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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:
  1. No formed-number should start with 0.
  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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