Said Limbic~Region:

As I indicated in another reply, I considered changing the method to change what elements appeared in each group for every iteration. Without that change, it should be easy to see that the re-ordering isn't random.

As I kept saying on IRC, it doesn't matter how you change the method, because the results won't be random regardless of what method you choose. If you only have enough entropy to randomize 8 things, and you try to generate random permutations of 64 things, you are not going to be able to generate all possible permutations.

You asked for a method that I could use to tell that your method wasn't working. Here's one: give me a sample of 200,000 of the "randomized" 64-element lists. You said "(Assume we have 64 elements in our list but only enough entropy to truly randomize 8)." Then I can tell there's something fishy because the sample you gave me will have lots of duplicate lists in it. You have only about log(8!) = 15.3 bits of entropy, so in the 200,000 items, there will be several that appear five times each. But the probability of such an occurrence in a truly random sample of 200,000 shuffled lists is about 10^-356. So it'll be totally obvious that you're cheating.

This will be true regardless of what clever method you are using to mix up the items. You can't get blood from a stone. As I said last night, what you're doing is like taking a 16-bit PRNG and then using it to generate 32-bit numbers by concatenating two 16-bit numbers together. If you hand someone a list of your "random" 32-bit numbers, it will be clear that they aren't really random.