# Revisiting card shuffling after seeing a talk by Persi Diaconis

Last week I took the boys to a talk given by Persi Diaconis at MIT:

Despite most of the talk going over their heads, the boys were really excited after the talk and had lots of different “shuffling” ideas that they wanted to explore.

Since this was going to be a long project, I divided it into two pieces – studying “pick up 52” with my younger son last night and studying shaking the cards in a box with both kids this morning.

The idea of using Shannon Entropy to study how random the shuffles are is something that we explored in these two prior projects:

Chard Shuffling and Shannon Entropy

Chard Shuffling and Shannon Entropy part 2

The original idea for those projects came from on an old Stackexchange post (well, the first comment) here:

See the first comment on this Stackexchange post

So, I kicked off the project last night with my younger son. Here are his thoughts about the Diaconis talk and about his “pick up 52” idea

We did 4 trials without re-sorting the cards in between. Here are some quick thoughts about how the deck was getting mixed up between the 2nd and 3rd throws:

After we finished I had my son do a few minutes of riffle shuffling to completely mix up the deck (starting from where the deck was after the 4th pick up 52). While he was doing that I entered the numbers from our throws into a spreadsheet.

The surprise was that even after the first throw the cards were really mixed up. I was even more surprised by this because he basically threw the deck in the air rather than what (to me anyway) is the normal way of throwing cards for pick up 52.

This morning we continued the project with my older son’s idea of putting the cards in a box and shaking the box. Here the introduction to that idea:

Here are some thoughts from my older son after the first mixing. He didn’t think they were all that mixed up. We did a total of 3 more mixings – the 3rd and 4th were off camera.

Finally, we wrapped up by reviewing the numbers for mixing the cards up in the box. The first mixing had more entropy than we thought, and after the 2nd mixing the cards appear to be pretty close to as mixed up as you can get (equivalent to about 10 riffle shuffles, I think).

This was a really fun project. The math you need to describe what’s going on here is much to advanced for kids (and worthy of a math lecture at MIT!), but kids can still have a lot of fun exploring some of the ideas. The seemingly simple idea of how can you measure how mixed up a shuffle is is a pretty interest idea all by itself.