Here's some code for finding paths of a given length starting from a given spot. You just need to run through all the starting spots and find the paths of the desired lengths. There will be a lot of them.

my %adjacency_list = (
                       1 => [2,5,6],
                       2 => [1,3,5,6,7],
                       3 => [2,4,6,7,8],
                       4 => [3,7,8],
                       5 => [1,2,6,9,10],
                       6 => [1,2,3,5,7,9,10,11],
                       7 => [2,3,4,6,8,10,11,12],
                       8 => [3,4,7,11,12],
                       9 => [5,6,10,13,14],
                      10 => [5,6,7,9,11,13,14,15],
                      11 => [6,7,8,10,12,14,15,16],
                      12 => [7,8,11,15,16],
                      13 => [9,10,14],
                      14 => [9,10,11,13,15],
                      15 => [10,11,12,14,16],
                      16 => [11,12,15]
                     );


sub find_path {
  my ($start_at, $length, $been_there) = (@_, {});
  if ($length <= 1) {
    return [$start_at];
  }
  else {
    my @try_these = grep { ! $been_there->{$_} } @{$adjacency_list{$st
+art_at}};
    return map {
      my @cdr_list = find_path($_, $length-1, {%$been_there, $start_at
+ => 1});
      map [$start_at, @$_], @cdr_list;
    } @try_these;
  }
}

# Example usage:
print "@$_\n" for (find_path(3, 3));
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