I just learned the trick of Faro Shuffling.
Lets say you have a brand new deck of cards, each suit organized ace to king. A faro shuffle would be what you would consider a perfect shuffle. Split the deck into even halfs, then create a new deck using one card at a time from each half deck. Seems random enough, but if you were to do this 8 times - you would get your original order back.
I just tried this with 26 cards, with hearts and clubs. I laid them out on the table face up each time to witness the "randomness" of them. I thought I had made a mistake somewhere, but to my astonishment, they returned to their original order. I also doubted myself in the middle thinking "Maybe this only works with 52 cards." But my intuition told me that this would work with any even number of items. And so I have proven it to myself.
Any even number of items in a certain order, when faro shuffled 8 times, will reproduce that order.
I totally thought there would be a nice java applet somewhere on the net showing this amazing truth, but there is not. Google searches for faro shuffling produce card tricks and not math. I would write a program for this, but I'm not too good with arrays.