How do I automate multiple category loops?

How do I automate multiple category loops?

I've asked this question already on the WordPress forum, but it sank like a stone, which isn't at all unusual.
I've set up a general Articles page with multiple loops, in order to display the first three articles from any category within the custom post type "Articles." These are all hard-coded into the template, like the one below, changing only the category ID number:
<?php
  $args = array(
  'cat' => 13,
  'posts_per_page' => 3,
  'post_type' => Article
  );
  query_posts( $args );

  if( have_posts() ):?>
..and so on. What follows is all the necessary code to produce the title, the permalink, the excerpt, etc.
The loops work fine. The problem is that they're hard-coded into the template, which would require that with every new category added, I add a loop to the template file. I need to find a solution so that instead, every time a new category is created it automatically gets added as a block to the Articles page. Also, I need to have it create a link, just as though I were using wp_get_archives, so that each category block links to its own page which contains ALL of the posts within that category (with pagination, if necessary).
I can't think of any function which accomplishes this. Has anyone ever come across this problem before? Or does this require some heavy custom PHP coding that WordPress doesn't support unless hacked?