# Math question about poker chip shuffling

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.

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

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

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