We’ll use this page to keep track of what has happened each day in class. It won’t contain any of the nitty-gritty details, but will instead serve to summarize what has transpired each day.

**Wednesday, August 12:**First Day! We spent the majority of the time discussing logistics. In particular, we did a brief overview of Zoom (chat, raise/lower hand, thumbs up, breakout rooms, polling) and toured the course webpage and BbLearn. We spent a few minutes discussing the purpose of the course and the framework for homework, but I'm leaving the rest of the syllabus for students to read on their own. With the time we had left, I split the class up into several small groups and each was tasked with discussing Problem 1. We had JC and NC present some of their thoughts in the final couple minutes.**Friday, August 14:**Great first day of student presentations. After discussing a few logistics items, I split the class up into several breakout rooms. Each group was tasked with discussing/reviewing their solutions to Problems 2-4. After 10 or so minutes, we brought everyone back and launched into student presentations with me acting as scribe. We had NB, CF, and MK present Problems 2, 3, and 4, respectively. Along the way, MW, JC, and AS suggested alternate approaches and chimed in with ideas.

**Monday, August 17:**Busy day! We didn't quite get through everything, but we were productive nonetheless. After a few quick announcements, I revisited Problem 4 (Sunny Day Juice Stand) and attempted to convince everyone that the amounts would be the same in each. Next, we had MM, AR/NB, AS/KI, and MW present Problems 5(a), 5(b), 5(c), and 6, respectively. We'll tackle Problems 7 and 8 next time.**Wednesday, August 19:**Another good day.After a few announcements, I reviewed the strategies for Problems 5 and 6. Next, we had NC, BW/BT, and KB present Problems 7, 8, and 9, respectively, although we didn't quite finish Problem 9. We'll pick up there next time and tackle Problem 10, as well.**Friday, August 21:**Friday! We kicked off with KB wrapping up Problem 9 and then had AB and CB present Problems 10 and 11, respectively. We will catch up on Problem 12 next time.

**Monday, August 24:**We got caught up and finished all the new stuff. Yay! We had ZW, CT, KC, and KF present Problems 12, 13, 14, and 15, respectively (although, we didn't do them in that order). Lots of good discussion today, even though most of the class was kind of quiet.**Wednesday, August 26:**After chatting about the quiz on Friday and the plan for next week, we had UA, CH, TH, JV, and SV present Problems 16, 17(a), 17(b), 17(c), and 18, respectively.**Friday, August 28:**The students took Quiz 1.

**Monday, August 31:**First day having a mix of face-to-face students and remote students. I'd say it wasn't awful. Wearing a mask for two hours straight is tough (but necessary!). We cranked through quite a bit today. We had SG, AR, JY, DZ, KB, ZW, and TH present Problems 19, 20(a), 20(b), 20(c), 20(d), 20(d), and 21, respectively.**Wednesday, September 2:**Some battling with technology today, but we managed okay. After revisiting Problem 21 from last time, we had KF, AS, RR, KB, MK, CB, JC, and AR present Problems 22, 23(a), 23(b), 23(c), 24(b), 24(c), 24(d), and 24(a), respectively.**Friday, September 4:**We had UA, MW, CB, KI, MM, and NB present Problems 25, 26(a), 26(b), 26(c), 26(d), and 27, respectively.

**Monday, September 7:**Labor Day. No classes!**Wednesday, September 9:**We kicked off with BT revisiting Problem 24(c), so that we could address something that I said last time that wasn't quite right. Next, we had BW, JV, and CH present Problems 28, 29, and 30, respectively. We had some technical issues during Problem 30, but I think it worked out okay. At the end, I revisited Problem 27 to make some useful observations about winning versus losing positions in combinatorial games.**Friday, September 11:**The students took Quiz 2.

**Monday, September 14:**After discussing Problem B.3 from Quiz 2, we had AB, SG, and DZ/KB present Problems 31, 32, and 33, respectively.**Wednesday, September 16:**Lots of good stuff today! We had KF/SV, CF, RR, and JY present Problems 34(algebra/visual), 35, 36, and 37, respectively.**Friday, September 18:**We had CT, MK/BW/JV, and NC/KB present Problems 38, 39, and 40, respectively. Problem 39 was left hanging a little bit, but that was mostly my fault. I also spent some time discussing Problem 41.

**Monday, September 21:**After revisiting Problem 40 (pancake sort), we had UA/BT, CT, CH, KI/KF, and MM present Problems 41, 42(a), 42(b), 42(c), and 42(d), respectively.**Wednesday, September 23:**We had UA/NC/MM, ZW, MK, BW, and CB present Problems 43, 44(a), 44(bc), 44(d), and 45, respectively. With the time we had left, I quickly discussed Problem 44(e). We will revisit Problem 44(d) on Monday next week.**Friday, September 25:**The students took Quiz 3.

**Monday, September 28:**After quickly revisiting Problem 44 (safe packing), we had AB and JY present Problems 46 and 47, respectively. Next, we previewed Problem 48. With the time we had left, I (poorly) discussed the Futurama Theorem and how it relates to Problem 43 (soul swapping).**Wednesday, September 30:**Super productive day! We had AR/BT/SG/BW/CT, MK/KB, and MW tackle Problems 48, 49, and 50, respectively. There are some loose ends on Problem 48 and we should probably review Problem 50 since we did it quickly at the end of class.**Friday, October 2:**We started by revisiting Problems 50 and 48 and then had JV present Problem 51. With the time we had left, we started discussing Problem 52 (Circle-Dot).

**Monday, October 5:**We covered a lot of ground today. We had AB, CF, AS/CF, MW, SG, BT, KI/MK, and BW present Theorems D-K of Problem 52.**Wednesday, October 7:**We crushed the problems today! We had SV, NB, DZ/AR/MM, and RR/BW present Problems 53, 54, 55, and 56, respectively.**Friday, October 9:**The students took Quiz 4.

**Monday, October 12:**We had ZW, UA, CF, and CH present Problems 57, 58, 59, and 60, respectively. I was surprised we got through all of them as smoothly as we did. Well done!**Wednesday, October 14:**We had KF, JY/UA, and SV/NC present Problems 61, 62, and 63, respectively. I had an error in the wording on Problem 63. With the time we had left, I discussed the difficulty in finding Hamilton Paths/Circuits and the difficulty in finding an optimal weighted Hamilton Path/Circuit.**Friday, October 16:**We covered a lot of ground today. We had CB, CT, DZ, AS, and JV present Problems 64, 65, 66(a), 66(b), and 66(c). Along the way, I briefly discussed Catalan numbers. With the time we had available at the end of class, we started working on Problems 67 and 68.

**Monday, October 19:**We had BW, CB/KB, and NB present Problems 67, 68, and 69, respectively. In the last few minutes, I outlined the justification of why we found all the desired configurations in Problem 68.**Wednesday, October 21:**We had ZW/AS, SV/MK, and UA/KF/BW present Problems 70, 71, and 72, respectively. I added some comments about Problem 71 (Good Teacher) with the time we had left.**Friday, October 23:**The students took Quiz 5.

**Monday, October 26:**We kicked off by discussing B1 and B3 from Quiz 4. Next, KI and AR presented Problems 74 and 75, respectively. The rest of class was devoted to discussing Problem 73 (Martian Artifacts). We had DZ, CH, and MK/BT present the cases involving 2, 3, 4, and 5 spheres.**Wednesday, October 28:**We spent a good chunk of time revisiting Problem 73. In particular, we had BT, MK, and SV discuss the cases involving 5 and 6 spheres. We left the cases involving 7 and 8 spheres. Next, we had BW and CF present Problems 76 and 77, respectively. We will revisit Problem 77 on Friday.**Friday, October 30:**After discuss B2 on Quiz 5, we revisited Problem 77 and discussed an algebraic solution. Next, we had AB/ZW and SG present Problems 78 and 79, respectively. With the time we had left, we discussed most of Problem 80.

**Monday, November 2:**We stared off with discussing the solution to Problem B3 from Quiz 5. Net, we had KF, KI/JV, and MW present Problems 80, 81, and 82, respectively.**Wednesday, November 4:**We had MM, CH, and AS present Problems 83, 94, and 85, respectively, and had excellent discussions along the way.**Friday, November 6:**The students took Quiz 6.

**Monday, November 9:**After discussing B1 and B2 from Quiz 6, we had AB, AR, and BW present Problems 86(a), 86(b), and 87, respectively. We didn't have time for Problem 88, so we will revisit that one later.**Wednesday, November 11:**Veteran's Day. No classes.**Friday, November 13:**We kicked off with a quick discussion of B3 from Quiz 6 and then had JY and NC/BT/KF present Problems 89 and 90, respectively. With the time we had left at the end, we discussed a proof technique called induction.

**Monday, November 16:**I spent a good chunk of time presenting Problem 88 (left over from a while back) and Problem 93. Next, we had MK and CT/AR present Problems 94 and 95, respectively.**Wednesday, November 18:**Last day of classes! We had AB and KB present Problems 96 and 98, respectively. We spent quite a bit of time discussing Problem 96. With the time we had left, we had a quick discussion about how we could use induction to prove Problem 88 (Klingon senate problem).

Mathematics & Teaching

Northern Arizona University

Flagstaff, AZ

Website

928.523.6852

Twitter

Instagram

Facebook

Strava

GitHub

arXiv

ResearchGate

LinkedIn

Mendeley

Google Scholar

Impact Story

ORCID

MAT 411: Abstract Algebra

MAT 690: Genome Combinatorics

This website was created using GitHub Pages and Jekyll together with Twitter Bootstrap.

Unless stated otherwise, content on this site is licensed under a Creative Commons Attribution-Share Alike 4.0 International License.

The views expressed on this site are my own and are not necessarily shared by my employer Northern Arizona University.

The source code is on GitHub.

Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.