### Archives For mind-switching

On Friday, February 1, I gave a 30-minute talk titled “The Stargate Switch” during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). As the name of the seminar suggests, the target audience for FAMUS is undergraduates. Here is the abstract for my talk.

An episode of Stargate SG-1 features a two-body mind-switching machine which will not work more than once on the same pair of bodies. The plot centers around two disjoint pairs of individuals who swap minds but subsequently wish they could reverse the process. We will discuss the mathematics behind the solution to this problem, as well as some generalizations.

This was my third FAMUS talk of the academic year, but my first of the semester. My first FAMUS talk was titled On an open problem of the symmetric group and my second talk was titled Euler’s characteristic, soccer balls, and golf balls.

In the 1999 episode “Holiday” from season 2 of Stargate SG-1, the character Ma’chello tricks Daniel into swapping minds with him. In an attempt to save Daniel, Jack and Teal’c accidentally swap minds, after which they then discover a limitation: the machine will not work more than once on the same pair of bodies. Physicist Samantha Carter saves the day by improvising a sequence of 4 switches that brings everyone back to normal.

In two recent papers, R. Evans and L. Huang from the University of California, San Diego, study the mathematics behind the Stargate Switch Problem, and generalize the solution to permutations involving $m$ pairs of bodies.

• Evans, R., & Huang, L. (2012). The Stargate Switch. [arXiv]
• Evans, R., & Huang, L. (2012). Mind switches in Futurama and Stargate. [arXiv]

The limitation of the machine in Stargate SG-1 is the same as the one suffered by the mind-switching machine in Futurama’s 2010 episode “The Prisoner of Benda”. However, the Stargate Switch, and Evans and Huang’s generalization, is a special case of the problem posed in Futurama. In the general case, we need to introduce two outsiders to solve the problem. But the Stargate Switch can be solved without the addition of outsiders (as long as we have at least 2 pairs of bodies).

The solution to the dilemma in Futurama is known as the Futurama Theorem and was proved by Ken Keeler, who is one of the show’s writers and has a PhD in applied mathematics. I’ve given a few talks about the Futurama Theorem and if you want to know more, check out my recent blog post located here.

As with my previous talks about the Futurama Theorem, I utilized deck.js to create HTML slides and I used MathJax to typeset all of mathematical notation. One advantage of this approach is that it allows you to view the slides directly in your web browser. You can view the slides by going here. To advance the slides, just use your arrow keys. Also, you can get an overview of the slides by typing “m” or go to a specific slide by typing “g”.

On Tuesday, November 6, 2012 I gave a talk titled “The Futurama Theorem and some refinements” in the NAU Department of Mathematics and Statistics Colloquium. This was the fourth time that I’ve given a talk about the Futurama Theorem (also known as Keeler’s Theorem).

The Futurama Theorem is a theorem about the symmetric group that was developed for and proved in the episode “The Prisoner of Benda” for the TV show Futurama. The theorem was proved by show writer Ken Keeler, who has a PhD in applied mathematics from Harvard. During the episode, Professor Farnsworth and Amy invent a mind swapping machine and after swapping minds, they realize that the machine cannot be used on the same pair of bodies again. After several characters swap minds, they are confronted with the problem of putting everyone’s mind back where it belongs. The Futurama Theorem proves that regardless of how many mind swaps have been made, all minds can be restored to their original bodies using only two extra people.

As with my previous talks about the Futurama Theorem, I utilized deck.js to create HTML slides and I used MathJax to typeset all of mathematical notation. One advantage of this approach is that it allows you to view the slides directly in your web browser. You can view the slides by going here. To advance the slides, just use your arrow keys. Also, you can get an overview of the slides by typing “m” or go to a specific slide by typing “g”.

This most recent talk about the Futurama Theorem was very similar to previous versions. However, I did add a few new slides. In particular, I included a summary of some recent results by Evans, Huang, and Nguyen (University of California, San Diego) that provides some refinements of Keeler’s theorem. Their paper will appear in the American Mathematical Monthly, but you can also find it on the arXiv. Here is the abstract for the paper:

An episode of Futurama features a two-body mind-switching machine which will not work more than once on the same pair of bodies. After the Futurama community engages in a mind-switching spree, the question is asked, “Can the switching be undone so as to restore all minds to their original bodies?” Ken Keeler found an algorithm that undoes any mind-scrambling permutation with the aid of two “outsiders.” We refine Keeler’s result by providing a more efficient algorithm that uses the smallest possible number of switches. We also present best possible algorithms for undoing two natural sequences of switches, each sequence effecting a cyclic mind-scrambling permutation in the symmetric group $S_n$. Finally, we give necessary and sufficient conditions on $m$ and $n$ for the identity permutation to be expressible as a product of m distinct transpositions in $S_n$.

If you are interested in comparing, you can find the slides for my previous talks about the Futurama Theorem by following the links below:

A couple hours before my talk, one of my colleagues asked if I minded if his son watched my talk via Skype. I suggested we try a Google+ Hangout and open it to anyone that was interested in watching. Having never done this before, I wasn’t exactly sure how it would work out. I sent out a quick announcement about the Hangout via G+, Twitter, and Facebook. The plan was to record the Hangout using Hangouts on Air, but we couldn’t get this set up in time. It looks like 13 people stopped by the Hangout, but it appears some of them were random Hangout lurkers. Also, it looks like there was an issue with the audio that we didn’t know about until the end.

The feedback that I received after the talk was that it went well, but I felt like I wasn’t at my best. The video camera made me more nervous than I anticipated, and trying to remember to stand where the video could see me was distracting. In addition, the light from the projector felt like a furnace.

Anyway, the Hangout was a cool experience and next time I’ll plan things out a little better. For example, as Vincent Knight suggested, instead of hosting a public Hangout, it would be better to have a G+ Circle of people that are interested in viewing and just open the Hangout to them. Thanks to Richard Green, Hugh Denoncourt, Luis Guzman and Barbara Boschmans Beaudrie for stopping by the Hangout.

On December 7, 2011, I gave my second talk about the Futurama Theorem during a Plymouth State University Mathematics Seminar. The Futurama Theorem is a theorem about the symmetric group that was developed for and proved in the episode “The Prisoner of Benda” for the TV show Futurama. The theorem was proved by show writer Ken Keeler, who has a PhD in applied mathematics from Harvard.

The first time I gave a talk about this theorem was during the Mathematics Forum at Gordon College just few weeks earlier. You can find my blog post about my first talk by going here.

During the episode, Professor Farnsworth and Amy invent a mind swapping machine and after they swap minds, they realize that the machine cannot be used on the same pair of bodies again. After several characters swap minds, they are confronted with the problem of putting everyone’s mind back where it belongs. The Futurama Theorem proves that regardless of how many mind swaps have been made, all minds can be restored to their original bodies using only two extra people. If you want to know more, check out the slides.

The slides for the second talk are very similar to the first set, but there are a few differences:

• The second talk is shorter. I trimmed a few things from the first talk that were not completely necessary.
• I’ve improved the wording in a few spots.
• In the second talk, multiplication of permutations is right to left.

As with the first talk, I used deck.js to create the slides. This allows you to view the slides directly in your web browser. To advance the slides, just use your arrow keys. Also, note that I used MathJax to typeset all of mathematical notation.