Jim Propp published a terrific essay last week:

Here’s a direct link in case the Twitter link has problems:

Who knows two? by Jim Propp

One of the topics covered in the essay is a special type of card shuffle called the Faro shuffle. We have done a few projects on card shuffling projects previously, so I thought the kids would be interested in learning about the Faro shuffle. Here are our prior card shuffling projects:

Card Shuffling and Shannon Entropy

Chard Shuffling and Shannon Entropy part 2

Revisiting card shuffling after seeing a talk by Persi Diaconis

I started the project by asking the kids what they knew about cards. They remembered some of the shuffling projects and then introducing the idea of the Faro shuffle.

My younger son thought he saw a connection with pi, which was a fun surprise.

We continued studying the Faro shuffle with 8 cards and looked for patterns in the card numbers and positions. The boys noticed some neat patterns and were able to predict when we’d return to the original order of cards!

Next we looked at the paths taken by individual cards. My older son thought that there might be a connection with modular arithmetic (!!!) and the boys were able to find the pattern. I’d hoped that finding the pattern here would be within their reach, so it was a really nice moment when he brought up modular arithmetic.

Finally, we wrapped up by talking about how to extend the ideas to a 52 card deck and calculated how many Faro shuffles we’d need to get back to where we started.

I think that kids will find the idea of the Faro shuffle to be fascinating. Simply exploring the number patterns is a really interesting project, and there’s lots of really interesting math connected to the idea. I’m really thankful that Jim Propp takes the time to produce these incredible essays each month. They are a fantastic (and accessible) way to explore lots of fun mathematical ideas.

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