Iterating over Blocks of 2-Dim Array

Kozz has asked for the wisdom of the Perl Monks concerning the following question:

Esteemed monks:

I've got data in an n-by-n two-dimensional array. Assuming that my n is divisible by two, I need to be able to iterate over the items in sub-matrices. Let me explain, and see if you can help me with this algorithm. I've tried, but can't quite wrap my head around it, and the code I DO make always tends to be far uglier than I'd expect it to be.

Disclosure: This is homework related. However, iterating over the data in this manner is not the main thrust of it. My "work" was done in the assignment of the 2-dim array elements' values. I just chose to break up the output into more manageable blocks (perhaps badly?).

Supposing n = 4, then I've got 4 "rows" of indices 0..3, each row having a "column" with indices 0..3, like this (vertical bars and dashed line to represent a line dividing matrix into quadrants):

[ 00 01 | 02 03 ]
[ 10 11 | 12 13 ]
-----------------
[ 20 21 | 22 23 ]
[ 30 31 | 32 33 ]
[download]

I want to output the four elements in the upper-left quadrant of the matrix (00, 01, 10, 11, in that order), then upper right, lower left, lower right.

What trips me up is that I iterate over the first two cols of the first row, then increment row but reset cols. Then row gets reset and cols adjusted.... it's just a very awkward traversal of the matrix.

I know this is more of an algorithms question than a perl question, so smack me down with a -- if you must. But I know that there are monks that can likely whip up a very elegant solution for this.

Thanks for any insight you can provide.

Update: Code per request. This script was made just for a "simple" (ha-ha) base case to see if I could figure out how to iterate over pairs of indices in this manner. I know I've got an infinite loop, too, while I'm trying to figure out when to increment/decrement my rows & columns.

#!/usr/bin/perl

my $n = 2;
my $max_exp = 1;

use strict;

my $max_part = 2**$max_exp;

my $size = 2**$n;

print "Matrix is $size by $size.\n";
print "Max size I'm allowed to print is $max_part at a time.\n";

my $max_part_tiles = $max_part**2;

my $total_tiles = ($size)**2;
my $tiles_counted = 0;
my $part_tiles_counted = 0;

my ($row_num, $col_num, $end_col) = (0, 0);
my ($cols_displayed, $rows_displayed) = (0, 0);

while($tiles_counted < $total_tiles){

        $col_num=0;
        $end_col=$max_part;
        while($row_num < $size){
                while($col_num < $end_col){
                        print "$row_num $col_num\n";
                        $col_num++;
                        $part_tiles_counted++;
                }
                if($col_num < ($size-1) && (($row_num + 1) % $size > 0
+) ){
                        $row_num++;
                        $col_num-=$max_part;
                }
                if($col_num < ($max_part-1) && $row_num == ($size-1)){
                        # $col_num++;
                        $row_num-=$max_part;
                        $end_col+=$max_part;
                }
                if($col_num == ($size-1) && $row_num < ($size-1)){
                        $row_num++;
                        $col_num-=$max_part;
                }
                if($col_num == ($max_part-1) && $row_num == ($size-1))
+{
                        $row_num++;
                        $col_num=0;
                        $end_col=$max_part;
                }
                $tiles_counted += $part_tiles_counted;
        }

}
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