August 2011

Un-Random Shufflers

This is a great story about how statisticians at Stanford audited a new automatic shuffling machine and determined that the cards weren’t distributed randomly enough.

https://www.newscientist.com/blogs/onepercent/2011/07/shuffling.html

If a deck of cards is dealt one at a time, a knowledgeable observer, in theory, should be able to predict the next card dealt around 4.5 times per 52-card deck.  For example, by remembering which cards have been dealt, the observer will definitely know the final card, as it’s the only one that hasn’t been dealt.  Similarly, the observer will have a 1 in 2 chance of guessing the second-to-last card, and so on.  Calculations involving probability and expected value will give you the theoretical result.

For this particular shuffler, however, the statisticians from Stanford determined that an observer should be able to predict the next card 9.5 times per 52-card deck!  The shuffling machine manufacturer that hired them must have been pretty upset to hear this, but redesigning the machine is probably not as costly as selling casinos hundreds of predictable shufflers and then dealing with the consequences.

It should come as no surprise that Persi Diaconis is the lead author on the paper.  Diaconis is a living legend in the world of mathematics, having left home at an early age to become a sleight-of-hand artist, then returning to earn a PhD from Harvard in mathematical probability.  One of Diaconis first major results was proving that seven shuffles are necessary to “randomize” a standard 52-card deck.

The full paper from Stanford can be found here:

http://statistics.stanford.edu/~ckirby/techreports/GEN/2011/2011-08.pdf

www.MrHonner.com

By patrick honner, ago

Reading in Calculus Class

One classroom activity I struggle to make time for is reading.  As an activity that can be done on one’s own, I generally feel that reading is not an especially productive use of class time.  That being said, I do try to make space for it on occasion:  it’s a nice change-up from routine, good material can make for a good discussion, and students of all ages seem to enjoy being read to, by teachers or by peers.

Finding appropriate reading material for a math class can be difficult, but Steven Strogatz’s excellent book, “The Calculus of Friendship,” was a great fit for my calculus class.  The book is part memoir, part homage, and part introduction to advanced calculus.  As such, it offers a readable balance of engaging narrative and challenging mathematics.

As a class, we enjoyed reading together about Strogatz’s mathematical journeys as told through personal narrative, letters to and from his former teacher, and the presentation of some particularly interesting math problems.  We also enjoyed working through some of the more advanced mathematical material presented in the book, like Fourier Series and differential equations.

It was time well-spent with my senior class.  While giving them a little taste of what might lie ahead, it also prompted some reflection on where they had been, and it all happened in the framework of some great mathematics.

And who knows:  now that the idea is planted, maybe I’ll become involved in a fruitful mathematical correspondence with one of my students someday!

By patrick honner, ago

Math and Art: An Impossible Construction

A favorite pastime of mine is offering impossible problems to students as extra credit, like asking them to find the smallest perfect square that has a remainder of 3 when divided by 4.  I don’t tell them the problems are impossible, of course, as that would ruin the fun.  Usually it engages and confuses them, and it makes them suspicious of me.  That’s a win-win-win in my book.

So while discussing some three-dimensional geometry, I offered extra credit to anyone who could build a model of a Klein bottle.  The Klein bottle is a hard-to-imagine surface that has neither an inside nor an outside; it’s like a bag that is sealed up, but somehow the bag is inverted in on itself.  If you are familiar with the Mobius strip, the Klein bottle is basically a Mobius strip, one dimension up.

One reason that the Klein bottle is hard to visualize is that it can’t exist in three dimensions.  It needs a fourth dimension in order to twist around on itself, kind of like the way the Mobius strip (which itself is two-dimensional) needs that third dimension to twist through before you tape it back together.  So, I was pretty impressed with the student who made this:


Not bad at all, for someone who is dimensionally challenged.  Here’s a nice representation for comparison, although it’s still a cheat:  the Klein bottle really doesn’t intersect itself.

A nice example of student work!

By patrick honner, ago
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