MOVES, the Tower of Hanoi, and Mathigon

Welcome to this week’s Math Munch!

MOVESThe Math Munch team just wrapped up attending the first MOVES Conference, which was put on by the Museum of Math in NYC. MOVES is a recreational math conference and stands for Mathematics of Various Entertaining Subjects. Anna coordinated the Family Activities track at the conference and Paul gave a talk about his imbalance problems. I was just there as an attendee and had a blast soaking up wonderful math from some amazing people!

AnnaMOVES PaulMOVES JustinMOVES

Who all was there? Some of our math heroes—and familiar faces on Math Munch—like Erik Demaine, Tanya Khovanova, Tim and Tanya Chartier, and Henry Segerman, just to name just a few. I got to meet and learn from many new people, too! Even though I know it’s true, it still surprises me how big and varied the world of math and mathematicians is.

SuMOVES

Suzanne Dorée at MOVES.

One of my favorite talks at MOVES was given by Suzanne Dorée of Augsburg College. Su spoke about research she did with a former student—Danielle Arett—about the puzzle known as the Tower of Hanoi. You can try out this puzzle yourself with this online applet. The applet also includes some of the puzzle’s history and even some information about how the computer code for the applet was written.

danielle

Danielle Arett

A pice of a Tower of Hanoi graph with three pegs and four disks.

A piece of a Tower of Hanoi graph with three pegs and four disks.

But back to Su and Danielle. If you think of the different Tower of Hanoi puzzle states as dots, and moving a disk as a line connecting two of these dots, then you can make a picture (or graph) of the whole “puzzle space”. Here are some photos of the puzzle space for playing the Tower of Hanoi with four disks. Of course, how big your puzzle space graph is depends on how many disks you use for your puzzle, and you can imagine changing the number of pegs as well. All of these different pictures are given the technical name of Tower of Hanoi graphs. Su and Danielle investigated these graphs and especially ways to color them: how many different colors are needed so that all neighboring dots are different colors?

Images from Su and Danielle's paper. Towers of Hanoi graphs with four pegs.

Images from Su and Danielle’s paper. Tower of Hanoi graphs with four pegs.

 

Su and Danielle showed that even as the number of disks and pegs grows—and the puzzle graphs get very large and complicated—the number of colors required does not increase quickly. In fact, you only ever need as many colors as you have pegs! Su and Danielle wrote up their results and published them as an article in Mathematics Magazine in 2010.

Today Danielle lives in North Dakota and is an analyst at Hartford Funds. She uses math every day to help people to grow and manage their money. Su teaches at Ausburg College in Minnesota where she carries out her belief “that everyone can learn mathematics.”

Do you have a question for Su or Danielle—about their Tower of Hanoi research, about math more generally, or about their careers? If you do, send them to us in the form below for an upcoming Q&A!

UPDATE: We’re no longer accepting questions for Su and Danielle. Their interview will be posted soon! Ask questions of other math people here.

MathigonLast up, here’s a gorgeous website called Mathigon, which someone shared with me recently. It shares a colorful and sweeping view of different fields of mathematics, and there are some interactive parts of the site as well. There are features about graph theory—the field that Su and Danielle worked in—as well as combinatorics and polyhedra. There’s lots to explore!

Bon appetit!

10 responses »

  1. Tower of Hanoi is a favorite from my days as an undergrad in psychology. It is an important puzzle in the field of problem solving because you can only make progress by undoing some of your work. A classic used in cognitive experiments. I am glad that it is of interest to mathematicians too! I have my own Tower of Hanoi to challenge vistors to my apartment.

    • That’s interesting about making progress by (seemingly) undoing work. I wonder where else this shows up in mathematics? The example that pops to mind—though it’s more like pursuing an apparently fruitless path that suddenly becomes fruitful—is repeatedly using integration by parts and then solving when the original integrand turns up again. See Example 3 here: http://www.intmath.com/blog/integration-by-parts-twice/5396

      But there must be some other examples. Maybe solving a Rubik’s cube?

    • If you like to see new developments in the collaboration of psychologists and mathematicians on the Tower of Hanoi and related problems, have a look at
      A.M.Hinz, Graph theory of tower tasks, Behavioural Neurology 25(2012), 13-22.

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