### Archives For Mathematics Posts

On October 12th, I saw a post by Dan Christensen on Google+ about a list of five open problems posed by the mathematician John Conway that have monetary rewards associated with them. In particular, Conway is offering $\$1,000$for solutions (either positive or negative) to any of the problems. Here are the five problems (as stated by Conway): • Problem 1. Sylver coinage game (named after Sylvester, who proved it terminates): The game in which the players alternately name positive integers that are not sums of previously named integers (with repetitions being allowed). The person who names 1 (so ending the game) is the loser. The question is: If player 1 names 16, and both players play optimally thereafter, then who wins? • Problem 2. 99-Graph: Is there a graph with 99 vertices in which every edge (i.e., pair of joined vertices) belongs to a unique triangle and every nonedge (pair of unjoined vertices) to a unique quadrilateral? • Problem 3. The Thrackle Problem: A doodle on a piece of paper is called a thrackle if it consists of certain distinguished points called spots and some differentiable (i.e., smooth) curves called paths ending at distinct spots and so that any two paths hit once and only once, where hit means having a common point at which they have distinct tangents and which is either an endpoint of both or an interior point of both. The right hand figure shows a thrackle with six spots and six paths. But can a thrackle have more paths than spots? • Problem 4. Dead Fly Problem: If a set of points in the plane contains one point in each convex region of area 1, then must it have pairs of points at arbitrarily small distances? • Problem 5. Climb to a Prime: Let$n$be a positive integer. Write the prime factorization in the usual way, e.g.,$60 = 22 \cdot 3 \cdot 5$, in which the primes are written in increasing order, and exponents of 1 are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number$f(n)$. Now repeat.So, for example,$f(60) = f(22 \cdot 3 \cdot 5) = 2235$. Next, because$24235 = 3 \cdot 5 \cdot 149$, it maps, under$f$, to 35149, and since 35149 is prime, it maps to itself. Thus,$60 \to 2235 \to 35149
\to 35149$, so we have climbed to a prime, and we stop there forever. The conjecture, in which I (Conway) seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that$20 \to 225 \to 3252 \to 223271 \to \cdots$, eventually getting to more than one hundred digits without yet reaching a prime. If you solve one of these, you can reach Conway by sending snail mail (only) in care of the Department of Mathematics at Princeton University. Around the same time that I stumbled onto these problems, I was brainstorming ideas for a couple of upcoming talks that I was slated to give (one for undergraduates and one for high school students). I decided that discussing open problems with monetary rewards with an emphasis on Conway’s problems would likely make for a nice talk. Here is the abstract that I settled on for both talks. There is a history of individuals and organizations offering monetary rewards for solutions, either in the affirmative or negative, to difficult mathematically-oriented problems. For example, the Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. A correct solution to any of the problems results in a$\$1,000,000$ prize being awarded by the institute. To date, only one of the problems has been solved (the Poincaré Conjecture was solved by Grigori Perelman, but he declined the award in 2010). These are hard problems! The renowned mathematician John Conway (Princeton) maintains a list of open problems and for each problem on the list, he is offering $\$,1000$to the first person that provides a correct solution. In this talk, we will explore a few of Conway’s problems, and in the unlikely event we come up with a solution, we’ll split the money. On Friday, October 24, 2014, I gave a talk during the Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) at NAU. Speaking at FAMUS is always fun and my talk seemed to be well-recieved. After having a practice run during FAMUS, I was able to improve the slides I intended to use during my talk at the 2014 NAU High School Math Day, which took place a few days later on Tuesday, October 28, 2014. Here are my slides: I had a blast presenting to the high school students. It cracked me up that there were a few students that immediately started obsessing over the Sylver coinage problem and likely didn’t hear a word I said after that. My goal was to give an engaging and high energy talk. I also slid in some humor and I was happy that everyone laughed when they were supposed to. Interestingly, the thing I said that the students thought was the funniest was something that I didn’t intend to be humorous. When I stated that “If you solve one of these, you can reach Conway by sending snail mail (only) in care of the Department of Mathematics at Princeton University,” the audience burst into laughter. Requiring snail mail seemed so ridiculous to them, they thought it was a joke. As a side note, I used mtheme (available for free on GitHub) together with beamer/LaTeX to generate my slides. I’m really happy with the look of mtheme and thrilled to get away from the standard beamer themes. My colleague Nandor Sieben and I recently submitted for publication a paper titled “Impartial achievement and avoidance games for generating finite groups.” The current arXiv version of the article is located here. My typical pure mathematics research interests are in the combinatorics of Coxeter groups and their associated algebras, so while I have a background in group theory and combinatorics, this was my first research experience in combinatorial game theory. In fact, prior to working on this project, I knew next to nothing about the subject. In the year and a half we worked on the project, I learned a tremendous amount of new material, which was a lot of fun. It was exciting to branch out and try something new. Here is the abstract for the paper: We study two impartial games introduced by Anderson and Harary and further developed by Barnes. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. After the development of some general results, we determine the nim-numbers of these games for abelian and dihedral groups. We also present some conjectures based on computer calculations. Our main computational and theoretical tool is the structure diagram of a game, which is a type of identification digraph of the game digraph that is compatible with the nim-numbers of the positions. Structure diagrams also provide simple yet intuitive visualizations of these games that capture the complexity of the positions. The fundamental problem in the theory of impartial combinatorial games is the determination of the nim-number of the game. This allows for the calculation of the nim-numbers of game sums and the determination of the outcome of the games. The major aim of this paper is the development of some theoretical tools that allow the calculation of the nim-numbers of the achievement and avoidance games for a variety of familiar groups. In the paper, we introduce the structure diagram of a game, which is an identification digraph of the game digraph that is compatible with the nim-numbers of the positions. Structure diagrams also provide simple but intuitive visualizations of these games that capture the complexity of the positions. By making further identifications, we obtain the simplified structure diagram of a game, which is our main computational and theoretical tool in the paper. We developed a software package that computes the simplified structure digraph of the achievement and avoidance games. We used GAP to get the maximal subgroups and the rest of the computation is implemented in C++. The software is efficient enough to allow us to compute the nim-numbers for the smallest 100,000 groups, which includes all groups up to size 511. The result is available on our companion web page. In April of 2013, I gave two talks at the University of Nebraska at Omaha that introduce the two games that this paper is about, but did not elaborate on the nim-number aspect. I summarized those talks in this blog post. The past two years, Angie Hodge, Stan Yoshinobu, and I have organized an Inquiry-Based Learning Best Practices special session at MathFest. We’ve had a fantastic turn out in terms of speakers and attendees both years. This year we thought we would try something new and decided to organize a poster session instead. Here’s the abstract for the session: New and experienced instructors implementing inquiry-based learning methods are invited to share their experiences, resources, and insights in this poster session. The posters in this session will focus on IBL best practices. We seek both novel ideas and effective approaches to IBL. Claims made should be supported by data (student responses, sample work, test scores, survey results, etc.). This session will be of interest to instructors new to IBL, as well as experienced practitioners looking for new ideas. Presenters should have their materials prepared in advance and will be provided with a self-standing, trifold tabletop poster approximately 48 in wide by 36 in high. One of our goals of the poster session is to increase interaction between presenters and attendees. We hope that someone can wander around and gather a lot of information about implementing IBL in a short period of time. I’m not usually a fan of poster sessions, but I’m looking forward to this one. The poster session takes place on Thursday, August 7 at 3:30-5:00PM in the Hilton Portland, Plaza Level, Plaza Foyer. If you are attending MathFest, please stop by the poster session. Also, if you think you have something interesting to share, we encourage you to submit an abstract. The deadline for submission is Friday, June 13, 2014. Questions regarding this session should be sent to the organizers: Angie Hodge, University of Nebraska at Omaha Dana Ernst, Northern Arizona University Stan Yoshinobu, Cal Poly San Luis Obispo If you want to learn more about IBL, check my “What the Heck is IBL?” post over on the Math Ed Matters blog. A couple months ago, my colleague Jeff Rushall and I co-applied for a Center for Undergraduate Research in Mathematics (CURM) mini-grant to fund a group of undergraduate students to work on an academic-year research project. Jeff and I had both individually applied in the past, but neither of us were successful in our proposals. If you are interested, you can take a look at my previous proposal by going here. Jeff and I are both passionate about undergraduate research and work well together. We decided that a joint application would likely be stronger than two individual proposals. I’m happy to report that we recently found out that our proposal was funded. We’re thrilled! Here are a few more details. For the upcoming project, we recruited a diverse group of 7 talented undergraduates: Michael Hastings (one of my current research students), Emily White, Hanna Prawzinsky, Alyssa Whittemore, Levi Heath, Brianha Preston, and Nathan Diefenderfer. Students are expected to spend ten hours per week during the academic year working on the research project. In return, each student will earn a$\$3000$ stipend. Money from the grant will also be used to buyout a single course for both me and Jeff. In addition, Jeff and I will team-teach a topics course each semester that will include our research students but will also be open to other interested students. CURM will cover most of the travel expenses for Jeff and I to attend the Faculty Summer Workshop at Brigham Young University (BYU). CURM will also cover most of the travel expenses for the nine of us to attend the Student Research Conference at BYU.

A collaborative research program has many advantages over operating several disconnected projects, as Jeff and I have done in the past. One of our goals is to build a self-sustaining research group. Ideally, this group will consist of students at different stages of their education, each participating for multiple years. The opportunities afforded by a CURM mini-grant will provide a catalyst for our endeavor in several ways. First, the visibility of a student research group with a CURM mini-grant will help our department recruit mathematically inclined students. NAU has many such students, but some are enticed by existing research groups and grants in our science programs. Second, we would like to take advantage of the mentoring and training the CURM program provides for faculty. One weakness in my past projects is getting students to finish writing up their results for publication. I am hopeful that my involvement in CURM will help remedy this. Third, we believe that the stipend money will enable our students to forgo some of their part-time work and instead devote their time to mathematics. Lastly, we want to use the experience as a stepping stone towards obtaining an externally funded REU program.

Next year’s research project involves “prime labelings of graphs,” which is outside my typical research interests. Jeff and I believe that we have found a project that is accessible to undergraduates yet rich enough that we won’t even come close to running out of stuff to do. I’m really looking forward to branching out and exploring something new.

If you are interested, here is the project description that we submitted.

## Project Description

This research project is motivated by a conjecture in graph theory, first stated in a 1999 paper by Seoud and Youssef [1], namely:

All unicyclic graphs have prime labelings.

This is a viable choice as a research problem for undergraduates because it is interesting yet accessible, in large part due to the minimal amount of background information required. To wit, a unicyclic graph is a simple graph containing exactly one cycle. An $n$-vertex simple graph $G$ with vertex set $V(G)$ is said to have a prime labeling if there exists a bijection $f: V(G) \to \{1, 2, 3, \ldots, n\}$ such that the labels assigned to adjacent vertices of $G$ are relatively prime.

As discussed in Gallian’s “A Dynamic Survey of Graph Labeling” [2], many families of graphs have prime labelings; the “simpler” types of unicyclic graphs that are known to have prime labelings include cycles, helms, crowns, and tadpoles. The goal of our project will be to discover additional classes of unicyclic graphs with prime labelings, in hopes of bringing the aforementioned conjecture on all unicyclic graphs within reach. The families of graphs we will investigate include, but are not limited to:

2. irregular crowns (crowns with paths of different lengths attached to each cycle vertex);
3. unicyclic graphs with one or more trivalent trees attached to cycle vertices;
4. unicyclic graphs with one or more complete ternary trees attached to cycle vertices; and
5. unicyclic graphs with a specified number of non-cycle cubic vertices.

Seoud and Youssef have established necessary and sufficient conditions for some graphs to have prime labelings, but they are somewhat limited in scope. Seoud has also published an upper bound on the chromatic numbers of prime graphs. These and other results may be beneficial to our students as their research project progresses.

Ernst and Rushall have already made some progress on these specific cases. We will use these initial results as a starting point with our team of students. More precisely, the 7 students involved in this project will attend a 3-credit research seminar during both semesters of the 2014-2015 academic year. Ernst and Rushall will team-teach the seminar, but eventually the students will play an equal role in leading discussions, presenting research results, etc.

Our recent experience with Seoud’s Conjecture has indicated that this problem is ripe with potential and highly appropriate as an undergraduate research project. The students can begin productive work in a single afternoon, and yet we anticipate the students producing original results worthy of publication in refereed journals by the end of the academic year. Moreover, there appear to be a virtually unlimited number of families of graphs to investigate, which will hopefully lead to a sustainable research program for undergraduates in future years.

The 7 students we have recruited to work on this research project are mathematics majors in our department. In addition, they all have very good academic records, and have proven themselves to be hard-working, reliable and creative students in previous courses that we have taught, including vector calculus, linear algebra, abstract algebra, foundations, discrete mathematics and number theory. These students regularly attend our weekly departmental undergraduate seminar, so they are familiar with the rigors associated with research and are motivated to investigate deeper problems in mathematics.

It should be noted that several presentation venues (departmental, university-wide, as well as regional conferences) will be exploited to allow our students an opportunity to showcase their efforts during the 2014-2015 academic year.

## References

[1] M.A. Seoud and M.Z. Youssef, “On Prime Labeling of Graphs,” Congressus Numerantium, Vol. 141, 1999, pp. 203-215.

[2] J.A. Gallian, “A Dynamic Survey of Graph Labeling,” The Electronic Journal of Combinatorics, Vol. 18, 2011. http://www.combinatorics.org/Surveys/ds6.pdf

The Joint Mathematics Meetings took place last week in Baltimore, MD. The JMM is a joint venture between the American Mathematical Society and the Mathematical Association of America. Held each January, the JMM is the largest annual mathematics meeting in the world. Attendance in 2013 was an incredible 6600. I don’t know what the numbers were this year, but my impression was that attendance was down from last year.

As usual, the JMM was a whirlwind of talks, meetings, and socializing with my math family. My original plan was to keep my commitments to a minimum as I’ve been feeling stretched a bit thin lately. After turning down a few invitations to give talks, I agreed to speak as part of panel during a Project NExT session. The title of the panel was “Tried & True Practices for IBL & Active Learning”. Here, IBL is referring to inquiry-based learning. Last year I gave several talks and co-organized workshops about IBL, so I knew preparing for the panel wouldn’t be a huge burden.

The other speakers on the panel were Angie Hodge (University of Nebraska, Omaha) and Anna Davis (Ohio Dominican University). Each of us were given 15 minutes to discuss our approach to IBL and active learning and then we had about 20 minutes for questions. For my portion, I summarized my implementation of IBL in my proof-based courses followed by an overview of my IBL-lite version of calculus. Angie discussed in detail the way in which UNO is successfully adopting IBL in some of their calculus sections. Anna spent some time telling us about an exciting project called the One-Room Schoolhouse, which attempts to solve the following problem: Small colleges and universities often struggle to offer a variety of upper-level courses due to low enrollment. According to the ORS webpage, the ORS setting allows schools to offer low-enrollment courses every semester at no additional cost to the institution. ORS is made possible by flipping the classroom.

If you are interested in my slides, you can find them below.

I’m really glad that I took the time to speak on the panel. The audience asked some great questions and I had several fantastic follow-up conversations with people the rest of the week.

As a side note, one thing I was reminded of is that speaking early during the conference (which is when the panel occurred) is much better than speaking later. The past couple JMMs, I spoke on the last day or two, and in this case, my talk is always taking up bandwidth in the days leading up to it. It’s nice to just get it out of the way and then be able to concentrate on enjoying the conference.

One highlight of the conference was having a pair of my current undergraduate research students (Michael Hastings and Sarah Salmon) present a poster during the Undergraduate Student Poster Session. Michael and Sarah have been working on an original research project involving diagrammatic representations of Temperley-Lieb algebras. I’ve been extremely impressed with the progress that they have made and it was nice to be able to see them show off their current results. The title of their poster was “A factorization of Temperley–Lieb diagrams” (although this was printed incorrectly in the program). Here is their abstract:

The Temperley-Lieb Algebra, invented by Temperley and Lieb in 1971, is a finite dimensional associative algebra that arose in the context of statistical mechanics. Later in 1971, R. Penrose showed that this algebra can be realized in terms of certain diagrams. Then in 1987, V. Jones showed that the Temperley–Lieb Algebra occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type $A$ (whose underlying group is the symmetric group). This realization of the Temperley-Lieb Algebra as a Hecke algebra quotient was generalized to the case of an arbitrary Coxeter group. In the cases when diagrammatic representations are known to exist, it turns out that every diagram can be written as a product of “simple diagrams.” These factorizations correspond precisely to factorizations in the underlying group. Given a diagrammatic representation and a reduced factorization of a group element, it is easy to construct the corresponding diagram. However, given a diagram, it is generally difficult to reconstruct the factorization of the corresponding group element. In cases that include Temperley–Lieb algebras of types $A$ and $B$, we have devised an efficient algorithm for obtaining a reduced factorization for a given diagram.

Perhaps I’m biased, but I think their poster looks pretty darn good, too.

All in all, I would say that I had a successful JMM. I had a lot more meetings than I’ve had in the past, so I missed out on several talks I really would have liked to attend. There’s always next year.

On page 36 of the December 2013/January 2014 issue of the MAA FOCUS there is a short article about Math Ed Matters, which is a monthly blog/column sponsored by the Mathematical Association of America and coauthored by Angie Hodge and myself. The article, written by Katharine Merow, highlights a few of our recent posts and describes some potential future posts (I should write these down, so we remember to write the advertised posts!). Katharine wanted to include a picture of Angie and me for the article, and as you can see in the picture to the left, she chose one of us that was taken after we had finished a trail run. I’m also wearing some ridiculous looking socks! The socks are actually compression socks designed for running, but they look silly nonetheless. I’ve been getting some flak for wearing such tall socks, but I think it’s funny.

I knew this article was going to appear, but I wasn’t sure when. It was brought to my attention by Robert Jacobson via one of his Google+ posts. Robert is responsible for the photo.

The Fall semester of 2013 just ended and one of the classes I taught was abstract algebra. The course is intended to be an introduction to groups and rings, although, I spent a lot more time discussing group theory than the latter. A few weeks into the semester, the students were asked to prove the following theorem.

Theorem. If $G$ is a cyclic group, then all the subgroups of $G$ are cyclic.

As with all conditional theorems, I also encouraged my students to think about whether the converse of this theorem is true. That is, if all the subgroups of a group $G$ are cyclic, does this imply that $G$ is cyclic? Well, since $G$ is always a subgroup of itself, the answer is clearly yes. But this isn’t the interesting question to ask. Instead, we should ask:

If $G$ is a group such that all proper subgroups are cyclic, does this imply that $G$ is cyclic?

The answer is “no” and my students were able to quickly come up with a few counterexamples. In particular, they mentioned the dihedral group $D_3$ (symmetry group for an equilateral triangle), the Klein four-group $V_4$, and the Quarternion group $Q_8$. The groups $D_3$ and $Q_8$ are both non-abelian and hence non-cyclic, but each have 5 subgroups, all of which are cyclic. The group $V_4$ happens to be abelian, but is non-cyclic. Yet it has 4 subgroups, all of which are cyclic.

Following the discussion of these three examples, one of my students asked whether the question above is true for infinite groups. I responded with something like, “Uh…well…no, I don’t think so. Hmmm, let me think about it.” So, I thought about it off and on for a couple hours, but didn’t make much headway. I decided to roam the hallways and recruit some help. I ended up chatting with my colleagues Mike Falk, Jim Swift, and Jeff Rushall. Collectively, we all thought about it a little bit here and a little bit there. I was fairly confident that an internet search would provide some insight, but I was intentionally putting that off in the hopes that I could come up with an example.

A couple days later, I was meeting with one of my undergraduate research students and we chatted briefly about the problem. A few hours after we met, he sent me a link to a discussion on Math Stack Exchange, which contains a response that is precisely about the question above.

Without further ado, here’s an example that confirms that the answer to the question above is “no” even if the group is infinite.

Theorem. The group $G=\{a/2^k\mid a\in\mathbb{Z}, k\in\mathbb{N}\}$ is an infinite non-cyclic group whose proper subgroups are cyclic.

Note that any fixed prime will do for the denominator. Let’s sketch a proof.

First, it is clear that $G$ is an infinite subgroup of $\mathbb{Q}$ since the sum of any two elements from $G$ will be contained in $G$ and the additive inverse of any element from $G$ is also in $G$. To see that $G$ is not cyclic, let $a/2^k\in G$ such that $a$ is odd and consider $\langle a/2^k\rangle$. It’s quickly seen that $\langle a/2^k\rangle$ does not contain any rational numbers having denominators equal to $2^{n}$ for $n>k$, and hence $G$ is not cyclic. Now, suppose that $H$ is a proper subgroup of $G$. If $a/2^k\in H$, then $a/2^{k-1}=a/2^k+a/2^k\in H$, as well. It follows that if there is an element in $H$ with denominator equal $2^k$ (in reduced form), then $H$ also contains elements with denominators equal to $2^n$ (in reduced form) for all $n\leq k$. Since $H$ is a proper subgroup, there exists a smallest $m\in \mathbb{N}$ such that no element of $H$ has a denominator equal to $2^m$. Then it must be the case that $H$ is contained in $\langle 1/2^{m-1}\rangle$, and so $H$ is cyclic (since subgroups of cyclic groups are cyclic).

Cool.

On Thursday, October 17, I gave an hour long talk in our department seminar titled “An Iterated Prisoner’s Dilemma.” There were about 35 people in attendance, including undergraduates (mostly my calculus students), graduate students, and faculty from the Mathematics & Statistics Department at Northern Arizona University. I was pleased with the turnout since our seminars are usually on Tuesdays and I wasn’t sure how many people would come on a non-standard day. Here is the abstract for the talk:

The Prisoner’s Dilemma goes something like this. Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don’t have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police offer each prisoner a bargain. If A and B both confess the crime, each of them serves 2 years in prison. If A confesses but B denies the crime, A will be set free whereas B will serve 3 years in prison (and vice versa). If A and B both deny the crime, both of them will only serve 1 year in prison. In this talk, we will first discuss this classic game theoretic problem and then introduce an iterative version that consists of a round robin tournament of teams, where the winner is the team that spends the least amount of time in prison.

I pretty much lifted this straight from the the Wikipedia page on the Prisoner’s Dilemma. So, thanks to the author(s) of that page!

There were several motivating factors in choosing this topic. First, every once in a while, I like to give a talk about something that I don’t know much about. Doing this forces me to learn something new. Also, I’ve found that some of my best talks are on things that I am not an expert on. Certainly, one of the reasons why this is true is that I’m likely to pitch a talk at a lower level if I’m talking about something unfamiliar. I don’t know about you, but I much prefer sitting through a talk when I understand most of what’s going on. Culturally, it seems acceptable to give talks where most of the audience doesn’t understand most of the talk. I’m trying to give talks where this doesn’t happen.

It’s expected that our graduate students (we have a masters program at NAU) attend our weekly seminars, but lately their attendance has been poor. I wanted to pick a topic that might entice them to start coming.

I ended up choosing the Prisoner’s Dilemma as a topic because I wanted to learn more about game theory and I figured the topic would be accessible. Moreover, I was inspired by Google+ and blog posts by Vincent Knight and Paul Harper (both from Cardiff University). There was also an excellent Radiolab episode about the Prisoner’s Dilemma back in 2010 that planted a seed for me. I’d like to thank Vince and Paul for helping out while I was preparing my talk. In particular, my slides are a modification of Vince’s slides, which he discusses here.

Without further ado, here are the slides for my talk.

As you can see, the talk began with an activity involving the Two Thirds of Average Game. During the activity, the audience made two different guesses. While I was giving the rest of the talk, I had a volunteer enter all the guesses into a csv file on the Sagemath Cloud. At the end of the talk, I ran Vince’s python script on the csv file in the Sagemath cloud. The output told me who the winners were for both rounds of guessing and provided a dandy looking graph, seen below.

I provided the winners with some chocolate.

Around slide 18, the plan was to conduct an Iterated Prisoner’s Dilemma tournament involving 4 teams, but I was a little worried about running out of time. So, I decided to wait until the end of the talk and do it if I had time. I ended up squeezing in a 3-team tournament that we probably flew through too quickly to get much out of, but it was fun nonetheless. The three team names were the United States, North Korea, and Russia. North Korea ended up winning, but only by a small margin.

Several weeks ago, links to a survey article by Jeffrey Lagarias about Euler’s work and its modern developments and a blog post by Richard J. Lipton that discusses Lagarias’ paper were circulated on Google+. I’d like to thank Luiz Guzman and Joerg Fliege for first bringing these items to my attention.

Lagarias’ paper is full of lots of yummy goodies, but my favorite part is his summary of Euler’s approach to research (see Section 2.6 of the paper or the end of Lipton’s post).

## Euler’s Research Rules

Taken directly from the paper, here is Lagarias’ summary of Euler’s research rules.

1. Always attack a special problem. If possible solve the special problem in a way that leads to a general method.
2. Read and digest every earlier attempts at a theory of the phenomenon in question.
3. Let a key problem solved be a father to a key problem posed. The new problem finds its place on the structure provided by the solution of the old; its solution in turn will provide further structure.
4. If two special problems solved seem cognate to each other, try to unite them in a general scheme. To do so, set aside the differences, and try to build a structure on the common features.
5. Never rest content with an imperfect or incomplete argument. If you cannot complete and perfect it yourself, lay bare its flaws for others to see.
6. Never abandon a problem you have solved. There are always better ways. Keep searching for them, for they lead to a fuller understanding. While broadening, deepen and simplify.

Lipton’s blog post also lists “Euler’s Research Rules.” My main motivation for reposting them here is to remind myself to follow them!

On Friday, September 20, I gave a 30-minute talk titled “Proofs without Words” during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). As the name of the seminar suggests, the target audience for FAMUS is undergraduates. I usually give a couple talks at FAMUS each semester and this was my first of the semester. Here is the abstract with words for my talk.

In this FAMUS talk, we’ll explore several cool mathematical theorems from a visual perspective.

The talk basically went like this. I displayed a figure or drawing and then the goal was for the audience to come up with the corresponding theorem. I had a ton of fun and the audience seemed to enjoy it. The initial idea for the talk came from the book Charming Proofs: A Journey into Elegant Mathematics by Claudi Alsina and Roger B. Nelsen. This is a wonderful book that incorporates lots of visual proofs. If you don’t have a copy, I highly recommend it. I borrowed lots of ideas from it when I taught a class titled “Introduction to Formal Mathematics” while I was at Plymouth State University.

My original plan was to recreate a lot of the figures I had in mind using TikZ, but I should have known that I wouldn’t have time for that. When I was brainstorming the talk a couple days before, I decided to do a Google search in the hope that I could find some figures to borrow that others had already made. In my search, I stumbled on several references to “proof without words”, which is what I ultimately named my talk. In fact, there is a Wikipedia entry and Roger B. Nelsen also wrote a book called Proofs without Words: Exercises in Visual Thinking. Moreover, I was thrilled to find lots of cool figures on the Internet. For my talk, I borrowed images and content from the following sources:

In the end, my slides ended up being a sample each of these three sources. If you are ready to see some cool figures, check out the slides below. I’ve left the pauses in so that you can ponder the theorem before you see it.

As anticipated, I had more figures than I had time to discuss, but we did get through most of them.

The custom at FAMUS is to interview a faculty member after the 30 minute talk. The usual FAMUS host, Jeff Rushall, was out of town, so I was filling in for him. I had the honor of interviewing Michael Falk. This was fun for me because Mike was my masters thesis advisor.