisor,': ',join('-',@Num),
+"\n"
if (DEBUG >= 1);

@Sol=CheckSolution(\@Num);
last if (\$#Sol > -1);
\$Divisor--;
}

print "\n";
if (\$#Sol > -1)
{
print 'Maximum divisor is ',\$Divisor,', solution is ',join('-',@Sol)
+,"\n";
}
else
{
print 'No solution found...',"\n";
}

# End of Script

Observation: Unless I really missed something, I could not find a good CPAN module which would generate permutations of n distinct numbers taken r at a time. This problem led to the GetNextPerm() sub which uses a series or r pointers which are moved through the array of numbers generating one permutation at a time.

In reply to Code to solve the "matrix formation" puzzle by Rhose

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