Understanding the Gilbreath Principle

Background: Those who know me know that I love magic, and I love mathematics. A while back—okay, over ten years ago—I created a sheet that explained "The Gilbreath Principle," a mathematical principle that is sometimes used in card magic. On some of the magic forums, I offered to send a PDF of the file to anyone who requested it. At first, there were many requests, but as the post moved further and further into the distant past, the number of requests dropped off. Every year or so, I would receive a request for the file, and I would send it off dutifully. Then many years past and I forgot about it, until just the other day when someone found the OP on the Magic Cafe and sent me a request. That's when I decided to post it here. I don't know if it's exactly Pop Void, since, the idea behind principles such as this is to not enter popular culture, but stay hidden from most people. Nonetheless, part of the mission of this blog is to explore the unexplored and tackle the willfully ignored. So, without further ado, this one's for the math and magic geeks out there.
The Gilbreath Principle
When two groups of cards in reversed sequential order are riffle shuffled together, the lower cards in one packet will force their complement out of the group at the top of the other packet. That is the Gilbreath Principle in a nutshell.
Here’s a visualized version of what happens:
Imagine you have a stack of five red blocks. The blocks are numbered one through five from the top down. The stack may contain exactly five blocks and no more. Next to the red set you have an identical set of green blocks, but these blocks are numbered in the reverse order, with block number five on top, and block number one on the bottom:

You are allowed to place as few or as many of the green blocks in the red stack as you wish. The only stipulation is that blocks must stay in their original sequential order. That is, green block number one must stay below green block number two, green block number two must stay below green block number three and so on. Likewise, red block number one must stay above red block number two, et cetera. In other words, the bottom block on the green stack (number one) must be the first block to be added to the stack of red blocks.

The same thing happens when you riffle shuffle two packet of cards together; they interlace, starting with the bottom cards of each packet, but they maintain their sequential order.
Now for the Gilbreath Principle in action:
When you put green block number one into the red set, red block number one is forced out of the set. Then result: five blocks numbered one through five. They are no longer in numeric order, and one of them is green, but there is still only one of each number in the five card stack:

 
If you put green block #2 into the red set, red block #2 is also forced out of the set:

Again, there is only one of each number in the set of five. As you can see, it does not matter how few or how many of the green blocks you put into the set, the resulting combination will always be only one with each number:

The same thing happens when you riffle shuffle the cards. It doesn’t matter if the sequence consists of only two numbers (which is the case for red-black combinations), or an entire suit.

Sometimes you will hear people talk about “The First Gilbreath Principle” and “The Second Gilbreath Principle.” This is a mistake; there is only one Gilbreath Principle. When Gilbreath first encountered this effect, he used it to shuffle red and black cards together. Later he found that the same thing worked with sequentially organized cards. At first he thought these were different principles, but he later realized that it was the same principle, just applied differently. Unfortunately, the word on “The Second Gilbreath Principle” got out there and is still talked about.

Okay, that's all there is to it. It's pretty simple really, and yet it holds so much promise. For some innovative work on using the principle, check out the card magic of Max Maven.