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.

Here’s a classic quote from RL Moore:

That student is taught the best who is told the least.

During his talk yesterday at the RL Moore Conference, David Clark provided a slight modification:

The student is taught the best who is told only enough to ensure that he or she will continue to work hard, stay engaged, and make progress.

I think David’s revision does an excellent job of capturing the experience I hope to provide the students in my inquiry-based learning (IBL) classes with.

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 few weeks ago, Stan Yoshinobu asked me to round up a few student quotes about their experience with inquiry-based learning (IBL). The intention is to use some of the material he gets for pamphlets and flyers for the Academy of Inquiry-Based Learning. I contacted a few of the students from the abstract algebra course that I taught in the fall and here is what they had to say.

“I’m a very shy person. Presenting math problems in front of an audience of math students was at first excruciating, but by the end of the course I realized I had gained an enormous amount of confidence. I truly feel that the IBL process has given me access to internal resources I didn’t realize I had available.”

“IBL created an environment for me where I felt comfortable enough to try proofs without the pressure of needing to be 100% right on the first try. So now in later upper division courses I am more comfortable with trying more complex problems, which ultimately lead me to do undergraduate research. And in all honesty, the classroom culture created by the IBL setup is what sold me on pure mathematics and has made me a better independent learner.”

“IBL helps prepare the student for the real world by teaching them how to create intuition. When you get to the real world or higher level mathematics courses, you will not always have someone there to tell you how to solve the problem.”

“By far, and without a doubt, inquiry-based learning is the best way to learn mathematics. Most methods for teaching math involve an instructor showing how to “do” various problems often involving computations and formulas, and then the students mimic the process for similar problems. IBL, however, asks the students to use what they know (or assume) to be true in order to create their own ways to solve problems or form logical arguments to validate other ideas. And logical arguments, not computations, not formulas, are the basis of all mathematics. Being able to form logical arguments is not something that can be mimicked, it must be discovered on one’s own, which is exactly how IBL works. Hence, when it comes to math, real math, and not just computations, IBL is the way to go.”

It would be a crime if I didn’t mention my all-time favorite student quote about IBL that was written on a course evaluation at the end of my introduction to proof course from the spring 2013 semester.

“Try, fail, understand, win.”

I believe that this last quote perfectly captures the essence of an effective IBL experience for a student. If you want to know about IBL, check out my post, What the Heck is IBL?, over on Math Ed Matters.

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 have recruited a diverse group of 7 talented undergraduates for this project: 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: 1. double-tailed tadpoles, triple-tailed tadpoles, etc.; 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. Here’s a quick run down on the annual stats for my blog according to Jetpack. For the complete report, go here. This blog was viewed about 16,000 times in 2013. My teaching page is a subdomain of my main site, so I’m guessing that a good chunk of that 16,000 is a result of my students clicking around (but I’m not sure about that). In 2013, there were 29 new posts, growing the total archive of this blog to 63 posts. The busiest day of the year was September 18th with 340 views. The most popular post that day was Free and Open-Source Textbooks. The top five most viewed posts in 2013 were: 1. LaTeX Homework Template, Aug 2012 2. Mathematics Education on the arXiv?, Jul 2013 3. Montessori Observations, Apr 2013 4. An infinite non-cyclic group whose proper subgroups are cyclic, Dec 2013 5. Euler’s Research Rules, Oct 2013 The top referring sites in 2013 were: I would have expected Google+ to be the top referrer and I’m a bit surprised that Facebook made the list at all. I’m happy to see that being a part of Booles’ Rings (a network of academic home pages/blogs) is bringing some traffic this way and I hope that I’m reciprocating at least a little. Visitors came from 114 different countries! Most visitors came from the United States, but Canada and the United Kingdom were not far behind. The most commented on post in 2013 was Mathematics Education on the arXiv? with 26 comments. These were the 5 most active commenters on this blog: 1. Dana Ernst, 26 Comments 2. François G. Dorais, 7 Comments 3. Bret Benesh, 4 Comments 4. Simon H., 3 Comments 5. Peter Krautzberger, 2 Comments I guess it’s no surprise that I was the top commenter. It is worth pointing out that François and Peter are also members of Booles’ Rings. Thanks for a fun 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.

Recently one of my former students at Plymouth State University, Zach Goldenberg, was awarded the 2013 Graduating Senior Award of Excellence. Zach is an amazing person and the award is well-deserved. I was fortunate to have Zach as a student in several of my classes, including Calculus I, Calculus II, Intro to Proof, Number Theory, and Abstract Algebra. As a freshman, he was the top student in both calculus courses. In the other courses, he consistently stood out as one of the top students. In addition, Zach was one of four undergraduate research students that worked on an original research project with me during the 2010–2011 academic year. Along with his fellow researchers, Zach presented the findings of their research at two regional conferences and presented a poster at the Joint Mathematics Meetings. Zach was instrumental in the success of this research group. Zach graduated summa cum laude with a double major in mathematics and political science. He was heavily involved with service opportunities, giving back in the local community at the Pemi Youth Center and internationally through his work with PSU’s Nicaragua Club and as an intern with Compas de Nicaragua, a non-profit organization dedicated to improving lives in urban and rural Nicaragua.

One of the things that I like best about Zach is his enthusiasm for life. His positive attitude is contagious and makes him an absolute pleasure to be around. He is constantly smiling and providing words of encouragement to those around him. Zach has been an exceptional role model for others by demonstrating that hard work pays off, and that the right attitude can make the difficulties and challenges enjoyable.

Below is a short video in which Zach was able to provide us with some words of wisdom.