• Persi Diaconis, Ronald Graham and William M. Kantor, The mathematics of perfect shuffles, *Advances in Applied Mathematics* **4** (1983), 175–196.

There are two kinds of perfect shuffles: **out** shuffles, where the top card stays on top, and **in** shuffles, where it doesn’t. Here’s a wacky result from this paper: you have a deck of 24 cards, the group of permutations generated by in and out shuffles is the semidirect product of and the Mathieu group

This Mathieu group is an amazing group with 12 × 11 × 10 × 9 × 8 = 95,040 elements. It acts in a **sharply quintuply transitive** way on a 12-element set, meaning that you can take any 5-tuple of distinct elements of that set and find a unique element of the group that maps it to any other 5-tuple of distinct elements! Apart from symmetric and alternating groups, there are only a few groups that are k-tuply transitive for k > 3, so this group is very special. This group is also ‘simple’, meaning that it doesn’t have any nontrivial normal subgroups. There’s also a cool description of this group in terms of 12 balls rolling on a ball of the same size!

Ronald Graham is the guy that Graham’s number is named after. Diaconis and Graham won a writing prize at the Joint Mathematics Meetings in San Diego this January. Upon accepting the prize they played a card trick. I later asked Graham about the somewhat mysterious story behind Graham’s number, and he clarified it.

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• Shuffling playing cards – research, Wikipedia.

A rough rule of thumb is ‘7 riffle shuffles are enough’. But the details are complex and fascinating.

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