Math question about poker chip shuffling

[Lazlo]

Here’s a problem that had us stumped at our poker game on Friday. It involves poker chip shuffling, so if you’re not familiar with it, you can see a nice video here. In fact, if you watch past the two minute mark, he actually demonstrates the problem I’m curious about.

By the way, all of our data is experimental

Take 10 poker chips, five red and five green. Shuffle the separate stacks together and if done properly, you have a stack of interlaced red and green chips. Break the stack in half, and reshuffle. After a total of five iterations, you’ll have your red and green stacks separate again.

Now, take eight chips, four of each color. It takes four shuffles total to interlace then return to the starting state.

Six chips - three of each color takes four shuffles as well.

We then decided to see what happens when we increase our stack size. I can’t remember what six chips of each color did, but 14 chips, seven of each color, only took four shuffles to return to the starting state. This is counterintuitive. Shouldn’t more chips take longer to sort back into separate colors?

I’ve never taken combinatorics, so I really don’t know how to approach this problem.

[Gangster_Octopus]

I know a deck of cards returns to its original state after 8 "perfect’ shuffles.

[MikeS]

From a page containing more than you ever wanted to know about shuffling:

The numbers of out-shuffles needed to return a deck of n = 2 , 4, … to its original order are 1, 2, 4, 3, 6, 10, 12, 4, 8, 18, 6, 11, … , which is simply the multiplicative order of 2 (mod n - 1). For example, a deck of 52 cards therefore is returned to its original state after eight out-shuffles, since 2[sup]8[/sup] = 1 (mod 51). The smallest numbers of cards that require 1, 2, 3, … out-shuffles to return to the deck’s original state are 1, 2, 4, 3, 16, 5, 64, 9, 37, 6, …

I also found a brief explanation of why this is true, written by someone who used this fact to design a scarf.

[Lazlo]

Cool. Thanks for the links

[Oslo_Ostragoth]

(Jumping in here to say)I saw Chris Ferguson take a new deck (suits all aligned), shuffle it once to randomize things, then shuffle it again to return it to its original state. :eek: Never play cards with a guy who has a nickname!

[Whiteknight]

I’d imagine that this has something to do with using the 6 chips (or whatever the number is) as a basic unit of shuffled chips. Increasing the number in the stack is essentially the same as having X separate stacks that just happen to be vertically aligned. Thus, the number of shuffles required to get the chips back to their original state won’t change as the number of chips increases.